question_content stringlengths 60 5.92k | think_content stringlengths 724 111k | solution_content stringlengths 21 10.5k | pseudoteacher_responses stringlengths 1.13k 251k | prompt_idx int64 0 16.7k |
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Return your final response within \boxed{}. 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)}\ \... | 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) \o... | Okay, so I need to solve this problem where there's a custom operation defined as a⊗b equals a squared divided by b. The question is asking for [(1⊗2)⊗3] minus [1⊗(2⊗3)], and they want me to choose the correct answer from the options given. Let me try to break this down step by step.
First, let me make sure I understa... | 0 |
Return your final response within \boxed{}. 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... | 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. Thi... | Okay, so Doug has this square window made up of 8 equal-sized panes of glass. The Asymptote code shows the structure, with borders around and between the panes that are 2 inches wide. The ratio of the height to the width for each pane is 5:2. The question is asking for the side length of the square window. The answer c... | 1 |
Return your final response within \boxed{}. $P(x)$ 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 t... | Okay, so I need to find the value of n for which the polynomial P(x) of degree 3n satisfies all those given conditions, and specifically P(3n+1) = 730. Let's start by understanding the problem step by step.
First, the polynomial P(x) is of degree 3n. That means the highest power of x in P(x) is x^{3n}, and its leading... | 2 |
Return your final response within \boxed{}. 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... | 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(... | Okay, so I need to find the value of 'a' given the function f(x) = a x² - √2, and the condition that f(f(√2)) = -√2. The answer choices are provided, so hopefully, after working through the problem, one of them will stick out. Let me break this down step by step.
First, let's understand the function f(x). It's a quadr... | 3 |
Return your final response within \boxed{}. 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 appli... | 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... | Okay, so I need to figure out how many cents are saved by buying the appliance from the television advertiser compared to the in-store price. Let me start by understanding both options.
First, the in-store price is $99.99. That seems straightforward. Now, the television commercial is offering the same product for thr... | 4 |
Return your final response within \boxed{}. 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 ... | 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 \(G\) i... | Okay, so I need to solve this geometry problem. Let me read it again carefully.
Points A, B, C, D, E, and F lie in that order on line segment AF, dividing it into five segments, each of length 1. So, AF is divided into five equal parts, each 1 unit long. Then, point G is not on line AF. Point H lies on GD, and point J... | 5 |
Return your final response within \boxed{}. 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\% \... | 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} \rig... | Okay, so Judy had 35 hits in her softball season. The problem is asking what percentage of her hits were singles. Let's see. The options are A through E, with percentages 28%, 35%, 70%, 75%, and 80%. Hmm.
First, I need to break down her hits into different types. The problem mentions she had 1 home run, 1 triple, and ... | 6 |
Return your final response within \boxed{}. 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... | 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 rota... | Okay, so I need to figure out the equation of the graph G' which is the graph of y = log₁₀x rotated 90 degrees counter-clockwise about the origin. The options are given, and I need to pick the correct one. Let me start by recalling how rotations affect functions.
First, rotating a graph 90 degrees counter-clockwise ar... | 7 |
Return your final response within \boxed{}. 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 d... | 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 = ... | Okay, so Jose, Thuy, and Kareem all start with the number 10. Each of them performs a different sequence of operations, and we need to figure out who ends up with the largest final answer. The answer choices are given from A to E, with options like Jose, Thuy, Kareem, or combinations. Let me try to work through this st... | 8 |
Return your final response within \boxed{}. 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 \qq... | 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 ... | Okay, so I need to solve this problem where a set of consecutive positive integers starting with 1 is written on a blackboard, one number is erased, and the average of the remaining numbers is 35 and 7/17. The question is asking which number was erased, with the options given. Let me try to figure this out step by step... | 9 |
Return your final response within \boxed{}. 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 fl... | 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 me... | Okay, so I need to find the fraction of the yard that's occupied by the flower beds. The yard is a rectangle with two congruent isosceles right triangles as flower beds, and the rest is a trapezoid. The parallel sides of the trapezoid are 15 and 25 meters. The answer choices are fractions, so I need to figure out the a... | 10 |
Return your final response within \boxed{}. 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... | 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 cir... | Okay, so I need to find all possible values of t between 0 and 360 degrees such that the triangle with vertices at (cos40°, sin40°), (cos60°, sin60°), and (cos t°, sin t°) is isosceles. Then, sum all those possible t values. The answer choices are given, from 100 to 380, but I need to figure out which ones are correct.... | 11 |
Return your final response within \boxed{}. 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 > ... | Okay, so I need to figure out which inequality among the options A to E is not always correct given that x > 0, y > 0, x > y, and z ≠ 0. Let me go through each option one by one and see if they hold true under all conditions.
Starting with option A: x + z > y + z. Hmm, if I subtract z from both sides of the inequality... | 12 |
Return your final response within \boxed{}. 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)}\ 20... | 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. **... | Okay, so I need to find the harmonic mean of 1 and 2016 and determine which integer it's closest to. Let me start by recalling what harmonic mean is. The problem says that the harmonic mean of two numbers is calculated as twice their product divided by their sum. Let me verify that formula because sometimes harmonic me... | 13 |
Return your final response within \boxed{}. 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 bet... | Okay, so I need to find the value of x in this arithmetic progression. The problem says the first three terms are x minus 1, x plus 1, and 2x plus 3. The options are given as A through E. Let me recall what an arithmetic progression is. An arithmetic progression, or arithmetic sequence, is a sequence where each term af... | 14 |
Return your final response within \boxed{}. 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 \tex... | 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... | Okay, so I need to solve this problem about geometric progressions. Let me start by understanding what's being asked. The problem says there's a geometric progression with n terms. The first term is 1, common ratio is r, and the sum is s. Then, they take the reciprocal of each term and create another geometric progress... | 15 |
Return your final response within \boxed{}. 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 o... | 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} = \fra... | Okay, so Alicia has two containers. The first one is 5/6 full of water, and the second one is empty. She pours all the water from the first into the second, and after that, the second container is 3/4 full. The question is asking for the ratio of the volume of the first container to the volume of the second container. ... | 16 |
Return your final response within \boxed{}. 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**: L... | Okay, so Misha is rolling a die until she gets the sequence 1-2-3 in that order consecutively. We need to find the probability that she rolls the die an odd number of times, and express it as a reduced fraction m/n, then find m + n. Hmm, let me think about how to approach this.
