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 |
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