problem stringlengths 11 5.47k | reasoning stringlengths 722 106k | deepseek_solution stringlengths 0 4.02k | ground_truth_solution stringlengths 152 7.1k ⌀ | deepseek_final_answer stringlengths 0 350 | ground_truth_final_answer stringlengths 0 350 | correct bool 2
classes | judge_reasoning stringlengths 0 30k | matched_dataset stringclasses 2
values | matched_text stringclasses 33
values | match_score float64 0 99.2 |
|---|---|---|---|---|---|---|---|---|---|---|
The operation $\otimes$ is defined for all nonzero numbers by $a\otimes b =\frac{a^{2}}{b}$. Determine $[(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)]$.
$\text{(A)}\ -\frac{2}{3}\qquad\text{(B)}\ -\frac{1}{4}\qquad\text{(C)}\ 0\qquad\text{(D)}\ \frac{1}{4}\qquad\text{(E)}\ \frac{2}{3}$ | Okay, let me try to figure out this problem. So, we have this operation defined as a⊗b = a²/b. And we need to compute [(1⊗2)⊗3] - [1⊗(2⊗3)]. Then choose the correct answer from the options given. Alright, let's break it down step by step.
First, I need to remember that the operation ⊗ is not associative, right? Becaus... | To determine the value of \([(1 \otimes 2) \otimes 3] - [1 \otimes (2 \otimes 3)]\) where the operation \(\otimes\) is defined by \(a \otimes b = \frac{a^2}{b}\), we proceed as follows:
First, compute \(1 \otimes 2\):
\[
1 \otimes 2 = \frac{1^2}{2} = \frac{1}{2}
\]
Next, use this result to compute \((1 \otimes 2) \oti... | 1. **Apply the operation $\otimes$ to the innermost parentheses first:**
\[
(1 \otimes 2) \otimes 3 = \left(\frac{1^2}{2}\right) \otimes 3 = \frac{1}{2} \otimes 3
\]
\[
1 \otimes (2 \otimes 3) = 1 \otimes \left(\frac{2^2}{3}\right) = 1 \otimes \frac{4}{3}
\]
2. **Calculate each part using the definit... | 2 | 2 | true | Both the solution and the ground truth solution follow the same steps and calculations resulting in the same final answer of -\frac{2}{3}. Therefore, the solution is correct. | 0 | ||
Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window?
[asy] fill((0,0)--(2,0)--(2,26)--(0,26)--cycle,gray); fill(... | Okay, let me try to figure out this problem. So Doug has a square window made up of 8 equal-sized panes of glass. Each pane has a height to width ratio of 5:2. There are borders around and between the panes that are 2 inches wide. The question is asking for the side length of the square window in inches. The answer cho... | Doug constructs a square window using 8 equal-size panes of glass, with a height to width ratio of 5:2 for each pane. The borders around and between the panes are 2 inches wide. We need to determine the side length of the square window.
1. **Arrangement of Panes**: The panes are arranged in 2 rows and 4 columns. This ... | 1. **Identify the dimensions of each pane**: Given that the ratio of the height to the width of each pane is $5:2$, let the height of each pane be $5x$ inches and the width be $2x$ inches.
2. **Calculate the total dimensions of the window**: The window is constructed with $8$ panes arranged in $2$ rows and $4$ columns... | A | \textbf{(A)}26 | true | The solution correctly follows the steps to determine the dimensions of the window by setting up the equations based on the arrangement of the glass panes and the border widths. The calculations for the total width and height, as well as the final determination of the window's side length, match the expected results. B... | 0 | ||
$P(x)$ is a polynomial of degree $3n$ such that
\begin{eqnarray*} P(0) = P(3) = \cdots &=& P(3n) = 2, \\ P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\ P(2) = P(5) = \cdots &=& P(3n-1) = 0, \quad\text{ and }\\ && P(3n+1) = 730.\end{eqnarray*}
Determine $n$. | Okay, let's see. I need to find the value of n given this polynomial P(x) of degree 3n that satisfies certain conditions at points 0, 1, 2, 3, ..., 3n, and also P(3n+1) = 730. Hmm, this seems a bit involved, but let me break it down step by step.
First, let me try to understand the problem. The polynomial P(x) has deg... | To determine the value of \( n \) given the polynomial \( P(x) \) of degree \( 3n \) that satisfies the conditions \( P(0) = P(3) = \cdots = P(3n) = 2 \), \( P(1) = P(4) = \cdots = P(3n-2) = 1 \), \( P(2) = P(5) = \cdots = P(3n-1) = 0 \), and \( P(3n+1) = 730 \), we need to consider the polynomial interpolation and the... | To solve for $n$, we start by analyzing the polynomial $P(x)$ given its values at specific points and its degree. We use Lagrange Interpolation Formula to express $P(x)$, and then evaluate it at $x = 3n+1$ to find $n$.
1. **Constructing the Polynomial Using Lagrange Interpolation:**
The polynomial $P(x)$ is defined... | 4 | 4 | true | The derived solution correctly identifies that the value of \( n \) is 4 and aligns with the given ground truth solution through polynomial conditions. Both the solution and ground truth solution use polynomial interpolation concepts effectively to reach the conclusion. | 0 | ||
Let $f$ be the function defined by $f(x)=ax^2-\sqrt{2}$ for some positive $a$. If $f(f(\sqrt{2}))=-\sqrt{2}$ then $a=$
$\text{(A) } \frac{2-\sqrt{2}}{2}\quad \text{(B) } \frac{1}{2}\quad \text{(C) } 2-\sqrt{2}\quad \text{(D) } \frac{\sqrt{2}}{2}\quad \text{(E) } \frac{2+\sqrt{2}}{2}$ | Okay, let's try to figure out this problem step by step. The function is given as f(x) = a x² - √2, and we need to find the positive value of a such that f(f(√2)) = -√2. The answer choices are given, so maybe once we solve for a, we can match it with one of the options. Let me start by breaking down what's needed here.... | Given the function \( f(x) = ax^2 - \sqrt{2} \) where \( a \) is a positive constant, we need to find \( a \) such that \( f(f(\sqrt{2})) = -\sqrt{2} \).
First, compute \( f(\sqrt{2}) \):
\[
f(\sqrt{2}) = a(\sqrt{2})^2 - \sqrt{2} = 2a - \sqrt{2}
\]
Next, substitute \( f(\sqrt{2}) \) into the function \( f \):
\[
f(f(... | 1. **Define the function and apply it to $\sqrt{2}$:**
Given the function $f(x) = ax^2 - \sqrt{2}$, we first find $f(\sqrt{2})$:
\[
f(\sqrt{2}) = a(\sqrt{2})^2 - \sqrt{2} = 2a - \sqrt{2}.
\]
2. **Apply the function $f$ to the result of $f(\sqrt{2})$:**
Next, we need to find $f(f(\sqrt{2}))$:
\[
f(... | D | 2 | true | The provided solution accurately follows the steps outlined in the ground truth solution. Both solutions correctly calculate the value of \( f(f(\sqrt{2})) \) and successfully solve for \( a \) in the equation set to equal \( -\sqrt{2} \). The final result, \( a = \frac{\sqrt{2}}{2} \), is consistent in both solutions,... | 0 | ||
At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are $11$ students in Mrs. Germain's class, $8$ students in Mr. Newton's class, and $9$ students in Mrs. Young's class taking the AMC $8$ this year. How many mathematics students at Euclid Middle School are taking the co... | Okay, so the problem here is about figuring out how many mathematics students at Euclid Middle School are taking the AMC 8 contest. The information given is that there are three teachers: Mrs. Germain, Mr. Newton, and Mrs. Young. Each has their own class with a certain number of students taking the contest. Specificall... | To determine how many mathematics students at Euclid Middle School are taking the AMC 8 contest, we need to consider the number of students from each of the three mathematics teachers' classes. The problem states that there are 11 students in Mrs. Germain's class, 8 students in Mr. Newton's class, and 9 students in Mrs... | 1. **Identify the number of students in each class:**
- Mrs. Germain's class: 11 students
- Mr. Newton's class: 8 students
- Mrs. Young's class: 9 students
2. **Assumption of no overlap in students:**
Since the problem does not mention any students being in more than one class, we assume that all students ... | 3 | 1 | false | The solution concludes with oxed{C}, which is not a correct representation of the answer. The numerical result should be given as oxed{28} instead of using a letter. Also, the final conclusion lacks clarity about the format of the answer. | 0 | ||
If $991+993+995+997+999=5000-N$, then $N=$
$\text{(A)}\ 5 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 20 \qquad \text{(E)}\ 25$ | Okay, let's see. The problem is asking me to find the value of N in the equation 991 + 993 + 995 + 997 + 999 = 5000 - N. The options are from A to E, with numbers 5, 10, 15, 20, and 25. Hmm, so I need to compute the sum of those five numbers and then subtract that sum from 5000 to get N.
First, let me write down the ... | To find the value of \( N \) in the equation \( 991 + 993 + 995 + 997 + 999 = 5000 - N \), we can use the following steps:
1. Recognize that the numbers form an arithmetic sequence with the first term \( 991 \), the last term \( 999 \), and a common difference of \( 2 \). The number of terms is \( 5 \).
2. Calculate t... | 1. **Identify the problem and express each term in a form that reveals a pattern:**
\[
991+993+995+997+999=5000-N
\]
We can rewrite each term as $1000$ minus a small number:
\[
(1000-9) + (1000-7) + (1000-5) + (1000-3) + (1000-1)
\]
2. **Simplify the expression by factoring out $1000$ and summing ... | E | E | false | The provided solution correctly calculates the value of N, but the boxed answer is formatted incorrectly. It states '\boxed{E}' instead of '\boxed{25}', which is the correct numerical answer. | 0 | ||
The total in-store price for an appliance is $\textdollar 99.99$. A television commercial advertises the same product for three easy payments of $\textdollar 29.98$ and a one-time shipping and handling charge of $\textdollar 9.98$. How many cents are saved by buying the appliance from the television advertiser?
$\mathr... | Okay, let's try to figure out this problem step by step. So, the question is about comparing the total cost of buying an appliance in-store versus through a TV ad, and then finding out how many cents are saved by choosing the TV advertiser. The answer choices are from 6 to 10 cents, so the difference can't be too large... | The in-store price for the appliance is $99.99. The television advertisement offers three easy payments of $29.98 each and a one-time shipping and handling charge of $9.98.
First, we calculate the total cost from the television advertiser:
- Three payments of $29.98:
\[
29.98 \times 3 = 89.94
\]
- Adding the s... | 1. **Calculate the total cost of buying the appliance from the television advertiser:**
The advertisement mentions three payments of $\textdollar 29.98$ each and an additional one-time shipping and handling charge of $\textdollar 9.98$. First, calculate the total of the three payments:
\[
3 \times 29.98 = 89.9... | 6 | 8 | true | The solution correctly calculates the total cost of the appliance from the television advertisement, compares it to the in-store price, and accurately determines the savings. Both the original and ground truth solutions yield savings of 7 cents, confirming they are consistent and accurate. | 0 | ||
Points $A,B,C,D,E$ and $F$ lie, in that order, on $\overline{AF}$, dividing it into five segments, each of length 1. Point $G$ is not on line $AF$. Point $H$ lies on $\overline{GD}$, and point $J$ lies on $\overline{GF}$. The line segments $\overline{HC}, \overline{JE},$ and $\overline{AG}$ are parallel. Find $HC/JE$.
... | Okay, let's try to tackle this geometry problem step by step. First, I need to visualize the setup. There are points A, B, C, D, E, and F lying on the line AF in that order, dividing it into five segments each of length 1. So, the total length from A to F is 5 units. Let me draw a rough sketch in my mind: A----B----C--... | Points \(A, B, C, D, E,\) and \(F\) lie on \(\overline{AF}\) in that order, dividing it into five segments each of length 1. We place these points on a coordinate system with \(A\) at \((0, 0)\), \(B\) at \((1, 0)\), \(C\) at \((2, 0)\), \(D\) at \((3, 0)\), \(E\) at \((4, 0)\), and \(F\) at \((5, 0)\). Point \(G\) is ... | 1. **Identify Key Points and Relationships**:
- Points $A, B, C, D, E,$ and $F$ are collinear on line $\overline{AF}$, and each segment between consecutive points is of length 1.
- Point $G$ is not on line $AF$, and points $H$ and $J$ lie on lines $\overline{GD}$ and $\overline{GF}$ respectively.
- Lines $\ov... | 5 | 0.5 | true | Both the solution and the ground truth arrive at the same conclusion regarding the ratio \( \frac{HC}{JE} = \frac{5}{3} \). The explanation in the ground truth solution supports the findings of the original solution with clear logical steps based on the properties of similar triangles and parallel lines, confirming the... | 0 | ||
During the softball season, Judy had $35$ hits. Among her hits were $1$ home run, $1$ triple and $5$ doubles. The rest of her hits were single. What percent of her hits were single?
$\text{(A)}\ 28\% \qquad \text{(B)}\ 35\% \qquad \text{(C)}\ 70\% \qquad \text{(D)}\ 75\% \qquad \text{(E)}\ 80\%$ | Okay, let's see. The problem is about Judy's hits during the softball season. She had a total of 35 hits. Among those, 1 was a home run, 1 was a triple, and 5 were doubles. The rest were singles. We need to find out what percent of her hits were singles. The options are given from A to E.
First, let me break down the ... | Judy had a total of 35 hits, with 1 home run, 1 triple, and 5 doubles. To find the number of singles, we subtract these extra-base hits from the total:
\[
\text{Singles} = 35 - (1 + 1 + 5) = 35 - 7 = 28
\]
Next, we calculate the percentage of hits that were singles:
\[
\text{Percentage} = \left( \frac{28}{35} \right... | 1. **Identify the total number of hits**: Judy had a total of 35 hits during the season.
2. **Determine the number of hits that were not singles**:
- Home runs: 1
- Triples: 1
- Doubles: 5
- Total non-single hits = 1 (home run) + 1 (triple) + 5 (doubles) = 7
3. **Calculate the number of singles**:
\[
... | E | E | true | The solution correctly calculates the number of singles and the percentage of hits that were singles. The steps taken in the solution mirror those in the ground truth solution, and it arrives at the same final answer of 80%, which is represented as oxed{E}. Therefore, the solution is correct. | 0 | ||
The graph, $G$ of $y=\log_{10}x$ is rotated $90^{\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. Which of the following is an equation for $G'$?
(A) $y=\log_{10}\left(\frac{x+90}{9}\right)$ (B) $y=\log_{x}10$ (C) $y=\frac{1}{x+1}$ (D) $y=10^{-x}$ (E) $y=10^x$ | Okay, let's tackle this problem step by step. So, the question is about rotating the graph of y = log₁₀x 90 degrees counter-clockwise about the origin and finding the equation of the new graph G'. The options are given from A to E. Hmm, let's see.
First, I need to recall how rotation affects a graph. Rotating a graph ... | To determine the equation of the graph \( G' \) obtained by rotating the graph \( G \) of \( y = \log_{10}x \) 90 degrees counter-clockwise about the origin, we need to apply the rotation transformation to each point on \( G \).
1. **Rotation Transformation**: A point \((x, y)\) on the original graph \( G \) is rotate... | 1. **Understanding the rotation**: Rotating a point $(x, y)$ $90^\circ$ counterclockwise about the origin results in the point $(-y, x)$. This can be verified using rotation matrices:
\[
\begin{bmatrix}
\cos(90^\circ) & -\sin(90^\circ) \\
\sin(90^\circ) & \cos(90^\circ)
\end{bmatrix}
\begin{bmatrix}
... | D | D | true | The solution provided accurately describes the process of rotating the graph of the function $y = \log_{10} x$ by 90 degrees counter-clockwise using appropriate transformations and derives the correct equation of the new graph $y = 10^{-x}$. Both the steps and conclusions match with the ground truth solution. | 0 | ||
Jose, Thuy, and Kareem each start with the number 10. Jose subtracts 1 from the number 10, doubles his answer, and then adds 2. Thuy doubles the number 10, subtracts 1 from her answer, and then adds 2. Kareem subtracts 1 from the number 10, adds 2 to his number, and then doubles the result. Who gets the largest fin... | Okay, so I need to figure out who ends up with the largest final answer among Jose, Thuy, and Kareem. They all start with 10, but each of them does different operations. Let me take it step by step for each person.
First, let's break down what each person does. Maybe I'll write down each step for all three so I can co... | Jose, Thuy, and Kareem each start with the number 10 and perform different operations. Let's break down their steps and compute their final results:
**Jose:**
1. Subtract 1 from 10: \(10 - 1 = 9\)
2. Double the result: \(9 \times 2 = 18\)
3. Add 2: \(18 + 2 = 20\)
**Thuy:**
1. Double the number 10: \(10 \times 2 = 20... | Let's calculate the final results for Jose, Thuy, and Kareem step by step:
1. **Jose's Calculation:**
- Start with 10.
- Subtract 1: \(10 - 1 = 9\).
- Double the result: \(9 \times 2 = 18\).
- Add 2: \(18 + 2 = 20\).
Thus, Jose's final answer is \(20\).
2. **Thuy's Calculation:**
- Start with 10.
... | C | C | true | The solution correctly follows the calculations and correctly identifies Kareem as having the largest final answer. All computations are verified and consistent with the ground truth. | 0 | ||
A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\frac{7}{17}$. What number was erased?
$\textbf{(A)}\ 6\qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9\qquad \textbf{(E)}\ \... | Okay, let me try to figure out this problem. So, we have a set of consecutive positive integers starting from 1, and one number is erased. The average of the remaining numbers is 35 and 7/17. We need to find out which number was erased. The options are from 6 to 9, or cannot be determined. Hmm.
First, let's parse the ... | Given a set of consecutive positive integers starting from 1, one number is erased, and the average of the remaining numbers is \(35 \frac{7}{17}\). We need to determine which number was erased.
1. **Convert the mixed number to an improper fraction**:
\[
35 \frac{7}{17} = \frac{602}{17}
\]
2. **Set up the eq... | 1. **Identify the sum of the first $n$ positive integers**: The sum of the first $n$ positive integers is given by the formula for the sum of an arithmetic series:
\[
S = \frac{n(n+1)}{2}
\]
2. **Expression for the average after erasing one number**: If one number $x$ is erased, the sum of the remaining numbe... | 1 | 2 | true | Both the provided solution and the ground truth solution correctly compute the erased number as 7 using the average given. They apply the correct arithmetic operations and logic to arrive at the same conclusion, confirming that the calculations regarding the sum of integers and the condition of the average hold true. | 0 | ||
Patio blocks that are hexagons $1$ unit on a side are used to outline a garden by placing the blocks edge to edge with $n$ on each side. The diagram indicates the path of blocks around the garden when $n=5$.