First, I remember that problems involvin... | 17 |
Return your final response within \boxed{}. 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 $A$... | 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... | Okay, so I need to find the height of the pillar at point E on a regular hexagon. The hexagon is ABCDEF, and the pillars at A, B, and C are 12, 9, and 10 meters tall respectively. The solar panel isn't parallel to the ground, so the heights vary in some way. Hmm, how do I approach this?
First, I remember that in a reg... | 18 |
Return your final response within \boxed{}. 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**:
- ... | Okay, so I need to solve this geometry problem. Let me read it again carefully.
The problem states that diameter AB of a circle is a two-digit integer. When you reverse its digits, you get the length of the perpendicular chord CD. The intersection point H of AB and CD is a positive rational number distance from the ce... | 19 |
Return your final response within \boxed{}. 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... | Okay, so I need to find the smallest positive number among the options given. Let me start by understanding each option. The problem lists five choices: A through E. Each involves either a positive or negative combination of a square root term and a constant. Since we're looking for the smallest positive number, I need... | 20 |
Return your final response within \boxed{}. 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
... | Okay, so I need to find the value(s) of k such that the three points (2, -3), (4, 3), and (5, k/2) are all on the same straight line. The answer choices are given, so maybe after solving, I can check which one matches. Let me think about how to approach this.
First, I remember that if three points are colinear, the sl... | 21 |
Return your final response within \boxed{}. 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 $q_n = Q(n)$ fo... | 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 \( q_n = Q(n) \) fo... | Okay, so I need to prove that if we have an infinite sequence of integers q₁, q₂, ... that satisfies two conditions: (a) the difference qₘ - qₙ is divisible by m - n for m > n, and (b) the absolute value of qₙ is bounded by some polynomial P(n), then the sequence qₙ is actually given by a polynomial Q(n). Hmm, interest... | 22 |
Return your final response within \boxed{}. 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 neares... | 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 pri... | Okay, so I need to solve this problem about gasoline prices changing over four months: January, February, March, and April. The question says that after these changes, the price at the end of April is the same as it was at the beginning of January. I need to find the percentage decrease in April, denoted as x%, and cho... | 23 |
Return your final response within \boxed{}. 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... | 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... | Okay, so I need to find the area of quadrilateral ABCD. Let me start by understanding the given information. The quadrilateral has four sides with some right angles and specific lengths. Let me note down the details:
- Angle ABC is 90 degrees.
- Angle ACD is 90 degrees.
- The length of diagonal AC is 20 units.
- The l... | 24 |
Return your final response within \boxed{}. 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... | 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... | 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 supposed to be of the form y = kx, and I have to find the value of k from the given options. Let me think about how to approach this.
First, I remember that an angle bisecto... | 25 |
Return your final response within \boxed{}. 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 ... | 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 give... | Okay, so I need to find the smallest number of points in set S, which is symmetric about the origin, both coordinate axes, and the line y = x. And we know that (2,3) is in S. The options are from 1 to 16, with 16 being the highest. Hmm, let's start by understanding each symmetry.
First, symmetry about the origin means... | 26 |
Return your final response within \boxed{}. 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,... | Okay, so I need to solve this problem where there are five positive consecutive integers starting with 'a', and their average is 'b'. Then, I have to find the average of five consecutive integers that start with 'b'. The answer choices are given from A to E, with possible answers a+3, a+4, etc. Let me try to break this... | 27 |
Return your final response within \boxed{}. 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\... | 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:
- ... | Okay, so I need to find the negation of the statement: "For all x of a certain set, x squared is greater than 0." The options are given from A to E. Let me think through this step by step.
First, let me recall what the negation of a universal quantifier ("for all") is. I remember that the negation of "for all x, P(x)"... | 28 |
Return your final response within \boxed{}. 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^{100... | Okay, so I need to find the greatest power of 2 that divides 10^1002 - 4^501. Hmm, let me start by understanding the problem. The question is asking for the highest exponent of 2 such that 2^k divides this expression. The options are up to 2^1006, so it's a pretty high exponent.
First, maybe I should write both terms... | 29 |
Return your final response within \boxed{}. 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 \(n\)-ele... | Okay, so I need to find the arithmetic mean of the least elements of all 1000-element subsets of the set {1, 2, 3, ..., 2015}. Then, express that mean as a reduced fraction p/q and find p + q. Hmm, let's break this down step by step.
First, I recall that the arithmetic mean is the total sum of all the values divided b... | 30 |
Return your final response within \boxed{}. 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: ... | 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 th... | Okay, so the problem is about a man who goes to dinner after 6:00 p.m. and before 7:00 p.m. The hands of his watch form a 110-degree angle both when he leaves and when he returns. We need to find out how many minutes he was away. The answer choices are given, so that might help in checking our work, but I should try to... | 31 |
Return your final response within \boxed{}. 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... | 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... | Okay, so I need to find the smallest possible sum of all the visible faces on a 3x3x3 cube made up of 27 dice. Each die has opposite sides that add up to 7. The answer choices are given from 60 to 96. Let me try to figure this out step by step.
First, let me visualize the cube. It's a 3x3x3 structure, so there are 27 ... | 32 |
Return your final response within \boxed{}. 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... | 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 \( ABA \) ... | Okay, so I need to solve this multiplication problem where A, B, C, D are different digits, and find A + B. Let me try to figure this out step by step.
First, let me write down the problem again to visualize it better:
```
A B A
× C D
-------
C D C D
```
Hmm, so the multiplication of the three-digit number ABA ... | 33 |
Return your final response within \boxed{}. 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 \(... | Okay, so I need to prove that a^m + a^n ≥ m^m + n^n, where a is defined as (m^{m+1} + n^{n+1})/(m^m + n^n), and m and n are positive integers. Hmm, let's start by understanding what a is.
First, let's write down the expression for a again: a = (m^{m+1} + n^{n+1}) / (m^m + n^n). Maybe I can simplify this expression? L... | 34 |
Return your final response within \boxed{}. 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{... | 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 \( x = 10y \) ... | Okay, so I need to solve this problem where the sum of two natural numbers is 17,402. One of the numbers is divisible by 10, and if you erase its units digit, you get the other number. The question is asking for the difference between these two numbers, and there are multiple-choice options provided. Let me try to brea... | 35 |
Return your final response within \boxed{}. 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 th... | Okay, so I need to solve this problem: 2^(0^(1^9)) + ((2^0)^1)^9. The answer choices are from A to E, with numbers 0 to 4. Let me break this down step by step.
First, let me parse the expression correctly. The expression is split into two parts by the plus sign. The first part is 2 raised to the power of (0 raised to... | 36 |
Return your final response within \boxed{}. 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... | 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 bo... | Okay, so I need to figure out which of the given options describes the graph of the equation (x + y)^2 = x^2 + y^2. Let me start by expanding the left side of the equation to see if there's a simpler form.
First, expanding (x + y)^2 gives x^2 + 2xy + y^2. So, substituting this back into the equation, we have:
x^2 + 2... | 37 |
Return your final response within \boxed{}. 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... | 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-digi... | Okay, so I need to solve this multiplication problem where a and b are digits, and the partial products and final product are given. The problem is set up like this:
```
2 a
× b 3
-------
6 9
9 2
-------
9 8 9
```
And we need to find a + b. The answer choices are given as A) 3, B)4, C)7, D)9, E)12.
Alrig... | 38 |
Return your final response within \boxed{}. 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... | Okay, so I need to find the angle between the hour and minute hands at 2:15. Let me think... Hmm, angles on a clock. I remember that each hour mark represents 30 degrees because a full circle is 360 degrees, and there are 12 hours. So 360 divided by 12 is 30. So each hour mark is 30 degrees apart. But wait, the hour ha... | 39 |
Return your final response within \boxed{}. 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://art... | 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 ... | Okay, so I need to solve this geometry problem about a right triangle with two circles outside the triangle. Let me try to visualize the setup first. The triangle is right-angled at C, with legs CA = 30 and CB = 16. So, triangle ABC is right-angled at C, with legs 30 and 16, and hypotenuse AB. Then, the legs CA and CB ... | 40 |
Return your final response within \boxed{}. 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 a... | 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_... | Okay, so I need to find the largest prime factor that always divides the sum S of such sequences. Let me start by understanding the problem better.
First, the sequence is a finite list of three-digit integers. The key property is that each term's tens and units digits become the hundreds and tens digits of the next te... | 41 |
Return your final response within \boxed{}. 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... | 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 3... | Okay, so the question is asking what time it was 2011 minutes after midnight on January 1, 2011. The options are given from A to E, and I need to figure out which one is correct. Let me break this down step by step.
First, I know that midnight on January 1 is 12:00 AM, and we need to add 2011 minutes to that time. The... | 42 |
Return your final response within \boxed{}. 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 \... | 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. Substitut... | Okay, so the problem is about a basketball player who made 5 baskets, each worth either 2 or 3 points. We need to figure out how many different total points they could have scored. The answer choices are from A to E, with the options being 2, 3, 4, 5, or 6. Let me try to work through this step by step.
First, let's ma... | 43 |
Return your final response within \boxed{}. 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)}... | 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 \(A(0, 0)\), \(B(s, 0)\), ... | Okay, so I need to find the ratio of the area of square EFGH to the area of square ABCD. The problem says that square EFGH has one vertex on each side of square ABCD, and point E is on side AB with AE equal to 7 times EB. Let me try to visualize this.
First, let me sketch square ABCD. Let's assume it's a unit square ... | 44 |
Return your final response within \boxed{}. 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... | 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... | Okay, so I need to figure out the result of these two transactions involving Mr. A and Mr. B selling the house back and forth with different percentages of loss and gain. Let me start by breaking down each transaction step by step.
First, the house is worth $9000. Mr. A sells it to Mr. B at a 10% loss. To calculate th... | 45 |
Return your final response within \boxed{}. 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 a... | 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:
\[
\frac{400}{16} = 25
\]
Th... | Okay, so Chandler wants to buy a mountain bike that costs $500. Let me see how much money he already has from his birthday gifts. His grandparents gave him $50, his aunt $35, and his cousin $15. Let me add those up.
First, the grandparents: $50. Then the aunt: $35. So, 50 + 35 is 85. Then his cousin gives $15, so add... | 46 |
Return your final response within \boxed{}. 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}\q... | 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 reci... | Okay, so I need to find the harmonic mean of the numbers 1, 2, and 4. Hmm, harmonic mean is the reciprocal of the average of their reciprocals. Let me make sure I remember that correctly. So the formula should be something like 1 divided by the average of 1 over each number. Let me write that down step by step to avoid... | 47 |
Return your final response within \boxed{}. 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 f... | 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... | Okay, so Julie needs to prepare a speech for her class, and the length has to be between one-half hour and three-quarters of an hour. Her ideal speaking rate is 150 words per minute. We need to figure out which of the options (A) 2250, (B) 3000, (C) 4200, (D) 4350, or (E) 5650 is appropriate for her speech. Let me thin... | 48 |
Return your final response within \boxed{}. 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 ... | 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**: ... | Okay, so I need to figure out what value of X will be printed by instruction 5 in this computer program. Let me read through the instructions carefully again to make sure I understand them correctly.
Starting with instruction 1: "START X AT 3 AND S AT 0." So initially, X is 3 and S is 0. Got that.
Instruction 2: "INC... | 49 |
Return your final response within \boxed{}. 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 a... | Okay, so I need to figure out which option is equivalent to the statement "If P is true, then Q is false." Hmm, let's start by understanding what this statement actually means. In logic, an "if...then..." statement is a conditional, and its structure is "If P, then Q," which is often written as P → Q. The contrapositiv... | 50 |
Return your final response within \boxed{}. 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 o... | Okay, so I need to find the least positive integer divisible by 17 where the digits sum to 17. Hmm, let's break this down. First, the number has to be divisible by 17, which means when I divide it by 17, there's no remainder. Also, the sum of its digits must be exactly 17. The goal is the smallest such number.
Let me... | 51 |
Return your final response within \boxed{}. 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 w... | 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 ... | Okay, so I need to solve this problem where a shopper is comparing three coupons to find the smallest and largest prices where Coupon A saves at least as much as Coupon B or Coupon C. Then, find the difference between those two prices. Let me break this down step by step.
First, let's note the problem's details. The i... | 52 |
Return your final response within \boxed{}. 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 \( B... | Okay, so I need to find the measure of angle ACB in triangle ABC where angle ABC is 45 degrees. Point D is on BC such that 2*BD = CD, and angle DAB is 15 degrees. The answer choices are given, so maybe I can use some trigonometric relationships or maybe construct some auxiliary lines to help find the angle. Let me star... | 53 |
Return your final response within \boxed{}. 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... | 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 +... | Okay, so Mary has a shelf with five books. 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 book width in centimeters. The answer choices are A through E, with the options 1, 2, 3, 4, or 5. Hmm, let me think step by step how to figure this out.