[AIME 2002 II Problem 4.gif](https://artofproblemsolving.com/wiki/index.php/File:AIME_2002_II_Problem_4.gif)
If ... | Okay, so I need to solve this problem where patio blocks shaped like hexagons with each side 1 unit are used to outline a garden. The arrangement is such that there are n blocks on each side, and we need to find the area of the garden enclosed by the path when n=202, not including the path itself. The answer should be ... | To solve the problem, we need to determine the area of the garden enclosed by a hexagonal path with \( n = 202 \) blocks on each side. The area of the garden is given in the form \( m \left( \sqrt{3}/2 \right) \), and we need to find \( m \) modulo 1000.
1. **Understanding the Hexagonal Path**:
- The path forms a h... | 1. **Identify the number of blocks in the path**: For $n > 1$, the path of blocks has $6(n-1)$ blocks in total. This is because there are six sides to the hexagon, and each side (except the first in each direction) contributes an additional block to the length of the previous side.
2. **Calculate the total number of b... | 203 | 7 | false | The provided solution calculates the area of the garden based on the area of a smaller hexagon inside the hexagonal path but does not address the ground truth solution's method of counting hexagons correctly and thus provides a different value for m. The ground truth calculates the number of hexagons correctly and arri... | 0 | ||
A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $15$ and $25$ meters. What fraction of the yard is occupied by the flower beds?
[asy] unitsize(2mm); defaultpen(... | Okay, let me try to figure out this problem. So, we have a rectangular yard with two flower beds that are congruent isosceles right triangles. The rest of the yard is a trapezoid with parallel sides of 15 and 25 meters. We need to find the fraction of the yard occupied by the flower beds. Hmm, okay.
First, let me visu... | The problem involves determining the fraction of a rectangular yard occupied by two congruent isosceles right triangle flower beds. The remaining part of the yard is a trapezoid with parallel sides of 15 meters and 25 meters.
1. **Visualizing the Yard and Flower Beds**:
- The yard is a rectangle with length 25 mete... | 1. **Identify the dimensions of the triangles:**
The problem states that the yard contains two congruent isosceles right triangles and the parallel sides of the trapezoidal remainder are $15$ meters and $25$ meters. The difference in length between the parallel sides of the trapezoid is $25 - 15 = 10$ meters. Since ... | 6 | 4 | true | The reasoning presented in the solution correctly identifies the areas of the isosceles right triangles, confirms the total area of the rectangular yard, and calculates the fraction of the yard occupied by the flower beds accurately. Both the solution and the ground truth arrive at the same conclusion of \\frac{1}{5}, ... | 0 | ||
What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are \[(\cos 40^\circ,\sin 40^\circ), (\cos 60^\circ,\sin 60^\circ), \text{ and } (\cos t^\circ,\sin t^\circ)\]
is isosceles?
$\textbf{(A)} \: 100 \qquad\textbf{(B)} \: 150 \qquad\textbf{(C)... | Okay, let's see. I need to find the sum of all possible values of t between 0 and 360 degrees such that the triangle formed by the points (cos40°, sin40°), (cos60°, sin60°), and (cos t°, sin t°) is isosceles. The answer choices are given, so maybe after solving, I can check which one matches.
First, let me recall that... | To determine the sum of all possible values of \( t \) between \( 0 \) and \( 360 \) such that the triangle formed by the points \((\cos 40^\circ, \sin 40^\circ)\), \((\cos 60^\circ, \sin 60^\circ)\), and \((\cos t^\circ, \sin t^\circ)\) is isosceles, we analyze the central angles between these points on the unit circl... | Let $A = (\cos 40^\circ, \sin 40^\circ)$, $B = (\cos 60^\circ, \sin 60^\circ)$, and $C = (\cos t^\circ, \sin t^\circ)$. We need to find the values of $t$ such that $\triangle ABC$ is isosceles. We consider three cases based on which sides of the triangle are equal.
**Case 1: $AB = AC$**
For $AB = AC$, $C$ must lie on... | 2 | 3 | true | The provided solution accurately identifies all valid angles for an isosceles triangle formed by points on the unit circle and correctly sums them to arrive at the total of 380. The cases and reflections considered align with standard geometric principles. Additionally, the result matches the ground truth solution, con... | 0 | ||
The geometric mean of any set of $m$ non-negative numbers is the $m$-th root of their product.
$\quad (\text{i})\quad$ For which positive integers $n$ is there a finite set $S_n$ of $n$ distinct positive integers such that the geometric mean of any subset of $S_n$ is an integer?
$\quad (\text{ii})\quad$ Is there an inf... | Okay, so I need to figure out part (i) and (ii) of this problem about geometric means and sets of positive integers. Let me start with part (i).
The question is asking for which positive integers n there exists a finite set S_n of n distinct positive integers such that the geometric mean of any subset of S_n is an in... | To determine for which positive integers \( n \) there exists a finite set \( S_n \) of \( n \) distinct positive integers such that the geometric mean of any subset of \( S_n \) is an integer, we start by analyzing small values of \( n \).
For \( n = 1 \), any singleton set \(\{a\}\) trivially satisfies the condition... |
#### Part (i)
For which positive integers $n$ is there a finite set $S_n$ of $n$ distinct positive integers such that the geometric mean of any subset of $S_n$ is an integer?
To solve this, consider the set $S_n = \{2^a, 2^{2a}, 2^{3a}, \ldots, 2^{na}\}$ for some integer $a$. The geometric mean of any subset of $S_n$... | 2 | Nosuch\inftyinitesetSexists | false | The provided solution evaluates the condition for the geometric mean to be an integer effectively and provides constructions for finite sets, however, it does not align perfectly with the ground truth solution. The ground truth specifically suggests using a construction based on powers of 2, which is not addressed in t... | 0 | ||
In the adjoining figure, points $B$ and $C$ lie on line segment $AD$, and $AB, BC$, and $CD$ are diameters of circle $O, N$, and $P$, respectively. Circles $O, N$, and $P$ all have radius $15$ and the line $AG$ is tangent to circle $P$ at $G$. If $AG$ intersects circle $N$ at points $E$ and $F$, then chord $EF$ has le... | Alright, let me try to figure out this geometry problem. Let's start by understanding the given information and visualizing the figure.
We have points B and C on line segment AD. The segments AB, BC, and CD are each diameters of circles O, N, and P respectively. All three circles have a radius of 15. Since the radius... | Given points \( B \) and \( C \) on line segment \( AD \), where \( AB \), \( BC \), and \( CD \) are diameters of circles \( O \), \( N \), and \( P \) respectively, each with radius 15. The line \( AG \) is tangent to circle \( P \) at \( G \) and intersects circle \( N \) at points \( E \) and \( F \). We need to fi... | 1. **Identify the Configuration**: Given that $AB$, $BC$, and $CD$ are diameters of circles $O$, $N$, and $P$ respectively, and all circles have a radius of $15$. Since $B$ and $C$ lie on $AD$, and each segment is a diameter, $AB = BC = CD = 30$. Therefore, $AD = AB + BC + CD = 30 + 30 + 30 = 90$.
2. **Position of $N$... | 24 | 6\sqrt{21} | false | The provided solution concludes that the length of chord EF is 24, while the ground truth solution calculates it to be 6√21. The discrepancy arises from the calculation of distances and the use of similar triangles, which the provided solution appears to have misapplied. Therefore, the provided solution is incorrect. | 0 | ||
Given: $x > 0, y > 0, x > y$ and $z\ne 0$. The inequality which is not always correct is:
$\textbf{(A)}\ x + z > y + z \qquad\textbf{(B)}\ x - z > y - z \qquad\textbf{(C)}\ xz > yz$
$\textbf{(D)}\ \frac {x}{z^2} > \frac {y}{z^2} \qquad\textbf{(E)}\ xz^2 > yz^2$ | Okay, let's see. The problem is asking which inequality is not always correct given that x and y are positive, x is greater than y, and z is not zero. The options are from A to E. I need to check each one to see if there's a scenario where the inequality doesn't hold. Let's go step by step.
Starting with option A: x +... | The inequality that is not always correct is determined by analyzing each option under the given conditions \( x > 0 \), \( y > 0 \), \( x > y \), and \( z \ne 0 \):
- **Option A**: \( x + z > y + z \). Subtracting \( z \) from both sides simplifies to \( x > y \), which is always true.
- **Option B**: \( x - z > y ... | We are given that $x > y$ and both $x$ and $y$ are positive. We need to determine which of the given inequalities is not always correct.
1. **Option (A): $x + z > y + z$**
- Subtracting $z$ from both sides of the inequality:
\[
x + z - z > y + z - z \implies x > y
\]
- Since $x > y$ is given, this... | C | 1 | true | Both the provided solution and the ground truth solution correctly analyze each option according to the conditions given. They both conclude that option C ( exttt{xz > yz}) fails when z < 0, making it the only inequality that is not always true. Thus, the solution is correct. | 0 | ||
The graph of the line $y=mx+b$ is shown. Which of the following is true?
[2004 AMC 12A Problem 5.png](https://artofproblemsolving.com/wiki/index.php/File:2004_AMC_12A_Problem_5.png)
$\mathrm {(A)} mb<-1 \qquad \mathrm {(B)} -1<mb<0 \qquad \mathrm {(C)} mb=0 \qquad \mathrm {(D)}$ $0<mb<1 \qquad \mathrm {(E)} mb>1$ | Okay, let's try to figure out this problem step by step. The question is about a line graph described by the equation y = mx + b. We need to determine which of the given options is true regarding the product mb. The options range from mb being less than -1 to greater than 1, so it's important to analyze both the slope ... | To determine the correct answer, we analyze the graph of the line \( y = mx + b \). The y-intercept \( b \) is positive since the line intersects the y-axis above the origin. The slope \( m \) is negative as the line decreases from left to right.
Key steps:
1. **Slope Calculation**: The line passes through points suc... | 1. **Identify the slope and y-intercept from the graph**:
From the graph, we observe that the slope $m$ of the line is negative and appears to be $-\frac{1}{2}$. The y-intercept $b$ is positive and appears to be $\frac{4}{5}$.
2. **Calculate the product $mb$**:
\[
m = -\frac{1}{2}, \quad b = \frac{4}{5}
\... | B | B | false | The solution provided inaccurately identified the y-intercept as 1 instead of the correct value of \( b = \frac{4}{5} \). This led to an incorrect calculation of the product \( mb \). The slope was identified correctly, but the final product and the overall analysis are wrong due to this mistake. | 0 | ||
The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of $1$ and $2016$ is closest to which integer?
$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 45 \qquad \textbf{(C)}\ 504 \qquad \textbf{(D)}\ 1008 \qquad \textbf{(E)}\ 2015$ | Okay, let's see. The problem is about finding the harmonic mean of 1 and 2016 and determining which integer it's closest to among the given options. The harmonic mean formula is given as twice their product divided by their sum. Alright, let me recall what harmonic mean is.
So, harmonic mean of two numbers a and b is... | The harmonic mean of two numbers \(a\) and \(b\) is calculated as \(\frac{2ab}{a + b}\). For the numbers 1 and 2016, we need to compute this harmonic mean and determine the closest integer.
1. **Calculate the numerator**: \(2 \times 1 \times 2016 = 4032\).
2. **Calculate the denominator**: \(1 + 2016 = 2017\).
3. **Di... | 1. **Identify the formula for the harmonic mean of two numbers, $a$ and $b$:**
\[
\text{Harmonic Mean} = \frac{2ab}{a+b}
\]
Here, $a = 1$ and $b = 2016$.
2. **Substitute the values into the formula:**
\[
\text{Harmonic Mean} = \frac{2 \times 1 \times 2016}{1 + 2016} = \frac{4032}{2017}
\]
3. **Si... | 2 | 2 | true | Both the provided solution and the ground truth solution correctly calculate the harmonic mean of 1 and 2016 as approximately 1.999, and both conclude that the closest integer is 2. Hence, the evaluation is accurate. | 0 | ||
The first three terms of an arithmetic progression are $x - 1, x + 1, 2x + 3$, in the order shown. The value of $x$ is:
$\textbf{(A)}\ -2\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ \text{undetermined}$ | Okay, so I have this problem here about an arithmetic progression. The first three terms are given as x - 1, x + 1, and 2x + 3. I need to find the value of x from the options provided. Let me try to work through this step by step.
First, I remember that in an arithmetic progression (AP), the difference between consecu... | Given the first three terms of an arithmetic progression (AP) as \( x - 1 \), \( x + 1 \), and \( 2x + 3 \), we need to find the value of \( x \).
In an AP, the difference between consecutive terms is constant. Therefore, the difference between the second term and the first term should be equal to the difference betwe... | 1. **Identify the common difference**:
Given the arithmetic progression (AP) terms are $x - 1$, $x + 1$, and $2x + 3$. In an AP, the difference between consecutive terms is constant. Therefore, the common difference $d$ can be calculated as:
\[
d = (x + 1) - (x - 1)
\]
Simplifying this, we get:
\[
... | 5 | \textbf{(B)}0 | true | The solution correctly identifies that the terms form an arithmetic progression, calculates the common difference accurately, solves for x correctly, and verifies the solution through both direct substitutions and properties of arithmetic progressions. Both the provided solution and the ground truth reach the same conc... | 0 | ||
Figure 1 is called a "stack map." The numbers tell how many cubes are stacked in each position. Fig. 2 shows these cubes, and Fig. 3 shows the view of the stacked cubes as seen from the front.
Which of the following is the front view for the stack map in Fig. 4? | Okay, so I need to figure out the front view for the stack map given in Figure 4. Let me start by recalling what a stack map is. From the problem statement, Figure 1 is a stack map where numbers indicate how many cubes are stacked in each position. Then Figure 2 shows the 3D representation of those cubes, and Figure 3 ... | To determine the front view of the stack map in Fig. 4, follow these steps:
1. **Analyze the Stack Map**: Each number in the grid represents the height of cubes stacked at a specific (x, y) position, where:
- **Columns (x)**: Left-to-right positions.
- **Rows (y)**: Front-to-back positions.
2. **Determine Front... | To determine the front view of the stack map in Figure 4, we need to identify the tallest stack of cubes in each vertical column when viewed from the front. This involves finding the maximum number of cubes in each column from the given top view in Figure 4.
1. **Identify the tallest stack in each column:**
- **Fir... | 11 | false | The provided solution incorrectly outlines the step to determine the maximum height for the front view based on the heights listed in the example. While it discusses maximum heights for unspecified data in the prior context, it mismatches the heights specified in the ground truth solution's sample analysis of columns. ... | 0 | |||
Let a geometric progression with n terms have first term one, common ratio $r$ and sum $s$, where $r$ and $s$ are not zero. The sum of the geometric progression formed by replacing each term of the original progression by its reciprocal is
$\textbf{(A) }\frac{1}{s}\qquad \textbf{(B) }\frac{1}{r^ns}\qquad \textbf{(C) }\... | Alright, let me try to figure out this problem. So, we have a geometric progression (GP) with n terms. The first term is 1, common ratio is r, and the sum is s. Then, we're supposed to find the sum of another GP formed by taking the reciprocals of each term from the original progression. The options are given, and I ne... | Given a geometric progression (GP) with \( n \) terms, where the first term is 1, the common ratio is \( r \), and the sum is \( s \). We need to find the sum of the GP formed by taking the reciprocals of each term of the original progression.
1. **Original GP Sum**:
The sum \( s \) of the original GP is given by:
... | 1. **Identify the sum of the original geometric progression (GP):**
The sum $s$ of a geometric progression with first term $a = 1$, common ratio $r$, and $n$ terms is given by the formula:
\[
s = \frac{a(1 - r^n)}{1 - r} = \frac{1(1 - r^n)}{1 - r}
\]
This simplifies to:
\[
s = \frac{1 - r^n}{1 - r}... | C | \textbf{(C)}\frac{s}{r^{n-1}} | true | The provided solution correctly follows the necessary steps to derive the sum of the geometric progression formed by the reciprocals of the original terms. It accurately identifies the sum of the original GP, constructs the reciprocal GP, and simplifies the calculations step by step. The final expression for the sum of... | 0 | ||
Alicia had two containers. The first was $\tfrac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\tfrac{3}{4}$ full of water. What is the ratio of the volume of the first container to the volume of the s... | Okay, let's see. Alicia has two containers. The first one is 5/6 full of water, and the second is empty. Then she pours all the water from the first into the second, and the second becomes 3/4 full. We need to find the ratio of the volume of the first container to the second. Hmm.
Alright, let's start by defining vari... | Alicia poured \(\frac{5}{6}\) of the first container's volume (\(V_1\)) into the second container, making it \(\frac{3}{4}\) full of the second container's volume (\(V_2\)). This relationship is expressed as:
\[
\frac{5}{6}V_1 = \frac{3}{4}V_2
\]
Solving for the ratio \(\frac{V_1}{V_2}\):
\[
\frac{V_1}{V_2} = \frac{... |
1. **Define Variables:**
Let the volume of the first container be $A$ and the volume of the second container be $B$.
2. **Set Up the Equation:**
Since Alicia poured $\frac{5}{6}$ of the first container into the second container, and this amount filled $\frac{3}{4}$ of the second container, we can write the equa... | D | \textbf{(D)}\frac{9}{10} | true | The solution provided by Alicia correctly follows the steps to derive the ratio \\( \frac{V_1}{V_2} = \frac{9}{10} \\), matching the ground truth solution. Hence, the solution is correct. | 0 | ||
Misha rolls a standard, fair six-sided die until she rolls $1-2-3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | Okay, let's tackle this problem step by step. So, Misha is rolling a standard six-sided die until she gets the sequence 1-2-3 in that order on three consecutive rolls. We need to find the probability that the number of rolls she makes is odd, and then compute m + n where the probability is expressed as a reduced fracti... | To solve the problem, we need to determine the probability that Misha will roll the die an odd number of times before rolling the sequence 1-2-3. We model this using states and recursive equations.
### States:
- **S0**: No progress (starting state or last roll not part of 1-2-3)
- **S1**: Last roll was 1
- **S2**: Las... | To solve this problem, we need to consider the probability that Misha rolls the die an odd number of times before rolling the sequence $1-2-3$ consecutively. We will use a recursive approach to count the number of valid sequences of die rolls and then calculate the probability.
#### Step 1: Define the Recursive Functi... | 4 | 45 | true | The solution correctly derives the probability that Misha will roll the die an odd number of times before achieving the sequence 1-2-3. It uses states to model the problem and derive equations based on transitions between these states. The final answer matches the ground truth solution, including the calculated value o... | 0 | ||
An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at $A$, $B$, and $C$ are $12$, $9$, and $10$ meter... | Alright, let's tackle this problem step by step. So, we have a regular hexagon ABCDEF with pillars at each vertex. The heights at A, B, and C are given as 12, 9, and 10 meters respectively. We need to find the height at E. The solar panel is flat but not parallel to the ground, which means it's a plane tilted in some w... | Given a regular hexagon \(ABCDEF\) with pillars at vertices \(A\), \(B\), and \(C\) having heights 12, 9, and 10 meters respectively, we need to determine the height of the pillar at \(E\). The solar panel forms a plane passing through the tops of these pillars.
1. **Coordinate Setup**:
- Place the hexagon center a... | 1. **Establishing the Coordinate System and Points**:
Let's assume the side length of the hexagon is $6$ meters for simplicity. We place the hexagon in a 3D coordinate system with $A$ at the origin, i.e., $A = (0, 0, 0)$. The coordinates of $B$ and $C$ can be calculated based on the geometry of a regular hexagon:
... | D | 0 | true | The height of the pillar at E has been correctly calculated as 17 meters, consistent with the solution method used. Both the solution and ground truth yield the same result for the height at point E. | 0 | ||
Diameter $AB$ of a circle has length a $2$-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$. | Okay, let's try to tackle this problem step by step. The problem says that diameter AB of a circle is a two-digit integer, and reversing its digits gives the length of a perpendicular chord CD. The distance from their intersection point H to the center O is a positive rational number. We need to find the length of AB.
... | The length of diameter \( AB \) is a two-digit integer, and reversing its digits gives the length of the perpendicular chord \( CD \). The distance from their intersection point \( H \) to the center \( O \) is a positive rational number. We need to determine the length of \( AB \).
1. **Setup and Equations**:
- Le... | 1. **Assign Variables to Lengths**: Let the length of diameter $AB$ be represented as $10x + y$, where $x$ and $y$ are the tens and units digits respectively. Since reversing the digits gives the length of chord $CD$, we have $CD = 10y + x$.
2. **Calculate Half-Lengths**: The center of the circle $O$ lies at the midpo... | 2 | 65 | true | The provided solution correctly follows through with the mathematical setup and derivation needed to arrive at the answer. Both the solution and the ground truth solution analyze the variables systematically, derive the formulas accurately, and find consistent results that satisfy the conditions. The conclusion that th... | 0 | ||
Find the smallest positive number from the numbers below.
$\textbf{(A)} \ 10-3\sqrt{11} \qquad \textbf{(B)} \ 3\sqrt{11}-10 \qquad \textbf{(C)}\ 18-5\sqrt{13}\qquad \textbf{(D)}\ 51-10\sqrt{26}\qquad \textbf{(E)}\ 10\sqrt{26}-51$ | Okay, so I need to find the smallest positive number among the given options. The options are all expressions involving square roots, so I guess I have to evaluate each one numerically to see which is the smallest positive value. Let me write down the options again to make sure I have them right:
A) 10 - 3√11
B) 3√11... | To find the smallest positive number among the given options, we need to evaluate each expression and check if it is positive.
**Option A: \(10 - 3\sqrt{11}\)**
- Calculate \(3\sqrt{11}\):
\[
3\sqrt{11} \approx 3 \times 3.3166 \approx 9.9498
\]
- Subtract from 10:
\[
10 - 9.9498 \approx 0.0502
\]
- This i... | To find the smallest positive number among the given options, we first need to simplify and compare each expression.
#### Step 1: Simplify each expression
- **Option (A)**: $10 - 3\sqrt{11}$
- **Option (B)**: $3\sqrt{11} - 10$
- **Option (C)**: $18 - 5\sqrt{13}$
- **Option (D)**: $51 - 10\sqrt{26}$
- **Option (E)**: $... | D | 1 | true | The solution correctly calculates the values of each option, determining the positivity and then identifies the smallest positive value between options A and D as option D, supporting it with accurate calculations and reasoning. | 0 | ||
The points $(2,-3)$, $(4,3)$, and $(5, k/2)$ are on the same straight line. The value(s) of $k$ is (are):
$\textbf{(A)}\ 12\qquad \textbf{(B)}\ -12\qquad \textbf{(C)}\ \pm 12\qquad \textbf{(D)}\ {12}\text{ or }{6}\qquad \textbf{(E)}\ {6}\text{ or }{6\frac{2}{3}}$ | Okay, so I need to find the value of k such that the three points (2, -3), (4, 3), and (5, k/2) lie on the same straight line. Hmm, let me think. Since they are on the same line, the slope between any two pairs of points should be the same. That's the key idea here. Let me recall, the slope formula between two points (... | To determine the value of \( k \) such that the points \((2, -3)\), \((4, 3)\), and \((5, k/2)\) are collinear, we need to ensure that the slopes between any two pairs of points are equal.
1. **Calculate the slope between \((2, -3)\) and \((4, 3)\):**
\[
\text{slope} = \frac{3 - (-3)}{4 - 2} = \frac{6}{2} = 3
... | To determine the value(s) of $k$ such that the points $(2,-3)$, $(4,3)$, and $(5, k/2)$ lie on the same straight line, we need to ensure that the slope between any two pairs of these points is the same.
1. **Calculate the slope between the points $(2,-3)$ and $(4,3)$:**
The slope $m$ of a line through points $(x_1,... | 2 | 12 | true | Both the provided solution and the ground truth solution arrive at the same conclusion that the value of \( k \) is \( 12 \). The calculations are consistent in demonstrating that the points are collinear by checking slopes and by the equation of the line. Therefore, the original solution is correct. | 0 | ||
Suppose $q_1,q_2,...$ is an infinite sequence of integers satisfying the following two conditions:
(a) $m - n$ divides $q_m - q_n$ for $m>n \geq 0$
(b) There is a polynomial $P$ such that $|q_n|<P(n)$ for all $n$.
Prove that there is a polynomial $Q$ such that $q_n = Q(n)$ for each $n$. | Okay, so I need to prove that if there's an infinite sequence of integers q₁, q₂, q₃,... that satisfies two conditions: (a) for any m > n ≥ 0, the difference m - n divides q_m - q_n, and (b) there exists a polynomial P such that the absolute value of q_n is less than P(n) for all n, then there must be some polynomial Q... | Given an infinite sequence of integers \( q_1, q_2, \ldots \) satisfying the conditions:
(a) \( m - n \) divides \( q_m - q_n \) for \( m > n \geq 0 \)
(b) There exists a polynomial \( P \) such that \( |q_n| < P(n) \) for all \( n \),
we need to prove that there is a polynomial \( Q \) such that \( q_n = Q(n) \) for ... |
#### Step 1: Constructing the Polynomial $Q$
Given that $P$ is a polynomial of degree $d$, we construct a polynomial $Q$ of degree at most $d$ such that $Q(n) = q_n$ for $0 \leq n \leq d$. Since the $q_n$ are integers, $Q$ can be determined uniquely by interpolation and will have rational coefficients. Let $k$ be the ... | 0 | 6 | true | The provided solution accurately follows the logical steps necessary to prove that the sequence \( q_n \) must indeed be a polynomial. It establishes the necessary framework through which both the divisibility condition and polynomial growth constraint work together to ensure \( q_n \) can be represented by a polynomia... | 0 | ||
In a certain year the price of gasoline rose by $20\%$ during January, fell by $20\%$ during February, rose by $25\%$ during March, and fell by $x\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is $x$
$\mathrm{(A)}\ 12\qqu... | Okay, so I need to figure out the value of x, the percentage that gasoline prices fell in April, such that the price at the end of April is the same as it was at the beginning of January. The problem gives me a series of percentage changes each month: a 20% increase in January, a 20% decrease in February, a 25% increas... | Let the initial price of gasoline be \( P \).
1. **January**: The price increases by 20%, so the new price is:
\[
P \times 1.20
\]
2. **February**: The price falls by 20%, so the new price is:
\[
P \times 1.20 \times 0.80 = P \times 0.96
\]
3. **March**: The price increases by 25%, so the new price... | 1. **Assume the initial price**: Let's assume the initial price of gasoline at the beginning of January is $P_0 = 100$ dollars.
2. **Price after January's increase**: The price increased by 20% in January. Therefore, the new price at the end of January is:
\[
P_1 = P_0 + 0.20 \times P_0 = 100 + 20 = 120 \text{ d... | B | \textbf{(B)}17 | true | The provided solution and the ground truth both arrive at the same conclusion regarding the percentage decrease needed in April to return the price to its original value. Both solutions correctly calculate the intermediate steps and ultimately conclude that the required percentage decrease to restore the price is appro... | 0 | ||
Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC=20,$ and $CD=30.$ Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E,$ and $AE=5.$ What is the area of quadrilateral $ABCD?$
$\textbf{(A) } 330 \qquad \textbf{(B) } 340 \qquad \textbf{(C) } 350 \qquad \textbf{(D) } 360 \qquad \tex... | Alright, let me try to figure out how to solve this geometry problem. Okay, so we have quadrilateral ABCD with two right angles at ABC and ACD. The diagonal AC is 20 units long, and CD is 30 units. The diagonals AC and BD intersect at point E, and AE is 5 units. We need to find the area of quadrilateral ABCD. The answe... | Given quadrilateral \(ABCD\) with \(\angle ABC = 90^\circ\) and \(\angle ACD = 90^\circ\), \(AC = 20\), \(CD = 30\), and diagonals \(\overline{AC}\) and \(\overline{BD}\) intersecting at point \(E\) with \(AE = 5\). We need to find the area of quadrilateral \(ABCD\).
1. **Coordinates Setup**:
- Place \(C\) at the o... | 1. **Assign Variables and Use Pythagorean Theorem in $\triangle ABC$:**
Let $AB = x$ and $BC = y$. Since $\angle ABC = 90^\circ$, by the Pythagorean theorem, we have:
\[
x^2 + y^2 = AC^2 = 20^2 = 400.
\]
2. **Calculate Area of $\triangle ACD$:**
Since $\angle ACD = 90^\circ$, the area of $\triangle ACD$... | D | 360 | true | The area of quadrilateral ABCD calculated in the solution is 360, which matches the area calculated in the ground truth solution. Both solutions agree on the area, affirming that the statement of the problem is correctly solved. | 0 | ||
The angle bisector of the acute angle formed at the origin by the graphs of the lines $y = x$ and $y=3x$ has equation $y=kx.$ What is $k?$
$\textbf{(A)} \ \frac{1+\sqrt{5}}{2} \qquad \textbf{(B)} \ \frac{1+\sqrt{7}}{2} \qquad \textbf{(C)} \ \frac{2+\sqrt{3}}{2} \qquad \textbf{(D)} \ 2\qquad \textbf{(E)} \ \frac{2+\sqrt... | Okay, so I need to find the equation of the angle bisector of the acute angle formed at the origin by the lines y = x and y = 3x. The bisector is given as y = kx, and I need to determine the value of k. The answer choices are all different expressions involving square roots, so I guess it's not a simple fraction. Let m... | To find the equation of the angle bisector of the acute angle formed at the origin by the lines \( y = x \) and \( y = 3x \), we start by recognizing that the angle bisector will pass through the origin and have a positive slope \( k \).
We use the formula for the angle bisector between two lines \( L_1: a_1x + b_1y =... | 1. **Identify the Lines and Their Intersection**:
The lines given are $y = x$ and $y = 3x$. Both lines pass through the origin and form an acute angle there.
2. **Calculate the Slopes of the Lines**:
- The slope of the line $y = x$ is $1$.
- The slope of the line $y = 3x$ is $3$.
3. **Determine the Angle Be... | 2 | 4 | true | The solution correctly identifies the slopes of the lines, uses the appropriate formulas to derive the slope of the angle bisector, and arrives at the same result as the ground truth solution. The calculations agree, and the final answer for the slope is correctly boxed as \( \frac{1 + \sqrt{5}}{2} \), which is the sam... | 0 | ||
A set $S$ of points in the $xy$-plane is symmetric about the origin, both coordinate axes, and the line $y=x$. If $(2,3)$ is in $S$, what is the smallest number of points in $S$?
$\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 8\qquad \mathrm{(E) \ } 16$ | Okay, let's see. The problem says that a set S is symmetric about the origin, both coordinate axes, and the line y = x. If the point (2,3) is in S, what's the smallest number of points in S? The options are 1, 2, 4, 8, or 16. Hmm.
First, I need to recall what symmetry about each of these elements means. So, symmetry a... | Given a set \( S \) of points in the \( xy \)-plane that is symmetric about the origin, both coordinate axes, and the line \( y = x \), we need to determine the smallest number of points in \( S \) if the point \( (2, 3) \) is in \( S \).
1. **Symmetry about the x-axis**: Reflecting \( (2, 3) \) over the x-axis gives ... | 1. **Identify Symmetry Requirements**: The problem states that the set $S$ is symmetric about the origin, both coordinate axes, and the line $y=x$. This implies:
- Symmetry about the origin: If $(a, b) \in S$, then $(-a, -b) \in S$.
- Symmetry about the $x$-axis: If $(a, b) \in S$, then $(a, -b) \in S$.
- Symm... | 0 | \mathrm{(D)}8 | true | The solution correctly identifies all eight unique points generated by applying the required symmetries to the point (2, 3). It accurately describes how each symmetry leads to new points, and it confirms that all points are included, respecting the given symmetries. The conclusion aligns with the ground truth solution,... | 0 | ||
Five positive consecutive integers starting with $a$ have average $b$. What is the average of $5$ consecutive integers that start with $b$?
$\textbf{(A)}\ a+3\qquad\textbf{(B)}\ a+4\qquad\textbf{(C)}\ a+5\qquad\textbf{(D)}\ a+6\qquad\textbf{(E)}\ a+7$ | Okay, let's tackle this problem step by step. So, we have five positive consecutive integers starting with 'a', and their average is 'b'. Then, we need to find the average of five consecutive integers starting with 'b'. The answer choices are from A to E, with options like a+3, a+4, up to a+7.
First, let me make sure... | The problem involves finding the average of five consecutive integers starting with \( b \), where \( b \) is the average of five consecutive integers starting with \( a \).
1. **Calculate the average \( b \) of the first five consecutive integers starting with \( a \):**
- The five consecutive integers are \( a, a... | 1. **Define the sequence and calculate the average $b$:**
The five consecutive integers starting with $a$ are $a, a+1, a+2, a+3, a+4$. The average of these integers, $b$, is calculated as follows:
\[
b = \frac{a + (a+1) + (a+2) + (a+3) + (a+4)}{5} = \frac{5a + 10}{5} = a + 2
\]
2. **Determine the new seq... | 5 | 30 | true | The provided solution correctly calculates the average of five consecutive integers starting with both $a$ and $b$. The reasoning and calculations match those in the ground truth solution, confirming that the final result of the average starting with $b$ is indeed $a + 4$. Thus, both solutions agree that the answer is ... | 0 | ||
Which of the following is the negation of the statement: For all $x$ of a certain set, $x^2>0$?
$\textbf{(A)}\ \text{For all x}, x^2 < 0\qquad \textbf{(B)}\ \text{For all x}, x^2 \le 0\qquad \textbf{(C)}\ \text{For no x}, x^2>0\qquad \\ \textbf{(D)}\ \text{For some x}, x^2>0\qquad \textbf{(E)}\ \text{For some x}, x^2 \... | Okay, so I need to figure out the negation of the statement "For all x of a certain set, x² > 0." Let's break this down step by step.
First, the original statement is a universal quantification. It's saying that every element x in the set satisfies the condition x² being greater than zero. The logical form of this is... | The original statement is a universal quantification: "For all \( x \), \( x^2 > 0 \)." The negation of a universal statement \( \forall x, P(x) \) is an existential statement \( \exists x, \neg P(x) \). Here, \( P(x) \) is \( x^2 > 0 \), so its negation \( \neg P(x) \) is \( x^2 \le 0 \).
Analyzing the options:
- **... | 1. **Identify the original statement and its structure**: The original statement is "For all $x$ of a certain set, $x^2 > 0$". This is a universal quantification statement, which asserts that a certain property (in this case, $x^2 > 0$) holds for every element $x$ in a specified set.
2. **Understand the negation of a ... | E | 1 | true | The provided solution correctly identifies the original universal statement and properly applies the logical rules for negating universal quantification, ultimately arriving at the correct answer that matches the ground truth solution. | 0 | ||
At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values... | Okay, let's see. The problem says that at noon, Minneapolis is N degrees warmer than St. Louis. Then, by 4:00, Minneapolis's temperature has fallen by 5 degrees, and St. Louis's temperature has risen by 3 degrees. At that time, the difference between their temperatures is 2 degrees. We need to find the product of all p... | Let \( S \) be the temperature in St. Louis at noon. Then, the temperature in Minneapolis at noon is \( S + N \).
By 4:00, the temperature in Minneapolis has fallen by 5 degrees, making it \( (S + N) - 5 \). The temperature in St. Louis has risen by 3 degrees, making it \( S + 3 \).
The difference in temperatures a... | 1. **Define the variables:**
Let $M$ represent the temperature in Minneapolis at noon, and $L$ represent the temperature in St. Louis at noon. Given that Minneapolis is $N$ degrees warmer than St. Louis at noon, we can express this relationship as:
\[
M = L + N
\]
2. **Temperature changes by 4:00 PM:**
... | 2 | 2 | false | The provided solution gives the answer in the form of \\boxed{C} which does not indicate the numerical answer mentioned in the ground truth solution, which is \\boxed{60}. Therefore, the output format is incorrect and the correct answer is 60, not C. | 0 | ||
What is the greatest power of $2$ that is a factor of $10^{1002} - 4^{501}$?
$\textbf{(A) } 2^{1002} \qquad\textbf{(B) } 2^{1003} \qquad\textbf{(C) } 2^{1004} \qquad\textbf{(D) } 2^{1005} \qquad\textbf{(E) }2^{1006}$ | Okay, let's try to figure out the greatest power of 2 that divides \(10^{1002} - 4^{501}\). Hmm, so we need to find the highest power of 2 that is a factor of this expression. The answer choices go up to \(2^{1006}\), which is quite high. Let me break this down step by step.
First, let's rewrite the terms to see if th... | To determine the greatest power of \(2\) that is a factor of \(10^{1002} - 4^{501}\), we start by expressing the terms in terms of powers of 2 and 5:
\[
10^{1002} = (2 \times 5)^{1002} = 2^{1002} \times 5^{1002}
\]
\[
4^{501} = (2^2)^{501} = 2^{1002}
\]
Factoring out \(2^{1002}\) from both terms, we get:
\[
10^{1002}... | 1. **Expression Simplification**:
Start by rewriting the expression $10^{1002} - 4^{501}$:
\[
10^{1002} - 4^{501} = (2 \cdot 5)^{1002} - (2^2)^{501} = 2^{1002} \cdot 5^{1002} - 2^{1002}.