First, I remember t... | 54 |
Return your final response within \boxed{}. 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 ... | Okay, so I need to solve the equation 2^(2x) - 8*2^x + 12 = 0. Hmm, let me think. The equation has 2^(2x) and 2^x terms, so maybe I can substitute something to make it simpler. Let me try substituting y = 2^x. Then, 2^(2x) would be (2^x)^2 = y^2. So substituting that into the equation gives me y^2 - 8y + 12 = 0. That l... | 55 |
Return your final response within \boxed{}. 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 <... | 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** ... | Okay, so I need to solve this problem where the sum of the greatest integer less than or equal to x (which is the floor function, right?) and the least integer greater than or equal to x (which is the ceiling function) equals 5. The question is asking for the solution set of x. The options are given from A to E, and I ... | 56 |
Return your final response within \boxed{}. 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)}\... | 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... | Okay, so there's this problem about a powderman who sets a fuse for a blast that's supposed to happen in 30 seconds. He starts running away at 8 yards per second. Sound travels at 1080 feet per second. We need to figure out how far he had run when he heard the blast. The answer choices are given, so maybe after solving... | 57 |
Return your final response within \boxed{}. 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 ma... | 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... | Okay, so the problem is about selecting committees from a student council. There are two types of committees: a two-person welcoming committee and a three-person planning committee. The key information given is that there are exactly 10 ways to select the two-person welcoming committee. The question is asking how many ... | 58 |
Return your final response within \boxed{}. 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 \qq... | 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 equatio... | Okay, so I need to solve this problem about Pete and Claire's ages. Let me read it again to make sure I understand the details.
The problem states: 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 age... | 59 |
Return your final response within \boxed{}. 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)}\ ... | 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\t... | Okay, so I need to solve this problem where θ is a constant between 0 and π, and x + 1/x = 2cosθ. Then, for each positive integer n, x^n + 1/x^n equals one of the given options. The options are all about powers of 2, cosθ, and trigonometric functions with θ and n. Hmm, let me think step by step.
First, I remember that... | 60 |
Return your final response within \boxed{}. 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}\qq... | 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 \( x \)-intercep... | Okay, so I need to solve this problem where two linear equations, y = ax + 5 and y = 3x + b, intersect the x-axis at the same point. The question is asking for the sum of all possible x-coordinates of these points of intersection. The answer choices are given, so that might help if I get stuck, but I should try to figu... | 61 |
Return your final response within \boxed{}. (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.
**Induct... | Okay, so I need to prove that for every positive integer n, there exists an n-digit number divisible by 5^n where all its digits are odd. Hmm, let's start by understanding the problem.
First, an n-digit number means a number between 10^{n-1} and 10^n - 1. All digits must be odd, so each digit can be 1, 3, 5, 7, or 9.... | 62 |
Return your final response within \boxed{}. 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 c... | 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 \... | Okay, so I need to solve this geometry problem. Let me read it again carefully.
There are four circles, none congruent, with centers A, B, C, D. Points P and Q lie on all four circles. The radius of circle A is 5/8 of circle B's radius. Similarly, circle C's radius is 5/8 of circle D's. The distances from A to B and f... | 63 |
Return your final response within \boxed{}. 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 equ... | Okay, so I need to figure out which of the options (A) through (E) correctly states that a figure is an equiangular parallelogram if and only if it is that shape. The options are rectangle, regular polygon, rhombus, square, or trapezoid. Let me start by recalling what each of these terms means.
First, an equiangular p... | 64 |
Return your final response within \boxed{}. 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 t... | Okay, so I need to find the minimum value of sqrt(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 defined by 5x + 12y = 60. Because sqrt(x² + y²) is the distance from the point (x, y) to the origin, right? So this is essentially finding th... | 65 |
Return your final response within \boxed{}. 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)}\ 4... | 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:
\[
\frac{4}{7} = \frac{28}{x} \implies 4x = 196 \implies x = \frac{196}{4} = 49
\]
Bo... | Okay, so I need to solve this problem where the dealership sells sports cars and sedans in a ratio, and they predict selling 28 sports cars next month. The question is asking how many sedans they expect to sell. The answer choices are given from A to E. Let me think through this step by step.
First, the problem states... | 66 |
Return your final response within \boxed{}. 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 pro... | 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... | Okay, so I need to solve this problem about two dice with at least 6 faces each. The key points are:
1. The probability of rolling a sum of 7 is 3/4 of the probability of rolling a sum of 10.
2. The probability of rolling a sum of 12 is 1/12.
3. Find the least possible number of faces on the two dice combined.
First,... | 67 |
Return your final response within \boxed{}. 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 pas... | 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 sing... | Okay, so I need to find the number of ways to make all the roads one-way in a system of five towns connected by exactly one road each pair, such that you can still get from any town to any other town using the roads. That sounds like a directed graph problem where all edges are directed, and the graph must be strong, m... | 68 |
Return your final response within \boxed{}. 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}... | Okay, so 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 reduced to its lowest terms. The options are given, so maybe after solving, I can check which one matches. Let me think step by step.
First, repeating decimals can often be con... | 69 |
Return your final response within \boxed{}. 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\righ... | Okay, so I need to solve this problem where x and y are non-zero numbers such that x equals 1 over y. Then I have to find the value of the expression (x - 1/x)(y + 1/y) and choose the correct option from A to E. Let me start by writing down what I know.
First, the problem states that x = 1/y. That means y is the recip... | 70 |
Return your final response within \boxed{}. 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 + b... | Okay, so I need to find the values of k for which the quadratic equation 2x² - kx + x + 8 = 0 has real and equal roots. Hmm, let me recall the conditions for a quadratic equation to have real and equal roots. I think it has something to do with the discriminant.
First, let me write down the general form of a quadrati... | 71 |
Return your final response within \boxed{}. 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 \(n^3\) ... | Okay, so I need to figure out how many perfect cubes are between \(2^8 + 1\) and \(2^{18} + 1\), inclusive. Let me start by understanding the problem step by step.
First, the question is asking for the count of perfect cubes in a specific range. A perfect cube is a number that can be expressed as \(n^3\) where \(n\) i... | 72 |
Return your final response within \boxed{}. 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)}... | 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 \... | Okay, so I need to find the perimeter of an equilateral triangle formed by a line passing through the origin, intersecting both the line x = 1 and the line y = 1 + (√3/3)x. The answer choices are given, and I need to figure out which one is correct. Let me start by visualizing this problem.
First, let me sketch the co... | 73 |
Return your final response within \boxed{}. 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 \math... | 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 0.0... | Okay, so Alicia earns $20 per hour, and there's a local tax deduction of 1.45%. The question is asking how many cents per hour go towards local taxes. The answer choices are in cents, so the final answer needs to be in cents, which is a decimal of the dollar amount. Let me break this down step by step.
First, let's co... | 74 |
Return your final response within \boxed{}. 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 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 ... | Okay, so I need to solve this problem where the numbers -2, 4, 6, 9, and 12 are rearranged according to specific rules. Then, find the average of the first and last numbers. The answer choices are given, so hopefully, after working through it, I can pick the correct one. Let me break this down step by step.
First, let... | 75 |
Return your final response within \boxed{}. 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\... | 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: \... | Okay, let's try to solve this problem step by step. So, Mary thought of a positive two-digit number. Then she multiplied it by 3 and added 11. After that, she switched the digits of the result, and the new number was between 71 and 75, inclusive. We need to find out what Mary's original number was. The options are 11, ... | 76 |
Return your final response within \boxed{}. 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 $AD$... | 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... | Okay, so I need to solve this geometry problem about a trapezoid. Let me read it again and make sure I understand all the given information.
We have trapezoid ABCD with AB parallel to CD. The sides BC and CD are both equal to 43. Also, AD is perpendicular to BD. The diagonals AC and BD intersect at point O, and P is t... | 77 |
Return your final response within \boxed{}. 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} = \fr... | Okay, so I need to solve this problem where the expression (1^{4y-1}) divided by (5^{-1} + 3^{-1}) is given. The options are from A to E, and I have to choose the correct one. Let me break this down step by step.
First, let's look at the numerator: 1 raised to the power of (4y - 1). Hmm, 1 to any power is always 1, ri... | 78 |
Return your final response within \boxed{}. 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 ex... | 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 ... | Okay, so Carrie has this rectangular garden that's 6 feet by 8 feet. She's planting strawberry plants all over it, right? Let me make sure I understand the problem correctly. She plants 4 strawberry plants per square foot, and each plant gives her an average of 10 strawberries. The question is asking how many strawberr... | 79 |
Return your final response within \boxed{}. When the natural numbers $P$ and $P'$, with $P>P'$, are divided by the natural number $D$, the remainders are $R$ and $R'$, respectively. When $PP'$ and $RR'$ are divided by $D$, the remainders are $r$ and $r'$, respectively. Then:
$\text{(A) } r>r' \text{ always}\quad \text... | Alright, let me try to tackle this problem step by step. Okay, so we have two natural numbers P and P', where P is greater than P'. When both are divided by another natural number D, they leave remainders R and R' respectively. Then, we're looking at what happens when we divide the products PP' and RR' by D. The remain... |
When natural numbers \( P \) and \( P' \) (with \( P > P' \)) are divided by \( D \), they leave remainders \( R \) and \( R' \), respectively. By modular arithmetic, \( P \equiv R \pmod{D} \) and \( P' \equiv R' \pmod{D} \). Multiplying these congruences gives \( PP' \equiv RR' \pmod{D} \). This implies that the rem... | Okay, so I need to solve this problem about remainders when dividing natural numbers. Let me read it again carefully.
The problem says: When the natural numbers P and P', with P > P', are divided by the natural number D, the remainders are R and R', respectively. Then, when PP' and RR' are divided by D, the remainders... | 80 |
Return your final response within \boxed{}. The sides of a triangle have lengths $6.5$, $10$, and $s$, where $s$ is a whole number. What is the smallest possible value of $s$?
$\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 7$ | Okay, so I need to find the smallest possible whole number value for the third side of a triangle where the other two sides are 6.5 and 10. The answer choices are from 3 to 7, so I guess the answer is one of those. Let me remember the triangle inequality theorem. I think it states that the sum of any two sides of a tri... |
To determine the smallest possible whole number value \( s \) for the third side of a triangle with sides 6.5 and 10, we apply the triangle inequality theorem. The theorem states that the sum of any two sides must be greater than the third side.
We need to check the following inequalities:
1. \( 6.5 + 10 > s \)
2. ... | Okay, so I need to find the smallest possible whole number value of s for a triangle with sides 6.5, 10, and s. The triangle inequality theorem says that the sum of any two sides must be greater than the third side. Let me recall the exact conditions. For three sides a, b, c, the inequalities must hold: a + b > c, a + ... | 81 |
Return your final response within \boxed{}. One proposal for new postage rates for a letter was $30$ cents for the first ounce and $22$ cents for each additional ounce (or fraction of an ounce). The postage for a letter weighing $4.5$ ounces was
$\text{(A)}\ \text{96 cents} \qquad \text{(B)}\ \text{1.07 dollars} \qqua... | Okay, let's see here. The problem is about calculating the postage cost for a letter that weighs 4.5 ounces. The options are given, and I need to figure out which one is correct. The pricing structure is 30 cents for the first ounce and 22 cents for each additional ounce or fraction of an ounce. Hmm, so fractions of an... |
The problem involves calculating the postage cost for a letter weighing 4.5 ounces with a pricing structure of 30 cents for the first ounce and 22 cents for each additional ounce or fraction thereof.
1. **Initial Analysis**: The first ounce costs 30 cents. The remaining weight after the first ounce is 4.5 - 1 = 3.5... | Okay, so I need to figure out the postage cost for a letter that weighs 4.5 ounces. Let me read the problem again to make sure I understand it correctly. The proposal is 30 cents for the first ounce, and then 22 cents for each additional ounce or fraction of an ounce. The options are given in cents and dollars, so I ne... | 82 |
Return your final response within \boxed{}. Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that
\[ (2 + i)^n = a_n + b_ni \]for all integers $n\geq 0$, where $i = \sqrt{-1}$. What is\[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?\]
$\textbf{(A) }\frac 38\qquad\textbf{(B) }\frac7{16}\qquad\textbf{(C) }\frac12\qq... | Okay, let me try to figure out this problem. So we have two sequences, a_n and b_n, defined by the equation (2 + i)^n = a_n + b_n i for all integers n ≥ 0. We need to find the sum from n=0 to infinity of (a_n b_n)/7^n. The answer choices are fractions, so maybe it simplifies to a nice fraction. Let's break it down step... |
Given the sequences of real numbers \((a_n)\) and \((b_n)\) defined by \((2 + i)^n = a_n + b_n i\) for all integers \(n \geq 0\), we need to find the sum \(\sum_{n=0}^\infty \frac{a_n b_n}{7^n}\).