\]
Factor out $2^{1002}$:
\[
2^{1002} \cdot 5^{1002} - 2^{1002} = 2^{1002}(5^{1002} - 1).
\]
2. **Factorizat... | D | \textbf{(D)}\2^{1005} | true | The evaluation of the solution is correct. It outlines the steps for simplifying the expression, factoring, using Lifting The Exponent (LTE) correctly, and counting the powers of 2 precisely. Both the provided solution and the ground truth arrive at the same final result: the highest exponent of 2, which is 1005. The n... | 0 | ||
A given convex pentagon $ABCDE$ has the property that the area of each of the five triangles $ABC$, $BCD$, $CDE$, $DEA$, and $EAB$ is unity. Show that all pentagons with the above property have the same area, and calculate that area. Show, furthermore, that there are infinitely many non-congruent pentagons having the a... | Okay, so I need to show that all convex pentagons with the property that each of the five triangles ABC, BCD, CDE, DEA, and EAB has area 1 have the same total area, and then calculate that area. Also, I have to show that there are infinitely many such non-congruent pentagons. Hmm, let's start by understanding the probl... | To show that all convex pentagons \(ABCDE\) with the property that each of the five triangles \(ABC\), \(BCD\), \(CDE\), \(DEA\), and \(EAB\) has area 1 have the same area, we start by considering the coordinates of the vertices and the areas of the triangles.
1. **Assigning Coordinates and Setting Up Equations**:
... |
1. **Lemma Verification**:
The lemma states that in a convex pentagon $A_0A_1A_2A_3A_4$, the area of triangles $[A_0A_1A_2] = [A_1A_2A_3] = [A_2A_3A_4] = [A_3A_4A_0] = [A_4A_0A_1]$ if and only if $\overline{A_{n - 1}A_{n + 1}}\parallel\overline{A_{n - 2}A_{n + 2}}$ for $n = 0, 1, 2, 3, 4$ (indices taken mod 5).
... | \frac{3+\sqrt{5}}{2} | 5 | false | The calculated area in the SOLUTION is \\( rac{3 + \\sqrt{5}}{2}\\\), while the GROUND TRUTH SOLUTION concludes with an area of \\( rac{5 + \\sqrt{5}}{2}\\\). Since these two areas are not equivalent, the SOLUTION is incorrect. | 0 | ||
Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | Okay, so I have this problem here: Consider all 1000-element subsets of the set {1, 2, 3, ..., 2015}. From each such subset, choose the least element. Then we need to find the arithmetic mean of all these least elements, express it as a reduced fraction p/q, and then find p + q. Hmm, that seems a bit complex, but let's... | To find the arithmetic mean of the least elements of all 1000-element subsets of the set \(\{1, 2, 3, \ldots, 2015\}\), we start by considering the expectation of the minimum element in a random 1000-element subset.
The key insight is to use the known formula for the expectation of the minimum of a random \(n\)-eleme... |
To solve this problem, we need to find the arithmetic mean of the least elements of all 1000-element subsets of the set $\{1, 2, 3, \ldots, 2015\}$. We will use combinatorial arguments to derive the solution.
#### Step 1: Counting subsets with a fixed least element
Let $i$ be the least element of a 1000-element subse... | 431 | 431 | true | Both the solution and the ground truth solution arrive at the same final answer of \( \boxed{431} \). The calculations performed in both solutions support the conclusion that the arithmetic mean of the least elements of all 1000-element subsets from the set \( \{1, 2, 3, \ldots, 2015\} \) is \( \frac{288}{143} \) and t... | 0 | ||
A man on his way to dinner short after $6: 00$ p.m. observes that the hands of his watch form an angle of $110^{\circ}$. Returning before $7: 00$ p.m. he notices that again the hands of his watch form an angle of $110^{\circ}$. The number of minutes that he has been away is:
$\textbf{(A)}\ 36\frac{2}3\qquad\textbf{(B)... | Okay, so I need to figure out how many minutes the man was away based on the times when the angle between the clock hands was 110 degrees. Let's start by recalling how clock angles work.
First, I remember that the formula to calculate the angle between the hour and minute hands is:
θ = |30H - 5.5M|
where θ is the a... | To determine the number of minutes the man was away, we need to find the times between 6:00 p.m. and 7:00 p.m. when the angle between the clock hands is \(110^\circ\). The formula for the angle between the hour and minute hands is:
\[
\theta = |30H - 5.5M|
\]
where \(H\) is the hour and \(M\) is the minutes past the ... | 1. **Define the angles of the hands**: Let $n$ be the number of minutes after 6:00 PM. The angle of the hour hand from the top (12:00 position) can be calculated as follows:
- In one hour, the hour hand moves $30^\circ$ (since $360^\circ / 12 = 30^\circ$).
- In $n$ minutes, the hour hand moves $\frac{30}{60}n = \... | 4 | 4 | true | The solution provided correctly calculates the angle between the hour and minute hands at the specified times and arrives at the correct time difference of 40 minutes. The method of solving the absolute value equation is valid and aligns with the ground truth solution. Therefore, the answer is indeed correct. | 0 | ||
A 3x3x3 cube is made of $27$ normal dice. Each die's opposite sides sum to $7$. What is the smallest possible sum of all of the values visible on the $6$ faces of the large cube?
$\text{(A)}\ 60 \qquad \text{(B)}\ 72 \qquad \text{(C)}\ 84 \qquad \text{(D)}\ 90 \qquad \text{(E)} 96$ | Alright, so I've got this problem here about a 3x3x3 cube made up of 27 dice. The question is asking for the smallest possible sum of all the visible faces on the six faces of the large cube. Each die has opposite sides that sum to 7. The answer choices are 60, 72, 84, 90, and 96. Let me try to figure this out step by ... | To determine the smallest possible sum of all visible values on the 6 faces of a 3x3x3 cube made of 27 dice, we need to consider the optimal orientation of each die to minimize the sum of visible faces. Each die's opposite sides sum to 7.
1. **Categorizing Dice Positions**:
- **Corner Dice**: 8 corners, each with 3... | 1. **Understanding the Cube Configuration**:
- A 3x3x3 cube consists of 27 smaller cubes.
- The smaller cubes on the corners have three faces visible.
- The smaller cubes on the edges have two faces visible.
- The smaller cubes in the center of each face have one face visible.
2. **Counting Visible Faces**... | D | 3 | true | The solution provided accurately summarizes the configuration of the dice on the cube, counts the visible faces correctly, and calculates the minimum possible sum based on showing the lowest values on the visible faces of the dice. The calculations for each type of die are correct, leading to the total sum of 90, match... | 0 | ||
In the multiplication problem below $A$, $B$, $C$, $D$ are different digits. What is $A+B$?
\[\begin{array}{cccc}& A & B & A\\ \times & & C & D\\ \hline C & D & C & D\\ \end{array}\]
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 9$ | Okay, so I need to solve this multiplication problem where A, B, C, D are different digits. The problem is set up like a vertical multiplication:
A B A
× C D
----------
C D C D
And the question is asking for A + B, with the answer choices given from 1 to 9. Let me try to figure this out step by step.
First, let ... | Given the multiplication problem where \( A \), \( B \), \( C \), and \( D \) are different digits:
\[
\begin{array}{cccc}
& A & B & A \\
\times & & C & D \\
\hline
C & D & C & D \\
\end{array}
\]
We need to determine the value of \( A + B \).
1. **Analyzing the structure**:
- The three-digit number \( ABA \) ca... | 1. **Identify the value of $A$:**
Given the multiplication problem:
\[
\begin{array}{cccc}
& A & B & A\\
\times & & C & D\\
\hline
C & D & C & D\\
\end{array}
\]
We observe that the product of $A$ and $D$ results in a number ending in $D$. This implies that $A \times D$ must be a num... | 11 | 2 | true | Both the provided solution and the ground truth arrive at the same conclusion regarding the values of A and B, resulting in A + B = 1. These calculations are consistent, and the logic used to derive these results is sound. | 0 | ||
Let $\, a =(m^{m+1} + n^{n+1})/(m^m + n^n), \,$ where $\,m\,$ and $\,n\,$ are positive integers. Prove that $\,a^m + a^n \geq m^m + n^n$.
[You may wish to analyze the ratio $\,(a^N - N^N)/(a-N),$ for real $\, a \geq 0 \,$ and integer $\, N \geq 1$.] | Okay, so I need to prove that for positive integers m and n, if a is defined as (m^{m+1} + n^{n+1})/(m^m + n^n), then a^m + a^n is greater than or equal to m^m + n^n. Hmm, let's start by understanding what a is. It looks like a weighted average of m and n, but with exponents involved.
First, let me compute a. The num... | Given \( a = \frac{m^{m+1} + n^{n+1}}{m^m + n^n} \) where \( m \) and \( n \) are positive integers, we need to prove that \( a^m + a^n \geq m^m + n^n \).
### Key Steps and Analysis:
1. **Definition of \( a \)**:
- \( a \) is a weighted average of \( m \) and \( n \) with weights proportional to \( m^m \) and \( n... | 1. **Assumption and Definitions**: Assume without loss of generality that $m \geq n$. Define $a = \frac{m^{m+1} + n^{n+1}}{m^m + n^n}$.
2. **Expressing $m-a$ and $a-n$**:
- For $m-a$, we have:
\[
m-a = \frac{m(m^m + n^n) - (m^{m+1} + n^{n+1})}{m^m + n^n} = \frac{m^m n^n - n^{n+1}}{m^m + n^n} = n^n \frac{m... | a^m+a^n\gem^m+n^n | -1 | true | The provided solution correctly establishes the inequality \( a^m + a^n \geq m^m + n^n \) by deriving the necessary expressions and analyzing the bounds of \( a \) in relation to \( m \) and \( n \). The steps taken in both the original solution and the ground truth solution are logically sound, affirming the result. T... | 0 | ||
Andrea and Lauren are $20$ kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of $1$ kilometer per minute. After $5$ minutes, Andrea stops biking because of a flat tire and waits for Lauren. After how many minutes from t... | Okay, let's try to figure out this problem step by step. So, Andrea and Lauren are 20 kilometers apart and biking towards each other. Andrea is going three times as fast as Lauren, and the distance between them decreases at 1 kilometer per minute. After 5 minutes, Andrea stops biking because of a flat tire and waits fo... | Andrea and Lauren start 20 kilometers apart and bike towards each other. Andrea's speed is three times Lauren's speed, and the distance between them decreases at a rate of 1 kilometer per minute.
1. **Combined Speed Calculation**:
- Let Lauren's speed be \( v \) km/min. Then Andrea's speed is \( 3v \) km/min.
-... | 1. **Define Variables:**
Let $v_A$ be Andrea's speed and $v_L$ be Lauren's speed, both in kilometers per hour.
2. **Set Up Equations:**
Given that Andrea travels at three times the speed of Lauren, we have:
\[ v_A = 3v_L \]
Also, the rate at which the distance between them decreases is $1$ kilometer per mi... | 3 | 3 | false | The provided solution incorrectly calculates the speeds of Andrea and Lauren in kilometers per minute instead of per hour. In the ground truth, their speeds were appropriately calculated in km/h and represented correctly. The speed was established to sum to 60 km/h due to the given rate of distance decrease. Consequent... | 0 | ||
The sum of two natural numbers is $17{,}402$. One of the two numbers is divisible by $10$. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?
$\textbf{(A)} ~10{,}272\qquad\textbf{(B)} ~11{,}700\qquad\textbf{(C)} ~13{,}362\qquad\textbf{(D)} ~14{,}238\q... | Okay, let's tackle this problem step by step. So, we have two natural numbers whose sum is 17,402. One of them is divisible by 10, which means it ends with a 0 in the units place. The other number is obtained by erasing the units digit of the first number. We need to find the difference between these two numbers. The o... | Let's denote the two natural numbers as \( x \) and \( y \), where \( x \) is divisible by 10. According to the problem, erasing the units digit of \( x \) gives \( y \). This implies \( x = 10y \). The sum of the two numbers is 17,402, so we can set up the equation:
\[
x + y = 17,402
\]
Substituting \( x = 10y \) in... | 1. **Identify the relationship between the two numbers**: Given that one number is divisible by $10$ and removing its units digit (which is $0$) gives the other number, we can denote the smaller number as $a$ and the larger number as $10a$.
2. **Set up the equation for their sum**: The sum of the two numbers is given ... | 6 | 2 | true | The solution correctly derives the values of the two numbers based on the given conditions and matches the steps outlined in the ground truth solution. Both solutions arrive at the same value for the smaller number and therefore the correct difference, confirming that the solution is accurate. | 0 | ||
What is the value of \[2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9?\]
$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$ | Okay, let me try to figure out this problem step by step. The question is asking for the value of this expression:
\[2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9\]
And the options are from A to E, with possible answers 0, 1, 2, 3, or 4. Hmm, let's break this down into parts.
First, I not... | The given expression is \[2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9.\]
**First Term Analysis:**
1. Compute the innermost exponent: \(1^9 = 1\).
2. Substitute into the next exponent: \(0^{1} = 0\).
3. Substitute into the outer term: \(2^{0} = 1\).
**Second Term Analysis:**
1. Compute the ... | To solve the expression \[2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9,\] we need to evaluate each term separately.
1. **Evaluate the first term: $2^{\left(0^{\left(1^9\right)}\right)}$**
- First, simplify the exponent's base: $1^9 = 1$.
- Then, calculate $0^1 = 0$.
- Finally, compu... | C | 2 | true | The solution correctly evaluates the terms of the expression and arrives at the correct total of 2. It matches the ground truth solution in both structure and final result. | 0 | ||
Which of the following describes the graph of the equation $(x+y)^2=x^2+y^2$?
$\textbf{(A) } \text{the\,empty\,set}\qquad \textbf{(B) } \textrm{one\,point}\qquad \textbf{(C) } \textrm{two\,lines} \qquad \textbf{(D) } \textrm{a\,circle} \qquad \textbf{(E) } \textrm{the\,entire\,plane}$ | Okay, let me try to figure out what the graph of the equation \((x + y)^2 = x^2 + y^2\) looks like. Hmm, so the question gives me multiple choices: empty set, one point, two lines, a circle, or the entire plane. I need to figure out which one it is.
First, maybe I should start by expanding the left side of the equatio... | The equation given is \((x + y)^2 = x^2 + y^2\). To analyze this, we start by expanding the left-hand side:
\[
(x + y)^2 = x^2 + 2xy + y^2.
\]
Substituting this back into the equation, we have:
\[
x^2 + 2xy + y^2 = x^2 + y^2.
\]
Subtracting \(x^2 + y^2\) from both sides simplifies to:
\[
2xy = 0.
\]
Dividing both... | 1. **Expand the given equation**: Start by expanding the left-hand side of the equation $(x+y)^2 = x^2 + y^2$.
\[
(x+y)^2 = x^2 + 2xy + y^2
\]
This simplifies to:
\[
x^2 + 2xy + y^2 = x^2 + y^2
\]
2. **Simplify the equation**: Subtract $x^2 + y^2$ from both sides to isolate terms involving $xy$.
... | 4 | 4 | true | The provided solution correctly expands the given equation, simplifies it, and concludes that the solution consists of the x-axis and y-axis (the two lines). It matches the ground truth in terms of method, reasoning, and conclusion regarding the graphical representation of the equation. | 0 | ||
If $\texttt{a}$ and $\texttt{b}$ are digits for which
$\begin{array}{ccc}& 2 & a\\ \times & b & 3\\ \hline & 6 & 9\\ 9 & 2\\ \hline 9 & 8 & 9\end{array}$
then $\texttt{a+b =}$
$\mathrm{(A)\ } 3 \qquad \mathrm{(B) \ }4 \qquad \mathrm{(C) \ } 7 \qquad \mathrm{(D) \ } 9 \qquad \mathrm{(E) \ }12$ | Okay, so I have this multiplication problem here, and I need to figure out the values of a and b, which are digits. Then, add them together to get the answer. Let me try to break this down step by step.
First, let me write down the problem as it's given:
```
2 a
× b 3
------
6 9
9 2
------
9 8 9
```
Hmm, okay. T... | Given the multiplication problem:
\[
\begin{array}{ccc}
& 2 & a \\
\times & b & 3 \\
\hline
& 6 & 9 \\
9 & 2 \\
\hline
9 & 8 & 9 \\
\end{array}
\]
We need to determine the values of \(a\) and \(b\) (digits) and then find \(a + b\).
1. **First Partial Product (69):** This is the result of multiplying the two-digit ... | 1. **Identify the multiplication setup**: The multiplication setup is given as:
\[
\begin{array}{ccc}
& 2 & a\\
\times & b & 3\\
\hline
& 6 & 9\\
9 & 2\\
\hline
9 & 8 & 9
\end{array}
\]
This represents the multiplication of a two-digit number $2a$ by another two-digit number $b... | 3 | 9 | true | The solution correctly calculates the values of a and b, determining a = 3 and b = 4. It accurately verifies each step of the multiplication and confirms that both the partial products align with the expected output of 989. Therefore, the addition of a and b is also correctly computed as 7. | 0 | ||
If $x, 2x+2, 3x+3, \dots$ are in geometric progression, the fourth term is:
$\textbf{(A)}\ -27 \qquad \textbf{(B)}\ -13\frac{1}{2} \qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 13\frac{1}{2}\qquad \textbf{(E)}\ 27$ | Okay, so I have this problem here: If the sequence x, 2x+2, 3x+3, … is a geometric progression, then the fourth term is one of the options given. The options are A through E, with numbers like -27, -13.5, 12, 13.5, and 27. Hmm, let me think about how to approach this.
First, I need to recall what a geometric progressi... | Given the sequence \( x, 2x + 2, 3x + 3, \ldots \) is in geometric progression (GP), we need to find the fourth term.
In a GP, the ratio between consecutive terms is constant. Therefore, we set up the equation for the common ratio \( r \):
\[
\frac{2x + 2}{x} = \frac{3x + 3}{2x + 2}
\]
Cross-multiplying and simplify... | 1. **Identify the nature of the sequence**: Given that $x, 2x+2, 3x+3, \dots$ are in geometric progression, the ratio between consecutive terms must be constant. Let's denote this common ratio by $r$.