First, we convert the complex number \(2 + i\) to polar form. The modulus is \(\sqrt{5}\) and the argument is \(\theta =... | Okay, so I need to find the sum from n=0 to infinity of a_n b_n divided by 7^n, where (2 + i)^n is expressed as a_n + b_n i. Hmm, that seems a bit complex, but maybe I can break it down step by step.
First, let me recall that complex numbers can be represented in polar form, which might be helpful here. Since we have ... | 83 |
Return your final response within \boxed{}. An $11\times 11\times 11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point?
$\text{(A) 328} \quad \text{(B) 329} \quad \text{(C) 330} \quad \text{(D) 331} \quad \text{(E) 332}$ | Okay, so I have this problem here: an 11×11×11 wooden cube made up of 11³ unit cubes. The question is asking for the greatest number of unit cubes that can be seen from a single point. The answer choices are from 328 to 332, so the answer is probably one of those. Let me think about how to approach this.
First, when y... |
To determine the greatest number of unit cubes that can be seen from a single point on an \(11 \times 11 \times 11\) wooden cube, we need to consider the visibility of the cubes from a corner viewpoint. When viewed from a corner, three adjacent faces of the cube are visible. However, some cubes are counted multiple t... | Okay, so I need to figure out the greatest number of unit cubes that can be seen from a single point on an 11×11×11 wooden cube. The answer choices are from 328 to 332, so it's a multiple choice question. Let me start by understanding the problem thoroughly.
First, the cube is made up of 11 layers along each axis, so ... | 84 |
Return your final response within \boxed{}. If each of two intersecting lines intersects a hyperbola and neither line is tangent to the hyperbola,
then the possible number of points of intersection with the hyperbola is:
$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 2\text{ or }3\qquad \textbf{(C)}\ 2\text{ or }4\qquad \textb... | Okay, so I need to figure out the possible number of points where two intersecting lines, neither of which is tangent to a hyperbola, can intersect the hyperbola. The options are given as A through E, with E being 2, 3, or 4. Let me start by recalling some properties about hyperbolas and lines intersecting them.
First... |
To determine the possible number of points of intersection between two intersecting lines and a hyperbola, where neither line is tangent to the hyperbola, we need to consider the following:
1. **Hyperbola and Line Intersections**: A line that is not tangent to a hyperbola can intersect it in 0, 1, or 2 points. Howev... | Okay, so I need to figure out the possible number of points where two intersecting lines can intersect a hyperbola, given that neither line is tangent to the hyperbola. The options are from A to E, with E being 2,3, or 4. Let me start by recalling some basics about hyperbolas and lines intersecting them.
First, a hype... | 85 |
Return your final response within \boxed{}. Are there integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both perfect cubes of integers? | Okay, so I need to figure out if there are integers a and b such that both a^5b + 3 and ab^5 + 3 are perfect cubes. Hmm, interesting problem. Let me start by understanding what the question is asking. We have two expressions: a^5b + 3 and ab^5 + 3. Both of these need to be perfect cubes. That means there exist some int... |
After a thorough exploration, we analyze the problem using modular arithmetic and factorization techniques.
1. **Modulo 9 Analysis**: Perfect cubes modulo 9 can only be 0, 1, or 8. For \(a^5b + 3\) and \(ab^5 + 3\) to be cubes, they must align with these residues. Through detailed modular computations, we find cont... | Okay, so I need to figure out if there exist integers a and b such that both \(a^5b + 3\) and \(ab^5 + 3\) are perfect cubes of integers. Hmm, that's a pretty interesting problem. Let me start by understanding what's being asked. We need to find integers a and b where each expression when combined with 3 results in a p... | 86 |
Return your final response within \boxed{}. If $\frac {1}{x} - \frac {1}{y} = \frac {1}{z}$, then $z$ equals:
$\textbf{(A)}\ y - x\qquad \textbf{(B)}\ x - y\qquad \textbf{(C)}\ \frac {y - x}{xy}\qquad \textbf{(D)}\ \frac {xy}{y - x}\qquad \textbf{(E)}\ \frac {xy}{x - y}$ | Okay, let's see here. The problem is asking to solve for z given the equation 1/x minus 1/y equals 1/z. The answer choices are labeled from A to E, and I need to figure out which one is correct. Alright, let's start by writing down the equation again to make sure I have it right:
1/x - 1/y = 1/z
Hmm, so we need to so... |
To solve for \( z \) given the equation \( \frac{1}{x} - \frac{1}{y} = \frac{1}{z} \):
1. **Combine the fractions on the left side**:
\[
\frac{1}{x} - \frac{1}{y} = \frac{y - x}{xy}
\]
This is achieved by finding a common denominator \( xy \), resulting in \( \frac{y - x}{xy} \).
2. **Set the combined f... | Okay, so I need to solve this algebra problem where 1/x minus 1/y equals 1/z, and then figure out what z is in terms of x and y. The answer choices are given from A to E, so I need to work through the equation step by step.
First, let me write down the equation again to make sure I have it right: 1/x - 1/y = 1/z. My g... | 87 |
Return your final response within \boxed{}. Ashley, Betty, Carlos, Dick, and Elgin went shopping. Each had a whole number of dollars to spend, and together they had $56$ dollars. The absolute difference between the amounts Ashley and Betty had to spend was $19$ dollars. The absolute difference between the amounts Betty... | Okay, let's see. There are five people: Ashley, Betty, Carlos, Dick, and Elgin. Each has a whole number of dollars, and the total they all have together is 56. The problem gives the absolute differences between each pair in this order: Ashley and Betty is 19, Betty and Carlos is 7, Carlos and Dick is 5, Dick and Elgin ... |
Let \( A, B, C, D, \) and \( E \) represent the amounts of money Ashley, Betty, Carlos, Dick, and Elgin had, respectively. We know the following:
1. \( A + B + C + D + E = 56 \)
2. \( |A - B| = 19 \)
3. \( |B - C| = 7 \)
4. \( |C - D| = 5 \)
5. \( |D - E| = 4 \)
6. \( |E - A| = 11 \)
We need to find the value of \(... | Okay, so there's this problem about Ashley, Betty, Carlos, Dick, and Elgin shopping. Each of them had a whole number of dollars, and together they had $56. There are a bunch of absolute differences between the amounts each person had: Ashley and Betty was 19, Betty and Carlos 7, Carlos and Dick 5, Dick and Elgin 4, and... | 88 |
Return your final response within \boxed{}. A palindrome between $1000$ and $10,000$ is chosen at random. What is the probability that it is divisible by $7$?