2. **Set up the equation for the common ratio**:
\[
\frac{2x+2}{x} = \frac{3x+3}{2x+2}
\]
This equation ar... | B | 0 | false | The solution incorrectly identifies the fourth term as \\textbf{(B)} which corresponds to \(-13.5\\), instead of simply stating the answer without context. The correct answer should be \(-13.5\\), and the solution should not link it to an answer choice as it lacks clarity. | 0 | ||
At $2: 15$ o'clock, the hour and minute hands of a clock form an angle of:
$\textbf{(A)}\ 30^{\circ} \qquad\textbf{(B)}\ 5^{\circ} \qquad\textbf{(C)}\ 22\frac {1}{2}^{\circ} \qquad\textbf{(D)}\ 7\frac {1}{2} ^{\circ} \qquad\textbf{(E)}\ 28^{\circ}$ | Okay, so I need to figure out the angle between the hour and minute hands at 2:15. Hmm, let's think. I remember that clocks can be a bit tricky because both hands are moving. The minute hand moves around the clock as the minutes pass, but the hour hand also moves, not just jumping from hour to hour. So, at 2:15, the mi... | At 2:15, the minute hand is at \(15 \times 6 = 90^\circ\) (since it moves \(6^\circ\) per minute). The hour hand starts at \(2 \times 30 = 60^\circ\) and moves \(15 \times 0.5 = 7.5^\circ\) (as it moves \(0.5^\circ\) per minute). Thus, the hour hand is at \(60 + 7.5 = 67.5^\circ\). The angle between them is \(90 - 67.5... | To find the angle between the hour and minute hands of a clock at a given time, we can use the formula:
\[
\text{Angle} = \left|\frac{60h - 11m}{2}\right|
\]
where $h$ is the hour and $m$ is the minute.
1. **Identify the values of $h$ and $m$:**
At $2:15$, $h = 2$ and $m = 15$.
2. **Substitute these values into th... | C | 22\frac{1}{2} | true | The provided solution correctly calculates the angle between the hour and minute hands at 2:15 using both methods: manually computing the positions of the hands and applying the angle formula. Both methods arrive at the same conclusion of 22.5 degrees, confirming the solution's accuracy. | 0 | ||
In [right triangle](https://artofproblemsolving.com/wiki/index.php/Right_triangle) $ABC$ with [right angle](https://artofproblemsolving.com/wiki/index.php/Right_angle) $C$, $CA = 30$ and $CB = 16$. Its legs $CA$ and $CB$ are extended beyond $A$ and $B$. [Points](https://artofproblemsolving.com/wiki/index.php/Point) $... | Okay, let's try to figure out this problem. So, we have a right triangle ABC with a right angle at C. The legs CA and CB are 30 and 16 units long, respectively. The legs are extended beyond A and B, and there are two circles with centers O₁ and O₂ in the exterior of the triangle. Both circles have the same radius, and ... | In right triangle \(ABC\) with right angle at \(C\), the legs \(CA = 30\) and \(CB = 16\). The hypotenuse \(AB\) is calculated using the Pythagorean theorem:
\[
AB = \sqrt{30^2 + 16^2} = \sqrt{900 + 256} = \sqrt{1156} = 34
\]
The circles with centers \(O_1\) and \(O_2\) are tangent to the hypotenuse \(AB\) and the ex... |
1. **Identify the triangle and its properties**:
Triangle $ABC$ is a right triangle with $CA = 30$, $CB = 16$, and right angle at $C$. Using the Pythagorean theorem, we find the hypotenuse $AB$:
\[
AB = \sqrt{CA^2 + CB^2} = \sqrt{30^2 + 16^2} = \sqrt{900 + 256} = \sqrt{1156} = 34.
\]
2. **Understanding t... | 737 | 1 | true | The presented solution correctly follows the geometry and calculations to determine the radius of the circles tangent to the hypotenuse and their respective extensions. The calculations for the radius using the Pythagorean theorem and the tangent segment lengths are accurate, leading to the same result for the sum of p... | 0 | ||
A finite [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digi... | Okay, let's try to tackle this problem. So, we have a finite sequence of three-digit numbers. Each term in the sequence has the property that the tens and units digits become the hundreds and tens digits of the next term. Also, the last term's tens and units digits loop back to become the first term's hundreds and tens... | The sequence forms a cyclic structure where each term's tens and units digits become the next term's hundreds and tens digits. Denote each term as \( T_i = 100a_i + 10b_i + c_i \). The cyclic property implies:
- \( a_{i+1} = b_i \) and \( b_{i+1} = c_i \) for \( 1 \leq i \leq n-1 \)
- \( a_1 = b_n \) and \( b_1 = c_n ... | 1. **Understanding the Sequence Property**: The sequence is such that the tens and units digits of each term become the hundreds and tens digits of the next term. This implies that each digit in a position (hundreds, tens, units) of one term will appear in the next position in the next term and eventually cycle back to... | D | 1 | true | The provided solution correctly identifies the cyclic nature of the sequences and accurately derives the sum of the terms, showing that it can be factored as 111k, leading to the conclusion that the largest prime factor is 37. The reasoning aligns with the ground truth solution and uses similar processes to arrive at t... | 0 | ||
What time was it $2011$ minutes after midnight on January 1, 2011?
$\textbf{(A)}\ \text{January 1 at 9:31PM}$
$\textbf{(B)}\ \text{January 1 at 11:51PM}$
$\textbf{(C)}\ \text{January 2 at 3:11AM}$
$\textbf{(D)}\ \text{January 2 at 9:31AM}$
$\textbf{(E)}\ \text{January 2 at 6:01PM}$ | Okay, so I need to figure out what time it was 2011 minutes after midnight on January 1, 2011. The options are from A to E, with different dates and times. Let me start by breaking down the problem step by step.
First, midnight on January 1 is 12:00 AM. Adding minutes to that time should give me the correct time later... | To determine the time 2011 minutes after midnight on January 1, 2011, we first convert minutes into hours and days:
1. **Convert 2011 minutes to hours**:
\[
2011 \div 60 = 33 \text{ hours and } 31 \text{ minutes.}
\]
This means 2011 minutes is equivalent to 33 hours and 31 minutes.
2. **Break down 33 ... | 1. **Convert minutes to hours and minutes**:
We start by converting the total minutes (2011) into hours and minutes. Since there are 60 minutes in an hour, we perform the division:
\[
2011 \div 60 = 33 \text{ remainder } 31
\]
This tells us that 2011 minutes is equivalent to 33 hours and 31 minutes.
2.... | D | \textbf{(D)}\January2at9:31AM | true | The provided solution accurately follows the steps required to convert 2011 minutes into hours and determine the resulting date and time. It reaches the correct conclusion of "January 2 at 9:31 AM" and is consistent with the ground truth solution. | 0 | ||
A basketball player made $5$ baskets during a game. Each basket was worth either $2$ or $3$ points. How many different numbers could represent the total points scored by the player?
$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$ | Okay, let's try to figure out this basketball problem. So, the player made 5 baskets, each worth either 2 or 3 points. We need to find out how many different total points they could have scored. The options are from A to E, with possible answers 2, 3, 4, 5, or 6. Hmm, let's break this down step by step.
First, each ba... | The basketball player made 5 baskets, each worth either 2 or 3 points. We need to determine how many different total points could be scored.
1. Let \( x \) be the number of 2-pointers and \( y \) be the number of 3-pointers. We know \( x + y = 5 \).
2. The total points can be expressed as \( 2x + 3y \).
3. Substitutin... | We will solve the problem by analyzing the possible scores the player can achieve based on the number of three-point and two-point baskets made.
1. **Define the variables**:
Let $x$ be the number of three-point shots made, and $5-x$ be the number of two-point shots made.
2. **Formulate the total points**:
The t... | 0 | \textbf{(E)}6 | true | The solution accurately follows the correct logical steps to derive the number of distinct total points scored by the player. It correctly defines the variables, derives the total score from the number of baskets made, and correctly calculates and lists the distinct values. The final conclusion matches the ground truth... | 0 | ||
Square $EFGH$ has one vertex on each side of square $ABCD$. Point $E$ is on $AB$ with $AE=7\cdot EB$. What is the ratio of the area of $EFGH$ to the area of $ABCD$?
$\text{(A)}\,\frac{49}{64} \qquad\text{(B)}\,\frac{25}{32} \qquad\text{(C)}\,\frac78 \qquad\text{(D)}\,\frac{5\sqrt{2}}{8} \qquad\text{(E)}\,\f... | Okay, let's see. The problem is about two squares: a larger square ABCD and a smaller square EFGH. Each vertex of EFGH is on a different side of ABCD. Specifically, point E is on side AB, and they tell us that AE is 7 times EB. We need to find the ratio of the area of EFGH to the area of ABCD. The answer choices are gi... | Square \(EFGH\) has one vertex on each side of square \(ABCD\). Point \(E\) is on \(AB\) with \(AE = 7 \cdot EB\). We need to find the ratio of the area of \(EFGH\) to the area of \(ABCD\).
1. **Assign Coordinates:**
- Let the side length of square \(ABCD\) be \(s\).
- Coordinates of \(A(0, 0)\), \(B(s, 0)\), \(... | 1. **Assigning Side Lengths:**
Let the side length of square $ABCD$ be $8s$. This choice is made to simplify calculations, as $AE = 7 \cdot EB$ implies a division of side $AB$ into 8 equal parts.
2. **Locating Point E:**
Since $AE = 7 \cdot EB$, and if we let $EB = x$, then $AE = 7x$. Given $AB = 8s$, we have $A... | B | 0.1 | true | Both the provided solution and the ground truth solution yield the same ratio of the areas of squares EFGH and ABCD, which is \( \frac{25}{32} \). The computations regarding the coordinates and areas are consistent between the two solutions. Thus, the final boxed answer of \( \frac{25}{32} \) is indeed correct. | 0 | ||
A house worth $ $9000$ is sold by Mr. A to Mr. B at a $10$ % loss. Mr. B sells the house back to Mr. A at a $10$ % gain.
The result of the two transactions is:
$\textbf{(A)}\ \text{Mr. A breaks even} \qquad \textbf{(B)}\ \text{Mr. B gains }$900 \qquad \textbf{(C)}\ \text{Mr. A loses }$900\\ \textbf{(D)}\ \text{Mr. A ... | Okay, let's see. The problem is about Mr. A selling a house to Mr. B at a 10% loss, and then Mr. B selling it back to Mr. A at a 10% gain. We need to figure out the result of these two transactions for both parties. The options given are A through E, with possible outcomes like breaking even, gaining or losing specific... | Mr. A initially sells the house worth $9000 to Mr. B at a 10% loss.
**First transaction:**
- Loss amount: \(10\% \times 9000 = 900\)
- Selling price: \(9000 - 900 = 8100\)
Mr. A receives $8100, incurring a $900 loss.
Mr. B then sells the house back to Mr. A at a 10% gain.
**Second transaction:**
- Gain a... | 1. **Initial Value of the House**: The house is initially valued at $9000$.
2. **First Transaction (Mr. A to Mr. B)**:
- Mr. A sells the house at a 10% loss.
- Calculation of the selling price:
\[
9000 \times (1 - 0.1) = 9000 \times 0.9 = 8100
\]
- Mr. A sells the house for $8100$.
3. **Second... | D | 2 | true | The solution provided matches the ground truth solution in all calculations and conclusions. Mr. A's loss of $810 is correctly calculated based on the sell and buy-back transactions with appropriate percentage adjustments. | 0 | ||
Chandler wants to buy a $500$ dollar mountain bike. For his birthday, his grandparents
send him $50$ dollars, his aunt sends him $35$ dollars and his cousin gives him $15$ dollars. He earns
$16$ dollars per week for his paper route. He will use all of his birthday money and all
of the money he earns from his paper rout... | Okay, let's see. Chandler wants to buy a mountain bike that costs $500. He received some money from his grandparents, aunt, and cousin for his birthday. Plus, he earns money from his paper route every week. The question is asking how many weeks he needs to work to save up enough for the bike, using all his birthday mon... | Chandler receives a total of $50 + $35 + $15 = $100 from his birthday gifts. The remaining amount he needs to buy the $500 mountain bike is $500 - $100 = $400. He earns $16 per week from his paper route, so the number of weeks required to save $400 is calculated by dividing $400 by $16:
\[
\frac{400}{16} = 25
\]
Thus... | 1. **Identify the total amount of money Chandler needs:** Chandler wants to buy a mountain bike that costs $500.
2. **Calculate the total amount of birthday money Chandler received:**
- From his grandparents: $50$
- From his aunt: $35$
- From his cousin: $15$
- Total birthday money: $50 + 35 + 15 = 100$
3... | B | \textbf{(B)}25 | true | The solution correctly calculates the total birthday money received, determines the remaining amount needed for the bike, and accurately computes the number of weeks required to save the necessary amount. The steps follow a logical sequence similar to the ground truth solution, ultimately leading to the same correct an... | 0 | ||
The harmonic mean of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?
$\textbf{(A) }\frac{3}{7}\qquad\textbf{(B) }\frac{7}{12}\qquad\textbf{(C) }\frac{12}{7}\qquad\textbf{(D) }\frac{7}{4}\qquad \textbf{(E) }\frac{7}{3}$ | Okay, let's see. The problem is asking for the harmonic mean of the numbers 1, 2, and 4. The answer choices are given, so once I figure it out, I can check which one matches. Hmm, harmonic mean... I remember it's different from the arithmetic mean. Let me recall the definition.
The harmonic mean of a set of non-zero n... | The harmonic mean of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. To find the harmonic mean of 1, 2, and 4, we follow these steps:
1. **Find the reciprocals of the numbers**:
- The reciprocal of 1 is \(1\).
- The reciprocal of 2 is \(\frac{1}{2}\).
- The recipr... | 1. **Calculate the reciprocals of the numbers**:
Given numbers are 1, 2, and 4. Their reciprocals are:
\[
\frac{1}{1}, \frac{1}{2}, \text{ and } \frac{1}{4}
\]
2. **Sum the reciprocals**:
\[
\frac{1}{1} + \frac{1}{2} + \frac{1}{4} = 1 + 0.5 + 0.25 = 1.75 = \frac{7}{4}
\]
3. **Calculate the avera... | 0 | 3 | true | The provided solution accurately follows the steps to calculate the harmonic mean, and it matches the ground truth solution in both the process and the final result. The harmonic mean of 1, 2, and 4 is correctly calculated to be \frac{12}{7}, which matches the conclusion in the ground truth. Therefore, the given soluti... | 0 | ||
Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks at the ideal rate, which of the following number of words would be an appropriate length for her speech?
$\text{(A)}\ 2250 \qquad \te... | Okay, so Julie needs to prepare a speech that's between half an hour and three-quarters of an hour long. The ideal speaking rate is 150 words per minute. The question is asking which number of words from the options given would be appropriate for her speech. Let me break this down step by step.
First, I need to figure... | Julie's speech must last between 30 minutes (0.5 hours) and 45 minutes (0.75 hours). At an ideal rate of 150 words per minute, the required word count range is calculated as follows:
- **Minimum words**: \(150 \, \text{words/min} \times 30 \, \text{min} = 4500 \, \text{words}\)
- **Maximum words**: \(150 \, \text{w... | 1. **Convert Time to Minutes**:
- One-half hour is $\frac{1}{2} \times 60 = 30$ minutes.
- Three-quarters of an hour is $\frac{3}{4} \times 60 = 45$ minutes.
2. **Calculate the Range of Words**:
- At the ideal rate of 150 words per minute, the minimum number of words Julie should speak is:
\[
30 \t... | 150 | 4 | true | The provided solution correctly calculates the range of words Julie's speech requires (4500 to 6750), identifies the rate of speech (150 words/min), and evaluates each given option to find that only option E (5650 words) falls within the calculated range. The matches in calculations and conclusions confirm the correctn... | 0 | ||
Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB'=3 \cdot AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC'=3 \cdot BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA'=3 \cdot CA$. What is the ratio of the... | Okay, let's see. The problem is about an equilateral triangle ABC, and each side is extended beyond one of its endpoints to create points B', C', and A'. The extensions are such that BB' is three times AB, CC' is three times BC, and AA' is three times CA. We need to find the ratio of the area of triangle A'B'C' to the ... | Let $ABC$ be an equilateral triangle with side length $s$. We extend each side beyond a vertex by three times the side length to form points $A'$, $B'$, and $C'$. We need to find the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$.
1. **Coordinates of Original Triangle**:
- $A(0, 0)$
- $B(... | 1. **Define the side length of $\triangle ABC$:** Let $AB = BC = CA = x$.
2. **Extend the sides of $\triangle ABC$:**
- Extend $AB$ to $B'$ such that $BB' = 3AB$. Thus, $AB' = AB + BB' = x + 3x = 4x$.
- Extend $BC$ to $C'$ such that $CC' = 3BC$. Thus, $BC' = BC + CC' = x + 3x = 4x$.
- Extend $CA$ to $A'$ suc... | 3 | \textbf{(B)}16 | false | The provided solution calculates the ratio of the areas of triangles incorrectly. The solution arrives at a ratio of 37, while the ground truth solution correctly finds the ratio to be 16. Therefore, the two ratios do not match, indicating that the provided solution is incorrect. | 0 | ||
If the following instructions are carried out by a computer, which value of $X$ will be printed because of instruction $5$?
1. START $X$ AT $3$ AND $S$ AT $0$.
2. INCREASE THE VALUE OF $X$ BY $2$.
3. INCREASE THE VALUE OF $S$ BY THE VALUE OF $X$.
4. IF $S$ IS AT LEAST $10000$,
THEN GO TO INSTRUCTION $5... | Okay, let me try to figure out this problem. So, the question is about a computer program with instructions 1 through 6, and we need to find the value of X that gets printed when instruction 5 is executed. The options are given from (A) 19 up to (E) 201. Alright, let's break this down step by step.
First, let's parse ... | The problem involves determining the value of \( X \) printed by a computer program after executing a series of instructions. Here is the systematic breakdown:
1. **Initialization**: Start with \( X = 3 \) and \( S = 0 \).
2. **Loop Execution**:
- **Instruction 2**: Increase \( X \) by 2.
- **Instruction 3**: In... | 1. **Initialization**: The program starts with $X = 3$ and $S = 0$.
2. **Loop Execution**:
- **Instruction 2**: Increase the value of $X$ by $2$.
- **Instruction 3**: Increase the value of $S$ by the current value of $X$.
- **Instruction 4**: Check if $S \geq 10000$. If true, go to instruction 5; otherwise, ... | E | E | true | The solution correctly identifies the initialization and loop execution. The arithmetic sequence of X values starting from 3 with an increment of 2 is accurately calculated. The realization that the sum S becomes 9999 at n=99 and needs to reach n=100 to become 201 is also correct. Hence, this matches the ground truth. | 0 | ||
Which of the following is equivalent to "If P is true, then Q is false."? | Okay, let's tackle this logic problem. The question is asking which of the following is equivalent to "If P is true, then Q is false." Hmm, first, I need to recall how to translate conditional statements into logical expressions. So, the original statement is "If P, then not Q." In symbolic terms, that would be P → ¬Q,... | The statement "If P is true, then Q is false" (symbolically \( P \rightarrow \neg Q \)) is logically equivalent to:
1. **Contrapositive**: "If Q is true, then P is false" (\( Q \rightarrow \neg P \)).
This follows because the contrapositive of \( P \rightarrow \neg Q \) is formed by reversing and negating the ant... | 1. **Understanding the Original Statement**: The original statement given is "If P is true, then Q is false." In logical notation, this can be written as:
\[
P \implies \neg Q
\]
where $\implies$ denotes "implies" and $\neg$ denotes "not".
2. **Finding the Contrapositive**: The contrapositive of a statemen... | D | true | The provided solution correctly identifies the contrapositive of the statement "If P is true, then Q is false" (symbolically, \(P \rightarrow \neg Q\)) as "If Q is true, then P is false" (\(Q \rightarrow \neg P\)). It also correctly describes the logical equivalence of the contrapositive to the original statement, whic... | 0 | |||
Letters $A,B,C,$ and $D$ represent four different digits selected from $0,1,2,\ldots ,9.$ If $(A+B)/(C+D)$ is an integer that is as large as possible, what is the value of $A+B$?
$\mathrm{(A) \ }13 \qquad \mathrm{(B) \ }14 \qquad \mathrm{(C) \ } 15\qquad \mathrm{(D) \ }16 \qquad \mathrm{(E) \ } 17$ | Okay, let's tackle this problem step by step. The question is about finding four different digits A, B, C, and D such that the sum (A + B) divided by (C + D) is an integer, and we want this integer to be as large as possible. Then, we need to determine the value of A + B from the given options. The answer choices are 1... | To solve the problem, we need to find four different digits \(A, B, C,\) and \(D\) such that \(\frac{A+B}{C+D}\) is an integer and as large as possible. The goal is to determine the value of \(A + B\).
1. **Maximizing \(A + B\)**: The largest possible sum of two different digits is \(9 + 8 = 17\). Therefore, we consid... | 1. **Objective**: We need to maximize the fraction $\frac{A+B}{C+D}$ where $A, B, C, D$ are distinct digits from $0$ to $9$. This involves maximizing $A+B$ and minimizing $C+D$.
2. **Maximizing $A+B$**:
- The maximum value for any digit is $9$. Thus, to maximize $A+B$, we should choose the largest available distin... | 10 | 4 | false | The solution provided ends with the answer as \( \boxed{E} \), but the ground truth solution indicates that \( A+B = 17 \) should be boxed as \( \boxed{17} \). The answer should directly reflect the numeric result rather than any labeling scheme. Therefore, the solution is incorrect. | 0 | ||
Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$. | Okay, let's see. I need to find the smallest positive integer that's divisible by 17 and whose digits add up to 17. Hmm, so m has two conditions: divisible by 17 and digit sum of 17. Alright, let's break this down step by step.
First, understanding the problem: We need the smallest number (least positive integer) such... | To find the least positive integer \( m \) divisible by 17 whose digits sum to 17, we need to check numbers systematically.
First, we check two-digit numbers with a digit sum of 17 (89 and 98), but neither is divisible by 17. Next, we consider three-digit numbers. The smallest three-digit numbers with a digit sum of ... | 1. **Understanding the Problem:**
We need to find the smallest positive integer $m$ such that:
- $m$ is divisible by $17$.
- The sum of the digits of $m$ equals $17$.
2. **Using Congruences:**
- Since $m$ is divisible by $17$, we have $m \equiv 0 \pmod{17}$.
- The sum of the digits of $m$ is $17$. By th... | 9 | 476 | true | Both solutions outline the process of finding the smallest positive integer divisible by 17 with a digit sum of 17, arriving at the same answer of 476. The solution and ground truth correctly detail the mathematical reasoning and verification necessary to find 476 as the desired number. | 0 | ||
A shopper plans to purchase an item that has a listed price greater than $\textdollar 100$ and can use any one of the three coupons. Coupon A gives $15\%$ off the listed price, Coupon B gives $\textdollar 30$ off the listed price, and Coupon C gives $25\%$ off the amount by which the listed price exceeds
$\textdollar 1... | Okay, let's try to figure out this coupon problem. So, we have an item that costs more than $100, and the shopper can use one of three coupons: A, B, or C. We need to find the smallest and largest prices where Coupon A is at least as good as B or C. Then subtract those two numbers to get the answer.
First, let's make ... | To determine the smallest and largest prices for which Coupon A saves at least as many dollars as Coupon B or C, we analyze the savings for each coupon:
- **Coupon A**: 15% off the listed price. Savings: \(0.15P\)
- **Coupon B**: $30 off the listed price. Savings: $30
- **Coupon C**: 25% off the amount by which the li... | Let the listed price be $P$, where $P > 100$. We can express $P$ as $P = 100 + p$ where $p > 0$.
1. **Calculate the savings from each coupon:**
- **Coupon A:** This coupon gives $15\%$ off the listed price. Therefore, the savings from Coupon A are:
\[
0.15P = 0.15(100 + p) = 15 + 0.15p
\]
- **Coup... | A | \textbf{(A)}50 | true | The provided solution correctly analyzes the savings of each coupon, establishes the necessary inequalities to compare them, and accurately solves for the values of P that satisfy those conditions. The calculations lead to the correct conclusion that the smallest price is 200 and the largest is 250, resulting in a diff... | 0 | ||
In $\triangle ABC$, $\angle ABC=45^\circ$. Point $D$ is on $\overline{BC}$ so that $2\cdot BD=CD$ and $\angle DAB=15^\circ$. Find $\angle ACB.$
$\text{(A) }54^\circ \qquad \text{(B) }60^\circ \qquad \text{(C) }72^\circ \qquad \text{(D) }75^\circ \qquad \text{(E) }90^\circ$ | Okay, let me try to solve this geometry problem. Hmm, so we have triangle ABC where angle ABC is 45 degrees. Point D is on BC such that 2 times BD equals CD, and angle DAB is 15 degrees. We need to find angle ACB. The answer choices are 54°, 60°, 72°, 75°, and 90°.
First, let me draw a rough sketch of triangle ABC. L... | Given triangle \( \triangle ABC \) with \( \angle ABC = 45^\circ \). Point \( D \) is on \( \overline{BC} \) such that \( 2 \cdot BD = CD \) and \( \angle DAB = 15^\circ \). We need to find \( \angle ACB \).
1. **Identify Ratios and Angles**:
- \( BD : CD = 1 : 2 \), so \( BD = x \) and \( CD = 2x \), making \( BC ... | 1. **Given Information and Setup:**
- In $\triangle ABC$, $\angle ABC = 45^\circ$.
- Point $D$ is on $\overline{BC}$ such that $2 \cdot BD = CD$.
- $\angle DAB = 15^\circ$.
- Let $\angle ACB = \theta$.
2. **Using the Angle Bisector Theorem:**
- Since $2 \cdot BD = CD$, $D$ is the midpoint of $BC$.
- ... | 0 | 0 | true | The solution correctly follows the method outlined in the ground truth solution, utilizing geometric properties, the Law of Sines, and trigonometric identities to arrive at the correct angle for \( \angle ACB \). Both solutions arrive at the angle \( \angle ACB = 75^ extcirc \) appropriately. | 0 | ||
Mary's top book shelf holds five books with the following widths, in centimeters: $6$, $\dfrac{1}{2}$, $1$, $2.5$, and $10$. What is the average book width, in centimeters?
$\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 2 \qquad \mathrm{(C)}\ 3 \qquad \mathrm{(D)}\ 4 \qquad \mathrm{(E)}\ 5$ | Okay, so Mary has five books on her top shelf, and their widths are given as 6 cm, 1/2 cm, 1 cm, 2.5 cm, and 10 cm. The question is asking for the average width of these books. The answer choices are A through E, with the numbers 1 to 5. Hmm, let me think about how to approach this.
First, I remember that the average ... | To find the average width of Mary's books, we first convert all measurements to decimals for easier addition:
- \(6\) cm remains \(6.0\)
- \(\dfrac{1}{2}\) cm converts to \(0.5\)
- \(1\) cm remains \(1.0\)
- \(2.5\) cm remains \(2.5\)
- \(10\) cm remains \(10.0\)
Next, we sum these widths:
\[
6.0 + 0.5 + 1.0 + 2.5 + 1... | 1. **List the widths of the books**: The widths of the books are given as $6$, $\dfrac{1}{2}$, $1$, $2.5$, and $10$ centimeters.
2. **Calculate the total sum of the widths**:
\[
6 + \dfrac{1}{2} + 1 + 2.5 + 10 = 6 + 0.5 + 1 + 2.5 + 10
\]
\[
= 6 + 0.5 + 1 + 2.5 + 10 = 20
\]
3. **Count the number of b... | D | D | true | The solution provided correctly lists the widths of the books, calculates the total and average widths accurately, and arrives at the same final conclusion regarding the average width of the books as the ground truth solution. | 0 | ||
The equation $2^{2x}-8\cdot 2^x+12=0$ is satisfied by:
$\text{(A) } log(3)\quad \text{(B) } \tfrac{1}{2}log(6)\quad \text{(C) } 1+log(\tfrac{3}{2})\quad \text{(D) } 1+\frac{log(3)}{log(2)}\quad \text{(E) none of these}$ | Okay, let's see. I need to solve the equation \(2^{2x} - 8 \cdot 2^x + 12 = 0\) and check which of the options A to E is correct. Hmm, the equation looks a bit complicated with exponents. Maybe I can simplify it by substituting a variable for \(2^x\). That often helps with exponential equations. Let me try that.
Let’s... | To solve the equation \(2^{2x} - 8 \cdot 2^x + 12 = 0\), we start by substituting \(y = 2^x\). This transforms the equation into a quadratic equation:
\[
y^2 - 8y + 12 = 0
\]
Factoring the quadratic equation, we get:
\[
(y - 6)(y - 2) = 0
\]
This gives the solutions \(y = 6\) and \(y = 2\). Substituting back \(y = ... | 1. **Substitute and Simplify**: Let $2^x = a$. Then, $2^{2x} = (2^x)^2 = a^2$. The given equation $2^{2x} - 8 \cdot 2^x + 12 = 0$ can be rewritten using this substitution:
\[
a^2 - 8a + 12 = 0
\]
2. **Factorize the Quadratic**: The quadratic equation $a^2 - 8a + 12 = 0$ can be factored as:
\[
(a - 6)(a ... | 2 | D | true | The provided solution correctly follows the steps to solve the equation by substituting and simplifying. It accurately identifies the quadratic equation and factors it, leading to correct solutions for both scenarios of $2^x$. Both solutions, $x = 1$ and $x = 1 + rac{ ext{log}(3)}{ ext{log}(2)}$, are valid. The first ... | 0 | ||
The sum of the greatest integer less than or equal to $x$ and the least integer greater than or equal to $x$ is $5$. The solution set for $x$ is
$\textbf{(A)}\ \Big\{\frac{5}{2}\Big\}\qquad \textbf{(B)}\ \big\{x\ |\ 2 \le x \le 3\big\}\qquad \textbf{(C)}\ \big\{x\ |\ 2\le x < 3\big\}\qquad\\ \textbf{(D)}\ \Big\{x\ |\ ... | Alright, let me try to figure out this problem step by step. So, the question is about finding all real numbers x such that the sum of the greatest integer less than or equal to x and the least integer greater than or equal to x is equal to 5. The answer choices are given from A to E, and I need to determine which one ... | To solve the problem, we need to find all real numbers \( x \) such that the sum of the greatest integer less than or equal to \( x \) (floor function \( \lfloor x \rfloor \)) and the least integer greater than or equal to \( x \) (ceiling function \( \lceil x \rceil \)) is 5.
1. **Case 1: \( x \) is an integer**
... | 1. **Understanding the Problem:**
The problem asks us to find the set of values for $x$ such that the sum of the greatest integer less than or equal to $x$ (denoted $\lfloor x \rfloor$) and the least integer greater than or equal to $x$ (denoted $\lceil x \rceil$) equals 5.
2. **Analyzing the Floor and Ceiling Func... | E | \textbf{(E)}\\Big{x\|\2<x<3\Big} | true | The solution provided is correct as it accurately identifies and analyzes both cases for the variable x (whether x is an integer or not). It correctly concludes that there are no integers that satisfy the equation and precisely determines the interval for non-integer solutions. Furthermore, it verifies that the calcula... | 0 | ||
A powderman set a fuse for a blast to take place in $30$ seconds. He ran away at a rate of $8$ yards per second. Sound travels at the rate of $1080$ feet per second. When the powderman heard the blast, he had run approximately:
$\textbf{(A)}\ \text{200 yd.}\qquad\textbf{(B)}\ \text{352 yd.}\qquad\textbf{(C)}\ \text{300... | Okay, let's try to figure out this problem step by step. So, the powderman sets a fuse for 30 seconds and starts running away at 8 yards per second. Then, when he hears the blast, we need to find out how far he has run. The options are given, but I need to work through it myself first.
First, let me parse the informat... | The powderman sets a fuse for 30 seconds and runs away at 8 yards per second. Sound travels at 1080 feet per second, which converts to 360 yards per second.
1. **Distance run by the powderman in 30 seconds**:
\[
8 \, \text{yards/second} \times 30 \, \text{seconds} = 240 \, \text{yards}
\]
2. **Let \( t_2 \... | 1. **Define the problem in terms of equations:**
- Let $p(t)$ be the distance in feet that the powderman is from the starting point at time $t$ seconds after the fuse is lit. Since he runs at $8$ yards per second, and there are $3$ feet in a yard, he runs $8 \times 3 = 24$ feet per second. Therefore, $p(t) = 24t$.
... | D | \textbf{(D)}\245yd | true | The solution provided correctly follows the logic of setting up the problem in terms of distance run by the powderman and the distance traveled by sound, and correctly calculates both the time taken for the sound to reach him and the total distance he runs. The computed distance of approximately 245 yards aligns with t... | 0 | ||
A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly $10$ ways to select a two-person team for the welcoming committee. It is possible for students to serve on both committees. In how many different ways can a three-person plannin... | Okay, let's see. The problem is about selecting committees from a student council. Specifically, we need to find out how many ways there are to select a three-person planning committee, given that there are exactly 10 ways to select a two-person welcoming committee. Both committees can have overlapping members, so the ... | Given that there are exactly 10 ways to select a two-person welcoming committee, we need to determine the number of ways to select a three-person planning committee from the same student council members.
First, we use the combination formula to find the number of members \( n \) in the student council. The number of w... | 1. **Determine the number of students on the council**:
Let the number of students on the council be $x$. The number of ways to select a two-person welcoming committee is given by the combination formula $\binom{x}{2}$, which represents the number of ways to choose 2 students from $x$ without regard to order. This ... | 2 | 2 | true | The provided solution correctly identifies the number of student council members by solving the quadratic equation formed from the combination formula for selecting a two-person committee. It appropriately calculates $inom{5}{3}$ for selecting the three-person planning committee, ultimately arriving at the correct ans... | 0 | ||
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2$ : $1$ ?
$\textbf{(A)}\ 2 \qquad\textbf{(B)} \ 4 \qquad\textbf{(C)} \ 5 \qquad\textbf{(D)} \ 6 \qquad\textbf{(E)} \ 8$ | Okay, so I need to solve this age problem. Let me read it again carefully. The question says:
"Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be 2 : 1? The options are A through E: 2, 4, 5, 6, 8... | Let \( P \) be Pete's current age and \( C \) be Claire's current age.
From the problem, we know:
1. Two years ago, Pete was three times as old as Claire:
\[
P - 2 = 3(C - 2)
\]
2. Four years ago, Pete was four times as old as Claire:
\[
P - 4 = 4(C - 4)
\]
Expanding and simplifying these equations... | 1. **Define Variables:**
Let $p$ be Pete's current age and $c$ be Claire's current age.
2. **Translate the Problem into Equations:**
- From the statement "Two years ago Pete was three times as old as his cousin Claire," we have:
\[
p - 2 = 3(c - 2)
\]
- From the statement "Two years before that... | 2 | 14 | true | Both the solution and the ground truth solution arrive at the same conclusion about the ages of Pete and Claire, calculating that in 4 years the ratio of their ages will be 2:1. The steps involved, including formulating the equations, solving for ages, and determining the future age ratio, are consistent and correct in... | 0 | ||
Let $ABC$ be a triangle with $\angle A = 90^{\circ}$. Points $D$
and $E$ lie on sides $AC$ and $AB$, respectively, such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$. Segments $BD$ and
$CE$ meet at $I$. Determine whether or not it is possible for
segments $AB, AC, BI, ID, CI, IE$ to all have integer len... | Alright, so I came across this geometry problem, and it's pretty interesting. Let me try to break it down step by step. The problem is about a right-angled triangle ABC with the right angle at A. Points D and E are on sides AC and AB respectively, such that BD and CE are angle bisectors. Specifically, BD bisects angle ... | After a detailed analysis of the problem using coordinate geometry, angle bisector theorems, and properties of Pythagorean triples, it becomes evident that the required integer lengths for segments \(AB\), \(AC\), \(BI\), \(ID\), \(CI\), and \(IE\) cannot all be satisfied simultaneously. Key steps include:
1. **Incent... | 1. **Given Information and Assumptions**:
- Triangle $ABC$ is a right triangle with $\angle A = 90^\circ$.
- Points $D$ and $E$ lie on sides $AC$ and $AB$ respectively, such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$.
- Segments $BD$ and $CE$ meet at point $I$.
- We need to determine if it... | 0.2 | false | The provided solution and the ground truth solution both arrive at the conclusion that it is not possible for all segments to have integer lengths based on rational and Pythagorean properties. However, the reasoning provided in the ground truth is more detailed and accurately follows the implications of the angle bisec... | 0 | |||
If $\theta$ is a constant such that $0 < \theta < \pi$ and $x + \dfrac{1}{x} = 2\cos{\theta}$, then for each positive integer $n$, $x^n + \dfrac{1}{x^n}$ equals
$\textbf{(A)}\ 2\cos\theta\qquad \textbf{(B)}\ 2^n\cos\theta\qquad \textbf{(C)}\ 2\cos^n\theta\qquad \textbf{(D)}\ 2\cos n\theta\qquad \textbf{(E)}\ 2^n\cos^n\... | Okay, let's see. The problem says that if x + 1/x = 2 cos θ, where θ is between 0 and π, then for any positive integer n, what is x^n + 1/x^n equal to? The options are given as D, which is 2 cos nθ, and others. Hmm, I need to figure out why it's D. Let me think step by step.
First, I recall that expressions like x + 1... | Given the equation \( x + \dfrac{1}{x} = 2\cos{\theta} \) where \( 0 < \theta < \pi \), we need to determine the value of \( x^n + \dfrac{1}{x^n} \) for any positive integer \( n \).