$\textbf{(A)}\ \dfrac{1}{10} \qquad \textbf{(B)}\ \dfrac{1}{9} \qquad \textbf{(C)}\ \dfrac{1}{7} \qquad \textbf{(D)}\ \dfrac{1}{6} \qquad \textbf{(E)}\ \dfrac{1}... | Okay, so I need to find the probability that a randomly chosen palindrome between 1000 and 10,000 is divisible by 7. Hmm. Let me break this down step by step.
First, let's recall what a palindrome is. A palindrome is a number that reads the same forwards and backwards. Since we're dealing with four-digit numbers (bet... |
To find the probability that a randomly chosen palindrome between 1000 and 10,000 is divisible by 7, we start by noting that a four-digit palindrome has the form ABBA. This can be expressed numerically as \(1001A + 110B\), where \(A\) and \(B\) are digits with \(A\) ranging from 1 to 9 and \(B\) ranging from 0 to 9.
... | Okay, so I need to find the probability that a randomly chosen four-digit palindrome between 1000 and 10,000 is divisible by 7. The answer choices are fractions from 1/10 to 1/5. Let me break this down step by step.
First, let me recall what a palindrome is. A palindrome reads the same forwards and backwards. For a fo... | 89 |
Return your final response within \boxed{}. Let $P$ be an interior point of circle $K$ other than the center of $K$. Form all chords of $K$ which pass through $P$, and determine their midpoints. The locus of these midpoints is
$\textbf{(A)} \text{ a circle with one point deleted} \qquad \\ \textbf{(B)} \text{ a circle... | Okay, let's tackle this geometry problem. So, we have a circle K with an interior point P that's not the center. We need to consider all the chords passing through P and find the locus of their midpoints. The answer choices suggest it could be a circle, a semicircle, or some variation with points deleted. Let me think ... |
To determine the locus of the midpoints of all chords passing through an interior point \( P \) (other than the center) of a circle \( K \), we start by setting up a coordinate system with the center \( O \) of the circle at the origin \((0,0)\) and the radius of the circle as \( R \). Let point \( P \) be at \((h, 0... | Okay, so I need to figure out the locus of midpoints of all chords passing through a fixed interior point P of a circle K. Let me start by visualizing the problem. There's a circle K with center O, and a point P inside the circle but not the center. For any chord that goes through P, I need to find the midpoint of that... | 90 |
Return your final response within \boxed{}. What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is $\underline{not}$ a divisor of the product of the first $n$ positive integers?
$\textbf{(A) } 995 \qquad\textbf{(B) } 996 \qquad\textbf{(C) } 997 \qquad\textbf{(D) } ... | Okay, so I need to find the greatest three-digit number n where the sum of the first n positive integers (which is n(n+1)/2) is not a divisor of the product of the first n positive integers (which is n!). The options are from 995 to 999, so it's one of those. Let me think step by step.
First, I remember that the sum o... |
To determine the greatest three-digit positive integer \( n \) such that the sum of the first \( n \) positive integers is not a divisor of the product of the first \( n \) positive integers, we need to check when \( \frac{n(n+1)}{2} \) does not divide \( n! \). This is equivalent to checking when \( n+1 \) does not ... | Okay, so I need to find the greatest three-digit positive integer n where the sum of the first n positive integers is not a divisor of the product of the first n positive integers. The options are 995, 996, 997, 998, and 999. Hmm, let me start by understanding the problem.
First, the sum of the first n positive intege... | 91 |
Return your final response within \boxed{}. In $\triangle ABC$ with right angle at $C$, altitude $CH$ and median $CM$ trisect the right angle. If the area of $\triangle CHM$ is $K$, then the area of $\triangle ABC$ is
$\textbf{(A)}\ 6K\qquad\textbf{(B)}\ 4\sqrt3\ K\qquad\textbf{(C)}\ 3\sqrt3\ K\qquad\textbf{(D)}\ 3K... | Okay, let's tackle this geometry problem step by step. So, we have a right triangle ABC with a right angle at C. The altitude CH and the median CM trisect the right angle. The area of triangle CHM is K, and we need to find the area of triangle ABC in terms of K. The answer choices are given, so that might help later.
... |
Given a right triangle \( \triangle ABC \) with a right angle at \( C \), the altitude \( CH \) and the median \( CM \) trisect the right angle. The area of \( \triangle CHM \) is \( K \), and we need to find the area of \( \triangle ABC \).
1. **Trisection of the Right Angle**:
- The right angle at \( C \) is tr... | Okay, so I need to solve this geometry problem. Let me read it again carefully.
In triangle ABC, which is a right-angled triangle at C, the altitude CH and the median CM trisect the right angle. The area of triangle CHM is K, and I need to find the area of triangle ABC. The options are given from A to E.
First, let's... | 92 |
Return your final response within \boxed{}. Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?
$\textbf{(A)}\ 37\qquad\textbf{(B)}\ 63\qquad\textbf{(C)}\ 11... | Okay, let me try to figure out this problem step by step. So, Claire was adding the degree measures of the interior angles of a convex polygon and got a sum of 2017. But then she realized she forgot one angle. We need to find the measure of that forgotten angle. The answer choices are 37, 63, 117, 143, and 163. Alright... |
Claire calculated the sum of the interior angles of a convex polygon as 2017 degrees, but she missed one angle. The formula for the sum of the interior angles of a convex polygon with \( n \) sides is \((n-2) \times 180\) degrees. Let \( x \) be the measure of the forgotten angle. The correct sum of the interior angl... | Okay, so Claire was adding the degree measures of the interior angles of a convex polygon, but she forgot to include one angle. The sum she got is 2017, and the correct sum should be more. Then, the forgotten angle is the difference between the correct sum and 2017. The answer choices are 37, 63, 117, 143, and 163. Let... | 93 |
Return your final response within \boxed{}. Zara has a collection of $4$ marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this?
$\textbf{(A) }6 \qquad \textbf{(B) }8 ... | Okay, so Zara has four marbles: Aggie, Bumblebee, Steelie, and Tiger. She wants to arrange them in a row on a shelf. But there's a condition: she doesn't want the Steelie and the Tiger to be next to each other. I need to find out how many different ways she can do this. The answer choices are 6, 8, 12, 18, or 24. Let m... |
Zara has four marbles: Aggie (A), Bumblebee (B), Steelie (S), and Tiger (T). We need to find the number of ways to arrange these marbles such that Steelie and Tiger are not next to each other.