First, we recognize that \( x \) can be represented as a complex exponential \( e^{i\theta} \) or \( e^{-i\theta} \) because \( e^{i\the... | 1. **Given Equation and Transformation**:
Given that \( x + \frac{1}{x} = 2\cos{\theta} \), we multiply both sides by \( x \) to obtain:
\[
x^2 + 1 = 2x\cos{\theta}
\]
Rearranging terms, we get the quadratic equation:
\[
x^2 - 2x\cos{\theta} + 1 = 0
\]
2. **Solving the Quadratic Equation**:
... | D | \textbf{D} | true | The provided solution correctly establishes the relationships and manipulations necessary to derive that \( x^n + \frac{1}{x^n} = 2\cos(n\theta) \). It uses both complex exponentiation and mathematical induction to validate the result. The inductive proof is well-structured, confirming that \( y_n = 2\cos(n\theta) \) h... | 0 | ||
A square piece of paper, 4 inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle?
[asy] draw((0,8)--... | Okay, let me try to figure out this problem step by step. First, I need to visualize what's happening here. We have a square piece of paper that's 4 inches on each side. Then, it's folded in half vertically. Hmm, folding vertically probably means along the vertical midline, so the square which was originally 4x4 inches... | A square piece of paper, 4 inches on each side, is folded in half vertically. This results in a rectangle of dimensions 2 inches by 4 inches. Both layers of the folded paper are then cut in half parallel to the fold. Since the fold is vertical, the cuts are also vertical.
When the paper is folded, the vertical fold l... | 1. **Understanding the problem**: We start with a square piece of paper with each side measuring 4 inches. The paper is folded in half vertically, which means the fold divides the square into two equal rectangles each with dimensions $4 \times 2$ inches.
2. **Cutting the paper**: After folding, both layers of the pape... | 2 | 2 | false | The solution describes the cutting of the paper incorrectly. After folding the square (4x4) in half vertically, the resulting folded dimensions are 2x4. When cutting parallel to the fold, it should result in 4 layers of 2x1 rectangles, not 3 rectangles. Therefore, the claim of obtaining two 1x4 and one 2x4 is incorrect... | 0 | ||
Positive integers $a$ and $b$ are such that the graphs of $y=ax+5$ and $y=3x+b$ intersect the $x$-axis at the same point. What is the sum of all possible $x$-coordinates of these points of intersection?
$\textbf{(A)}\ {-20}\qquad\textbf{(B)}\ {-18}\qquad\textbf{(C)}\ {-15}\qquad\textbf{(D)}\ {-12}\qquad\textbf{(E)}\ {-... | Okay, so I need to solve this problem where positive integers a and b are such that the graphs of y = ax + 5 and y = 3x + b intersect the x-axis at the same point. Then, I have to find the sum of all possible x-coordinates of these intersection points. The answer choices are given from A to E. Let me try to break this ... | To solve the problem, we need to find the sum of all possible \( x \)-coordinates where the graphs of \( y = ax + 5 \) and \( y = 3x + b \) intersect the \( x \)-axis, given that \( a \) and \( b \) are positive integers and the graphs intersect the \( x \)-axis at the same point.
1. **Determine the \( x \)-intercepts... | 1. **Identify the points of intersection with the x-axis**:
The equations of the lines are given as $y = ax + 5$ and $y = 3x + b$. The points where these lines intersect the x-axis occur when $y = 0$.
2. **Set up equations for x-intercepts**:
For the line $y = ax + 5$, setting $y = 0$ gives:
\[
0 = ax + 5... | E | 2 | true | The solution systematically identifies the necessary steps to solve for the sum of the x-coordinates of the intersection points of the given lines. It correctly sets up the equations for the x-intercepts, equates them, and finds all pairs of positive integers satisfying the condition. The calculations for the x-interce... | 0 | ||
(Titu Andreescu)
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd. | Okay, so I need to prove that for every positive integer n, there's an n-digit number divisible by 5^n, and all its digits are odd. Hmm, let's start by understanding the problem better.
First, let's break it down. For each n, we need to construct a number with exactly n digits (so the first digit can't be zero), each... | To prove that for every positive integer \( n \) there exists an \( n \)-digit number divisible by \( 5^n \) with all digits odd, we use mathematical induction.
**Base Case (\( n = 1 \)):**
The number \( 5 \) is a 1-digit number divisible by \( 5^1 = 5 \), and its digit is odd. Thus, the base case holds.
**Inductiv... |
We will prove the statement using induction on $n$.
#### Base Case:
For $n=1$, the number $5$ is a one-digit number divisible by $5^1 = 5$, and all its digits (in this case, just one digit) are odd. Thus, the base case holds.
#### Inductive Step:
Assume that for some positive integer $k$, there exists a $k$-digit nu... | 2 | true | The provided solution correctly uses mathematical induction to prove the statement, stating the base case clearly and making valid deductions in the inductive step. Both solutions arrive at the same conclusion supporting the existence of an n-digit number divisible by 5^n with all odd digits. Therefore, the solution is... | 0 | |||
Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\tfrac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\tfrac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $... | Okay, let's try to tackle this problem step by step. First, I need to understand what's being asked here. We have four circles with centers at A, B, C, and D. None of the circles are congruent, which means all their radii are different. Points P and Q lie on all four circles, so P and Q are common intersection points o... | Given four circles with centers at \(A\), \(B\), \(C\), and \(D\), where points \(P\) and \(Q\) lie on all four circles, we need to find the sum of the distances from the midpoint \(R\) of \(\overline{PQ}\) to each center. The radii of circles \(A\) and \(C\) are \(\frac{5}{8}\) times the radii of circles \(B\) and \(D... | 1. **Understanding the Problem**: We are given four circles with centers at $A$, $B$, $C$, and $D$. Points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and similarly for circles $C$ and $D$. The distances $AB$ and $CD$ are both 39, and the length of segm... | 8 | 20 | true | The provided solution correctly follows the problem requirements and arrives at the conclusion that the sum of distances from the midpoint to each center equals 192. Analyzing the steps confirms that both the logic and calculations align with the ground truth solution. | 0 | ||
Medians $AD$ and $CE$ of $\triangle ABC$ intersect in $M$. The midpoint of $AE$ is $N$.
Let the area of $\triangle MNE$ be $k$ times the area of $\triangle ABC$. Then $k$ equals:
$\textbf{(A)}\ \frac{1}{6}\qquad \textbf{(B)}\ \frac{1}{8}\qquad \textbf{(C)}\ \frac{1}{9}\qquad \textbf{(D)}\ \frac{1}{12}\qquad \textbf{(E... | Okay, let's see. The problem is about medians AD and CE intersecting at M in triangle ABC. The midpoint of AE is N. We need to find the ratio of the area of triangle MNE to the area of triangle ABC, which is k. The answer choices are given from 1/6 to 1/16. Hmm.
First, I need to recall some properties about medians in... | To solve the problem, we start by noting that medians \(AD\) and \(CE\) of \(\triangle ABC\) intersect at the centroid \(M\), which divides each median into a 2:1 ratio. The midpoint of \(AE\) is \(N\). We need to determine the ratio of the area of \(\triangle MNE\) to the area of \(\triangle ABC\).
1. **Coordinate Sy... | 1. **Identify the relationships between the triangles**:
- Since $AD$ and $CE$ are medians, point $M$ is the centroid of $\triangle ABC$. The centroid divides each median into two segments, one of which is twice the length of the other, specifically, the segment connecting the centroid to the midpoint of a side is ... | D | 0.2112626 | false | The initial solution incorrectly calculated the ratio of the area of triangle MNE to triangle ABC as \frac{1}{12}. However, based on the corrected analysis, the ratio should be \frac{1}{18} based on the relationships between the areas produced by the centroid and midpoints. Therefore, the initial answer is not correct. | 0 | ||
A figure is an equiangular parallelogram if and only if it is a
$\textbf{(A)}\ \text{rectangle}\qquad \textbf{(B)}\ \text{regular polygon}\qquad \textbf{(C)}\ \text{rhombus}\qquad \textbf{(D)}\ \text{square}\qquad \textbf{(E)}\ \text{trapezoid}$ | Okay, let's tackle this problem step by step. The question is asking: "A figure is an equiangular parallelogram if and only if it is a..." with options given. The answer choices are rectangle, regular polygon, rhombus, square, trapezoid. Hmm. So first, let's parse the terms here.
An equiangular parallelogram. Let's re... | To determine the correct answer, we need to analyze the definitions and properties of the given options in relation to an equiangular parallelogram.
1. **Definitions and Properties**:
- A **parallelogram** is a quadrilateral with both pairs of opposite sides parallel.
- **Equiangular** means all angles are equal... | 1. **Definition and Analysis**: An equiangular parallelogram is defined as a parallelogram where all interior angles are equal. By the properties of a parallelogram, opposite angles are equal, and adjacent angles are supplementary. If all angles are equal in a parallelogram, each angle must be $90^\circ$.
2. **Examine... | 4 | 1 | true | The solution accurately follows the definitions and properties relevant to an equiangular parallelogram, correctly concluding that a rectangle is the appropriate answer. Both the provided solution and the ground truth solution arrive at the same conclusion, confirming the correctness of the solution. | 0 | ||
Find the minimum value of $\sqrt{x^2+y^2}$ if $5x+12y=60$.
$\textbf{(A)}\ \frac{60}{13}\qquad \textbf{(B)}\ \frac{13}{5}\qquad \textbf{(C)}\ \frac{13}{12}\qquad \textbf{(D)}\ 1\qquad \textbf{(E)}\ 0$ | Okay, so I need to find the minimum value of √(x² + y²) given that 5x + 12y = 60. Hmm, let me think. The problem is asking for the minimum distance from the origin to the line 5x + 12y = 60, right? Because √(x² + y²) is the distance formula from the origin to the point (x, y). So, if I can find the shortest distance fr... | To find the minimum value of \(\sqrt{x^2 + y^2}\) given that \(5x + 12y = 60\), we recognize that this is the distance from the origin to the line \(5x + 12y = 60\). The formula for the distance from a point \((x_0, y_0)\) to the line \(ax + by + c = 0\) is \(\frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}}\).
Rewriting the... |
#### Step 1: Understand the problem
We need to find the minimum value of $\sqrt{x^2 + y^2}$ given the constraint $5x + 12y = 60$. This expression $\sqrt{x^2 + y^2}$ represents the distance from the origin $(0,0)$ to the point $(x,y)$.
#### Step 2: Relate the problem to the distance from a point to a line
The equation... | A | \textbf{(A)}\\frac{60}{13} | true | The provided solution accurately follows a systematic approach to determine the minimum distance from the origin to the line defined by the constraint, using both geometric reasoning and calculus. The computations are correct, and the minimum distance derived is confirmed by multiple methods, matching the ground truth.... | 0 | ||
On average, for every 4 sports cars sold at the local dealership, 7 sedans are sold. The dealership predicts that it will sell 28 sports cars next month. How many sedans does it expect to sell?
$\textbf{(A)}\ 7\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 35\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 112$ | Okay, so I need to solve this problem where the dealership sells sports cars and sedans in a ratio, and they predict selling a certain number of sports cars next month. The question is asking how many sedans they expect to sell. Let me break this down step by step.
First, the problem says that for every 4 sports cars ... | The dealership's sales ratio is 4 sports cars to 7 sedans. For 28 sports cars, the scaling factor is \( \frac{28}{4} = 7 \). Multiplying the sedan part of the ratio by this factor:
\[
7 \times 7 = 49
\]
Verifying via proportion:
\[
\frac{4}{7} = \frac{28}{x} \implies 4x = 196 \implies x = \frac{196}{4} = 49
\]
Both... | 1. **Identify the given ratio**: The problem states that for every 4 sports cars sold, 7 sedans are sold. This sets up a ratio of sports cars to sedans as $4:7$.
2. **Set up the proportion**: Given that the dealership predicts selling 28 sports cars next month, we need to find the number of sedans expected to be sold,... | D | \textbf{(D)}49 | true | The solution correctly identifies the ratio of sports cars to sedans as 4:7 and successfully applies both a scaling method and a proportion to arrive at the conclusion that 49 sedans are expected to be sold. The final boxed answer \\boxed{D} matches the ground truth solution. | 0 | ||
Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\frac34$ of the probability of rolling a sum of $10,$ and the probability of rolling a sum of $12$ is $\frac{... | Okay, let me try to figure out this problem step by step. So, we have two fair dice, each with at least 6 faces. When rolled, the probability of getting a sum of 7 is three-fourths the probability of getting a sum of 10. Also, the probability of rolling a sum of 12 is 1/12. We need to find the least possible number of ... | Given two fair dice with at least 6 faces each, we need to find the least possible number of faces on the two dice combined, considering the following conditions:
1. The probability of rolling a sum of 7 is \(\frac{3}{4}\) of the probability of rolling a sum of 10.
2. The probability of rolling a sum of 12 is \(\frac{1... | 1. **Define Variables:**
Let the number of faces on the two dice be $a$ and $b$, respectively, with $a \geq b$. Assume each die has distinct integers from $1$ to the number of faces on that die.
2. **Analyze the Probability of Rolling a Sum of $7$:**
Since each die has at least $6$ faces, there are always $6$ wa... | 8 | 6 | true | Both the provided solution and the ground truth arrive at the same conclusion that the least possible number of faces on the two dice combined is 17. They correctly analyze the conditions given, such as the relationship between the probabilities and the counts of different rolling sums. The calculations regarding combi... | 0 | ||
Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way). | Okay, so I need to figure out the number of ways to make all roads between five towns one-way such that you can still get from any town to any other town. Hmm, let's start by understanding the problem better.
First, there are five towns, and each pair is connected by a single road. Since each pair has a road, this is ... | To determine the number of ways to make all roads between five towns one-way such that it is possible to travel from any town to any other town, we need to count the number of strongly connected tournaments on five vertices. A tournament is a complete oriented graph where every pair of vertices is connected by a single... | We are given a complete graph with 5 vertices (towns) and need to determine the number of ways to assign a direction to each edge (road) such that the resulting directed graph is strongly connected. This means that there is a directed path from any vertex to any other vertex.
#### Step 1: Understanding the Problem
A c... | 11 | 3 | true | The provided solution correctly follows the necessary steps to calculate the number of strongly connected directed graphs (tournaments) with 5 vertices, and it matches the ground truth solution's reasoning and conclusion exactly regarding the configurations and the use of inclusion-exclusion principle. | 0 | ||
Two long [cylindrical](https://artofproblemsolving.com/wiki/index.php/Cylinder) tubes of the same length but different [diameters](https://artofproblemsolving.com/wiki/index.php/Diameter) lie [parallel](https://artofproblemsolving.com/wiki/index.php/Parallel) to each other on a [flat surface](https://artofproblemsolvin... | Okay, let's try to tackle this problem. Hmm, so we have two cylindrical tubes lying parallel on a flat surface. The larger one has a radius of 72, and the smaller one has a radius of 24. The larger tube rolls toward the smaller one, goes over it, and continues rolling until it comes back to its starting position after ... | The problem involves a larger cylinder (radius 72) rolling over a smaller cylinder (radius 24) without slipping and continuing to roll until it completes one full revolution. We need to find the distance \( x \) from the starting point to the ending point expressed in the form \( a\pi + b\sqrt{c} \).
1. **Total Rotati... | 1. **Understanding the Problem:**
Two cylindrical tubes, one with radius 72 and the other with radius 24, are lying parallel on a flat surface. The larger tube rolls over the smaller tube and completes one full revolution, returning to the same point on its circumference, without slipping.
2. **Initial Setup:**
... | 179 | 195 | false | The provided solution incorrectly calculates the final values of a, b, and c, resulting in the wrong sum (179 instead of the correct 195). The solution fails to accurately account for the movements and distances involved in the interaction between the larger and smaller cylinders. | 0 | ||
The number $2.5252525\ldots$ can be written as a fraction.
When reduced to lowest terms the sum of the numerator and denominator of this fraction is:
$\textbf{(A) }7\qquad \textbf{(B) }29\qquad \textbf{(C) }141\qquad \textbf{(D) }349\qquad \textbf{(E) }\text{none of these}$ | Okay, let's see. I need to convert the repeating decimal 2.5252525... into a fraction and then find the sum of the numerator and denominator when it's in its simplest form. The options are given, but I should figure it out step by step.
First, I remember that repeating decimals can often be converted into fractions by... | Let \( x = 2.5252525\ldots \). The repeating part is two digits long, so we multiply both sides by 100:
\[
100x = 252.5252525\ldots
\]
Subtracting the original equation from this:
\[
100x - x = 252.5252525\ldots - 2.5252525\ldots
\]
This simplifies to:
\[
99x = 250
\]
Solving for \( x \):
\[
x = \frac{250}{99}
\... |
#### Step 1: Represent the repeating decimal as a fraction
Let $x = 2.5252525\ldots$. We can express $x$ in terms of itself shifted by two decimal places:
\[ 100x = 252.5252525\ldots \]
#### Step 2: Set up the equation
Subtract the original $x$ from $100x$:
\[ 100x - x = 252.5252525\ldots - 2.5252525\ldots \]
\[ 99x ... | 2 | 2 | true | The solution correctly represents the repeating decimal as a fraction, solves for it, and calculates the sum of the numerator and denominator. The answer matches the ground truth. | 0 | ||
For all non-zero numbers $x$ and $y$ such that $x = 1/y$, $\left(x-\frac{1}{x}\right)\left(y+\frac{1}{y}\right)$ equals
$\textbf{(A) }2x^2\qquad \textbf{(B) }2y^2\qquad \textbf{(C) }x^2+y^2\qquad \textbf{(D) }x^2-y^2\qquad \textbf{(E) }y^2-x^2$ | Okay, so I have this problem here. Let me read it again: For all non-zero numbers x and y such that x equals 1 over y, the expression (x minus 1/x) times (y plus 1/y) equals one of the given options. The options are A through E, with various expressions involving x squared and y squared. Alright, so my goal is to simpl... | Given the problem, we need to simplify the expression \(\left(x - \frac{1}{x}\right)\left(y + \frac{1}{y}\right)\) for non-zero numbers \(x\) and \(y\) such that \(x = \frac{1}{y}\).
First, we substitute \(y = \frac{1}{x}\) into the expression. This gives us:
\[
\left(x - \frac{1}{x}\right)\left(\frac{1}{x} + x\right)... | 1. **Substitute $y$ in terms of $x$:** Given $x = \frac{1}{y}$, we can rewrite $y$ as $y = \frac{1}{x}$.
2. **Substitute and simplify the expression:** We start by substituting $y = \frac{1}{x}$ into the expression $\left(x-\frac{1}{x}\right)\left(y+\frac{1}{y}\right)$:
\[
\left(x - \frac{1}{x}\right)\left(\fra... | 0 | \textbf{(D)}x^2-y^2 | true | The provided solution correctly follows the steps required to simplify the expression \( \left(x - \frac{1}{x}\right)\left(y + \frac{1}{y}\right) \) given \( x = \frac{1}{y} \). It accurately substitutes \( y = \frac{1}{x} \), applies the difference of squares formula, and reaches the correct simplification of the expr... | 0 | ||
The values of $k$ for which the equation $2x^2-kx+x+8=0$ will have real and equal roots are:
$\textbf{(A)}\ 9\text{ and }-7\qquad\textbf{(B)}\ \text{only }-7\qquad\textbf{(C)}\ \text{9 and 7}\\ \textbf{(D)}\ -9\text{ and }-7\qquad\textbf{(E)}\ \text{only 9}$ | Okay, let me try to solve this problem. The question is about finding the values of k for which the quadratic equation 2x² - kx + x + 8 = 0 has real and equal roots. The options are given from A to E, and I need to choose the correct one.
First, I remember that for a quadratic equation ax² + bx + c = 0 to have real a... | To determine the values of \( k \) for which the quadratic equation \( 2x^2 - kx + x + 8 = 0 \) has real and equal roots, we need to ensure the discriminant is zero.
First, we combine the like terms in the equation:
\[ 2x^2 - kx + x + 8 = 0 \]
\[ 2x^2 + (-k + 1)x + 8 = 0 \]
This is in the standard form \( ax^2 + bx ... | To find the values of $k$ for which the quadratic equation $2x^2 - kx + x + 8 = 0$ has real and equal roots, we first simplify the equation:
\[ 2x^2 - kx + x + 8 = 0 \]
\[ 2x^2 + (1-k)x + 8 = 0 \]
For a quadratic equation $ax^2 + bx + c = 0$, the condition for the roots to be real and equal is given by the discriminan... | 0 | \textbf{(A)}\9-7 | true | The provided solution correctly simplifies the quadratic equation, calculates the discriminant, sets it to zero, and finds the correct values for k leading to real and equal roots, which matches the ground truth solution. | 0 | ||
How many perfect cubes lie between $2^8+1$ and $2^{18}+1$, inclusive?
$\textbf{(A) }4\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }57\qquad \textbf{(E) }58$ | Alright, let me try to figure out how many perfect cubes lie between 2^8 + 1 and 2^18 + 1, inclusive. Hmm, okay. First, I need to understand the problem properly. So, we have two numbers: the lower bound is 2^8 + 1, and the upper bound is 2^18 + 1. We need to count how many perfect cubes are within this range, includin... | To determine how many perfect cubes lie between \(2^8 + 1\) and \(2^{18} + 1\), inclusive, we start by calculating the values of these bounds.
- \(2^8 = 256\), so \(2^8 + 1 = 257\).
- \(2^{18} = 262,144\), so \(2^{18} + 1 = 262,145\).
Next, we need to find the smallest and largest integers \(n\) such that \(n^3\) li... | 1. **Calculate the lower bound**: We start by calculating $2^8 + 1$:
\[
2^8 = 256 \quad \text{so} \quad 2^8 + 1 = 257.
\]
We need to find the smallest perfect cube greater than 257. We know $6^3 = 216$ and $7^3 = 343$. Since $343 > 257$, the smallest cube greater than 257 is $7^3 = 343$.
2. **Calculate the... | E | \textbf{(E)}58 | true | The solution correctly calculates the range of integers for which the cubes fall between the given bounds and arrives at the correct count of perfect cubes, which is 58. The logical steps are accurate, and the final boxed answer matches the expected ground truth solution. | 0 | ||
A line that passes through the origin intersects both the line $x = 1$ and the line $y=1+ \frac{\sqrt{3}}{3} x$. The three lines create an equilateral triangle. What is the perimeter of the triangle?
$\textbf{(A)}\ 2\sqrt{6} \qquad\textbf{(B)} \ 2+2\sqrt{3} \qquad\textbf{(C)} \ 6 \qquad\textbf{(D)} \ 3 + 2\sqrt{3} \qqu... | Okay, let's try to figure out this problem. So, we have a line that passes through the origin, and it intersects two other lines: x = 1 and y = 1 + (√3/3)x. These three lines form an equilateral triangle. We need to find the perimeter of this triangle. The answer choices are given, so maybe that can help check our work... | Given a line passing through the origin that intersects both the vertical line \( x = 1 \) and the line \( y = 1 + \frac{\sqrt{3}}{3}x \), we need to determine the perimeter of the equilateral triangle formed by these three lines.
1. **Equations of the Lines**:
- The line passing through the origin is \( y = mx \).... | 1. **Identify the slopes of the lines**:
- The line $y = 1 + \frac{\sqrt{3}}{3}x$ has a slope of $\frac{\sqrt{3}}{3}$.
- Since the triangle formed is equilateral, and one side is vertical ($x=1$), the line passing through the origin that forms another side of the triangle must have a slope that is the negative r... | D | 5 | true | The provided solution correctly identifies the necessary geometry and calculations to determine the perimeter of the equilateral triangle formed by the given lines. It accurately finds the vertices by determining the intersections of the relevant lines and correctly calculates the side lengths. Subsequentially, it comp... | 0 | ||
Alicia earns 20 dollars per hour, of which $1.45\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?
$\mathrm{(A) \ } 0.0029 \qquad \mathrm{(B) \ } 0.029 \qquad \mathrm{(C) \ } 0.29 \qquad \mathrm{(D) \ } 2.9 \qquad \mathrm{(E) \ } 29$ | Okay, let's see. The problem is about Alicia's wages and calculating how many cents per hour go to local taxes. Hmm. Let me start by breaking down the information given.
First, Alicia earns $20 per hour. The tax rate is 1.45%, which is deducted from her wages. The question is asking for how many cents per hour are use... | Alicia earns $20 per hour, and 1.45% of her wages are deducted for local taxes. To find how many cents per hour are used to pay local taxes, we start by calculating the tax amount in dollars:
1. Convert 1.45% to a decimal: \(1.45\% = 0.0145\).
2. Multiply Alicia's hourly wage by the tax rate:
\[
20 \times 0.014... | 1. **Convert Alicia's hourly wage to cents**: Alicia earns $20$ dollars per hour. Since there are $100$ cents in a dollar, her hourly wage in cents is:
\[
20 \text{ dollars} \times 100 \text{ cents/dollar} = 2000 \text{ cents}
\]
2. **Calculate the tax deduction in cents**: The local tax rate is $1.45\%$. To ... | E | \mathrm{(E)}29 | true | Both the provided solution and the ground truth solution correctly calculate the local tax deduction from Alicia's wages. They both conclude that the tax amount is 29 cents per hour. The way they approached the calculation is slightly different but leads to the same correct result. | 0 | ||
How many positive factors of 36 are also multiples of 4?
$\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6$ | Okay, let's try to figure out how many positive factors of 36 are also multiples of 4. Hmm, so the question is asking for numbers that both divide 36 evenly and are divisible by 4. Alright, let's start by understanding what factors of 36 are.
First, maybe I should list all the positive factors of 36. To do that, I ca... | To determine how many positive factors of 36 are also multiples of 4, we start by noting the prime factorization of 36: \(36 = 2^2 \times 3^2\).
A factor of 36 that is a multiple of 4 must include at least \(2^2\) in its prime factorization. Since 36 has \(2^2\) and \(3^2\), the exponent of 2 in such a factor must be... | To solve this problem, we first need to find the prime factorization of 36. Then, we will determine which factors of 36 are also multiples of 4.
1. **Prime Factorization of 36**:
\[
36 = 2^2 \cdot 3^2
\]
This factorization shows that 36 is composed of the square of 2 and the square of 3.
2. **Factors of 3... | B | 2 | false | The provided solution incorrectly states the final answer as \( \boxed{B} \), which seems to imply a format or choice that does not conform to the expected answer format of simply stating the number (3 in this case). The reasoning and calculations within the solution are accurate, confirming that there are indeed 3 pos... | 0 | ||
The numbers $-2, 4, 6, 9$ and $12$ are rearranged according to these rules:
1. The largest isn't first, but it is in one of the first three places.
2. The smallest isn't last, but it is in one of the last three places.
3. The median isn't first or last.
What is the average of the first and l... | Alright, let's tackle this problem step by step. So, we have the numbers -2, 4, 6, 9, and 12. We need to rearrange them according to three rules and then find the average of the first and last numbers. Let me try to break down each rule and see how they constrain the possible arrangements.
First, let's list the number... | The numbers \(-2, 4, 6, 9,\) and \(12\) need to be rearranged according to the given rules. Let's analyze each rule systematically:
1. **Largest number (12)** isn't first but must be in one of the first three positions. Thus, 12 can be in position 2 or 3.
2. **Smallest number (-2)** isn't last but must be in one of th... | 1. **Identify the largest and smallest numbers**: The largest number in the set $\{-2, 4, 6, 9, 12\}$ is $12$, and the smallest is $-2$.
2. **Apply rule 1**: The largest number, $12$, cannot be first but must be in one of the first three places. Thus, $12$ can be in the second or third position.
3. **Apply rule 2**: ... | 7 | 7 | true | The solution provided correctly analyzes the constraints and derives valid configurations based on the rules given. Both the solution and the ground truth arrive at the same final result of the average of the first and last numbers being 6.5, confirming the accuracy of the solution. | 0 | ||
The circumference of the circle with center $O$ is divided into $12$ equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$?
$\textbf{(A) }75\qquad\textbf{(B) }80\qquad\textbf{(C) }90\qquad\textbf{(D) }120\qquad\textbf{(E) }150$ | Okay, let's try to figure out this problem. So, we have a circle divided into 12 equal arcs, labeled from A to L. The question is asking for the number of degrees in the sum of angles x and y. The answer choices are 75, 80, 90, 120, and 150. Hmm, let's break this down step by step.
First, since the circumference is di... | The circumference of the circle with center \( O \) is divided into 12 equal arcs, each corresponding to a central angle of \( \frac{360^\circ}{12} = 30^\circ \). The problem requires finding the sum of the angles \( x \) and \( y \).
1. **Central and Inscribed Angles**: Each arc measures 30 degrees. Inscribed angles ... | 1. **Understanding the Circle and Arcs**: The circle is divided into 12 equal arcs, and each arc corresponds to a central angle at the center $O$. Since the circle's total degrees is $360^\circ$, each central angle measures:
\[
\frac{360^\circ}{12} = 30^\circ
\]
2. **Central Angles for $x$ and $y$**:
- If... | 1 | 6 | false | The solution provided does not arrive at the final boxed answer correctly. While both solutions correctly deduce that the sum of the angles is 90 degrees, the boxed format in the solution of 'C' is incorrect because it does not specify a numerical value. The ground truth solution clearly arrives at the sum of the angle... | 0 | ||
Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11$. Then she switched the digits of the result, obtaining a number between $71$ and $75$, inclusive. What was Mary's number?
$\textbf{(A)}\ 11\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 15$ | Okay, so Mary thought of a positive two-digit number. Let me call this number N. The problem says she multiplied it by 3 and then added 11. Then she switched the digits of the result, and that switched number is between 71 and 75, inclusive. We need to find Mary's original number from the options given: 11, 12, 13, 14,... | Mary thought of a positive two-digit number \( N \). She multiplied it by 3 and added 11. Then she switched the digits of the result, obtaining a number between 71 and 75, inclusive. We need to find Mary's original number.
1. Let \( N \) be the original two-digit number.
2. The operations performed on \( N \) are: \( ... | 1. **Define the problem in terms of algebra**: Let the two-digit number Mary thought of be $x$. According to the problem, she performs the following operations on $x$:
- Multiplies by $3$: $3x$
- Adds $11$: $3x + 11$
- Switches the digits of the result.
2. **Analyze the range of the final number**: The switch... | 2 | 8 | true | Both the provided solution and the ground truth solution arrive at the same conclusion that Mary's original two-digit number is 12. The steps taken in both solutions are logically sound and derive the solution correctly. Thus, the original solution is correct. | 0 | ||
Trapezoid $ABCD$ has $\overline{AB}\parallel\overline{CD},BC=CD=43$, and $\overline{AD}\perp\overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP=11$, the length of $AD$ can be written in the form $m\sqrt{n}$, whe... | Okay, so I need to solve this trapezoid problem. Let's break down the given information step by step.
First, we have trapezoid ABCD with AB parallel to CD. The sides BC and CD are both 43 units long. Also, AD is perpendicular to BD. The diagonals AC and BD intersect at point O, and P is the midpoint of BD. We're told... | Given trapezoid \(ABCD\) with \(\overline{AB} \parallel \overline{CD}\), \(BC = CD = 43\), and \(\overline{AD} \perp \overline{BD}\). The intersection of the diagonals \(AC\) and \(BD\) is \(O\), and \(P\) is the midpoint of \(\overline{BD}\) with \(OP = 11\). We need to find the length of \(AD\) in the form \(m\sqrt{n... | 1. **Identify the properties of the trapezoid**: Given that $ABCD$ is a trapezoid with $\overline{AB}\parallel\overline{CD}$ and $BC=CD=43$. Also, $\overline{AD}\perp\overline{BD}$, which implies that $\triangle ABD$ is a right triangle.
2. **Examine the diagonals and intersection**: The diagonals $\overline{AC}$ and ... | D | \textbf{(D)}194 | true | The provided solution correctly calculates the distance of segment AD in trapezoid ABCD following the properties of the trapezoid, using the conditions given and applying the relevant geometric concepts and theorems appropriately. All steps yield consistent results leading to the final expression for AD as \(4\sqrt{190... | 0 | ||
If $q_1 ( x )$ and $r_ 1$ are the quotient and remainder, respectively, when the polynomial $x^ 8$ is divided by
$x + \tfrac{1}{2}$ , and if $q_ 2 ( x )$ and $r_2$ are the quotient and remainder, respectively,
when $q_ 1 ( x )$ is divided by $x + \tfrac{1}{2}$, then $r_2$ equals
$\textbf{(A) }\frac{1}{256}\qquad \tex... | Alright, let's see. The problem is about dividing polynomials and finding remainders. Specifically, we have to divide x^8 by (x + 1/2) first, which gives a quotient q1(x) and remainder r1. Then, we take that quotient q1(x) and divide it again by (x + 1/2), resulting in another quotient q2(x) and remainder r2. We need t... | To solve the problem, we need to find the remainder \( r_2 \) when the quotient \( q_1(x) \) from dividing \( x^8 \) by \( x + \frac{1}{2} \) is further divided by \( x + \frac{1}{2} \).
1. **First Division (Dividing \( x^8 \) by \( x + \frac{1}{2} \))**:
- Using the Remainder Theorem, the remainder \( r_1 \) is \(... | 1. **Divide \(x^8\) by \(x + \frac{1}{2}\):**
We use synthetic division to divide \(x^8\) by \(x + \frac{1}{2}\). Setting \(x + \frac{1}{2} = 0\) gives \(x = -\frac{1}{2}\). We substitute \(x = -\frac{1}{2}\) into \(x^8\) to find the remainder:
\[
\left(-\frac{1}{2}\right)^8 = \frac{1}{256}
\]
Thus, the ... | B | \textbf{(B)}-\frac{1}{16} | false | In the provided solution, although the calculations for both the first and second division using synthetic division are generally correct, it incorrectly states that the final answer is \oxed{B} without the appropriate formatting for the response. The value of r_2 should explicitly match the format indicated in Ground... | 0 | ||
The expression $\frac{1^{4y-1}}{5^{-1}+3^{-1}}$ is equal to:
$\textbf{(A)}\ \frac{4y-1}{8}\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{15}{2}\qquad\textbf{(D)}\ \frac{15}{8}\qquad\textbf{(E)}\ \frac{1}{8}$ | Okay, let's see. I need to figure out which of the options A through E is equal to the expression (1^{4y-1})/(5^{-1} + 3^{-1}). Hmm, first, I should probably simplify both the numerator and the denominator step by step.
Starting with the numerator: 1 raised to any exponent is still 1, right? Because 1 multiplied by i... | The expression simplifies by recognizing that \(1^{4y-1} = 1\) for any \(y\). The denominator \(5^{-1} + 3^{-1}\) converts to \(\frac{1}{5} + \frac{1}{3}\). Finding a common denominator (15) gives \(\frac{3}{15} + \frac{5}{15} = \frac{8}{15}\). Taking the reciprocal, the expression becomes \(1 \div \frac{8}{15} = \frac... | 1. **Simplify the Numerator**:
The expression $1^{4y-1}$ simplifies to $1$ for any real number $y$, because any power of $1$ is $1$. Thus, the expression becomes:
\[
\frac{1^{4y-1}}{5^{-1} + 3^{-1}} = \frac{1}{5^{-1} + 3^{-1}}
\]
2. **Simplify the Denominator**:
We rewrite the terms in the denominator ... | D | \textbf{(D)}\\frac{15}{8} | true | The solution accurately follows the steps to simplify the expression just as outlined in the ground truth solution. It correctly simplifies the numerator and denominator and calculates the final result. Both solutions arrive at the same conclusion, therefore the given solution is correct. | 0 | ||
Carrie has a rectangular garden that measures $6$ feet by $8$ feet. She plants the entire garden with strawberry plants. Carrie is able to plant $4$ strawberry plants per square foot, and she harvests an average of $10$ strawberries per plant. How many strawberries can she expect to harvest?
$\textbf{(A) }560 \qquad \t... | Okay, let's try to figure out how many strawberries Carrie can expect to harvest from her garden. Hmm, let's start by understanding the problem step by step.
First, Carrie has a rectangular garden. The dimensions are 6 feet by 8 feet. So, the area of the garden must be calculated first because the number of plants dep... | Carrie's rectangular garden has an area of \(6 \, \text{feet} \times 8 \, \text{feet} = 48 \, \text{square feet}\). She plants \(4 \, \text{strawberry plants per square foot}\), resulting in \(48 \times 4 = 192 \, \text{plants}\). Each plant yields an average of \(10 \, \text{strawberries}\), so the total harvest is \(... | 1. **Calculate the area of the garden**:
The garden is a rectangle with dimensions $6$ feet by $8$ feet. The area \( A \) of a rectangle is given by the formula:
\[
A = \text{length} \times \text{width}
\]
Substituting the given dimensions:
\[
A = 6 \text{ ft} \times 8 \text{ ft} = 48 \text{ ft}^... | D | \textbf{(D)}1920 | true | The given solution accurately calculates the area of the garden, the number of plants, and the total yield of strawberries, matching the steps outlined in the ground truth solution and arriving at the exact same final answer. | 0 |
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