First, calculate the total number of permutations without any restrictions:
\[
4! = 24
\]
Next, calculate the number of per... | Okay, so Zara has 4 marbles: Aggie, Bumblebee, Steelie, and Tiger. She wants to arrange them in a row on a shelf. The catch is that she doesn't want the Steelie and the Tiger to be next to each other. The question is asking how many ways she can do this, and the options are from 6 to 24. Alright, let's see. Hmm, permut... | 94 |
Return your final response within \boxed{}. Ray's car averages $40$ miles per gallon of gasoline, and Tom's car averages $10$ miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rate of miles per gallon of gasoline?
$\textbf{(A)}\ 10 \qquad \textbf{(B)}\ 16 \qquad \... | Okay, let's see. The problem is about Ray and Tom's cars and their mileage. The question is asking for the combined rate of miles per gallon for both cars when they each drive the same number of miles. The answer choices are given, so I need to figure out which one is correct.
First, let me parse the information. Ray'... |
Ray's car averages 40 miles per gallon, and Tom's car averages 10 miles per gallon. Both drive the same number of miles. To find the combined rate of miles per gallon, we need to calculate the total miles driven and the total gallons used.
Let \( d \) be the distance each drives in miles.
- Gasoline used by Ray: \(... | Okay, so I need to figure out the combined miles per gallon rate of Ray's and Tom's cars. Let me start by understanding the problem again. Both Ray and Tom drive the same number of miles, but their cars have different fuel efficiencies. Ray's car is more fuel-efficient, averaging 40 miles per gallon, while Tom's is les... | 95 |
Return your final response within \boxed{}. Triangle $ABC$ has vertices $A = (3,0)$, $B = (0,3)$, and $C$, where $C$ is on the line $x + y = 7$. What is the area of $\triangle ABC$?
$\mathrm{(A)}\ 6\qquad \mathrm{(B)}\ 8\qquad \mathrm{(C)}\ 10\qquad \mathrm{(D)}\ 12\qquad \mathrm{(E)}\ 14$ | Okay, so I need to find the area of triangle ABC where points A and B are given, and point C lies on the line x + y = 7. Let me start by visualizing the problem.
First, let's recall the coordinates of the vertices. Point A is at (3, 0), and point B is at (0, 3). Point C is somewhere on the line x + y = 7. Since the l... |
To find the area of triangle \(ABC\) with vertices \(A = (3,0)\), \(B = (0,3)\), and \(C\) on the line \(x + y = 7\), we use the shoelace formula.
The coordinates of \(C\) satisfy \(x + y = 7\). Using the shoelace formula for the area of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):
... | Okay, so I need to find the area of triangle ABC where the vertices are A(3,0), B(0,3), and C is a point on the line x + y = 7. The answer choices are from 6 to 14, so the area isn't too big. Let me start by recalling how to find the area of a triangle given three vertices. The formula is half the base times height, or... | 96 |
Return your final response within \boxed{}. A team won $40$ of its first $50$ games. How many of the remaining $40$ games must this team win so it will have won exactly $70 \%$ of its games for the season?
$\text{(A)}\ 20 \qquad \text{(B)}\ 23 \qquad \text{(C)}\ 28 \qquad \text{(D)}\ 30 \qquad \text{(E)}\ 35$ | Okay, let's see. The problem is about a team that has won 40 out of their first 50 games. They want to know how many more games they need to win out of the remaining 40 games to have a total of 70% wins for the entire season. The answer choices are given from A to E. Hmm, let me break this down step by step.
First, le... |
The team has won 40 out of its first 50 games and has 40 remaining games. The total number of games in the season is \(50 + 40 = 90\). To determine how many games they need to win to achieve exactly 70% of the season games, we calculate 70% of 90:
\[
0.7 \times 90 = 63
\]
They need a total of 63 wins. Since they al... | Okay, so the problem is about a team that has already played 50 games and won 40 of them. They have 40 more games left in the season, and they want to know how many of those they need to win to end up with exactly 70% of all their games won for the entire season. The answer choices are given, so maybe I can work backwa... | 97 |
Return your final response within \boxed{}. For each real number $x$, let $\textbf{[}x\textbf{]}$ be the largest integer not exceeding $x$
(i.e., the integer $n$ such that $n\le x<n+1$). Which of the following statements is (are) true?
$\textbf{I. [}x+1\textbf{]}=\textbf{[}x\textbf{]}+1\text{ for all }x \\ \textbf{II.... | Okay, let's tackle this problem step by step. It's about the floor function, which is denoted here as [x]. The floor function, as I remember, gives the greatest integer less than or equal to x. So, for example, [3.7] would be 3, and [-1.2] would be -2. The question has three statements, I, II, and III, and we need to d... |
To determine which statements about the floor function are true, we analyze each statement systematically:
**Statement I: \([x + 1] = [x] + 1\) for all \(x\)**
- For any real number \(x\), adding 1 to \(x\) increments the integer part of \(x\) by 1 while the fractional part remains the same.
- Testing with various... | Okay, so I need to figure out which of the statements I, II, and III about the floor function are true. Let me start by recalling what the floor function does. The floor of x, written as [x], is the greatest integer less than or equal to x. For example, [3.2] is 3, and [-1.5] is -2. Got that.
Let me tackle each statem... | 98 |
Return your final response within \boxed{}. Positive integers $a$ and $b$ are each less than $6$. What is the smallest possible value for $2 \cdot a - a \cdot b$?
$\textbf{(A)}\ -20\qquad\textbf{{(B)}}\ -15\qquad\textbf{{(C)}}\ -10\qquad\textbf{{(D)}}\ 0\qquad\textbf{{(E)}}\ 2$ | Okay, so I have this problem here: Positive integers a and b are each less than 6. What is the smallest possible value for 2·a - a·b? The options are from A to E, with the answers given. Let me try to figure this out step by step.
First, let me make sure I understand the problem correctly. We need to find the smallest... |
To find the smallest possible value of \(2a - ab\) where \(a\) and \(b\) are positive integers less than 6, we start by rewriting the expression as \(a(2 - b)\).
We analyze the possible values of \(a\) and \(b\) which can be 1, 2, 3, 4, or 5. The term \(2 - b\) can take values 1, 0, -1, -2, or -3 depending on \(b\)... | Okay, so I need to find the smallest possible value of the expression 2a - ab, where a and b are positive integers each less than 6. The answer choices are given from (A) to (E), with the options being -20, -15, -10, 0, and 2. Let me try to figure this out step by step.
First, let's understand the problem. Both a and ... | 99 |
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