problem stringlengths 17 2.73k | answer stringlengths 0 210 | solution stringlengths 0 8.75k | id int64 0 17.6k | pass@32 int64 0 5 | prompt stringlengths 139 2.85k | pass@32_r1 int64 0 32 |
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Let $P(x)$ be a polynomial of degree $3n$ such that
\begin{align*} P(0) = P(3) = \dots = P(3n) &= 2, \\ P(1) = P(4) = \dots = P(3n+1-2) &= 1, \\ P(2) = P(5) = \dots = P(3n+2-2) &= 0. \end{align*}
Also, $P(3n+1) = 730$. Determine $n$. | 1 | To solve for $n$, we start by analyzing the polynomial $P(x)$ given its values at specific points and its degree. We use Lagrange Interpolation Formula to express $P(x)$, and then evaluate it at $x = 3n+1$ to find $n$.
1. **Constructing the Polynomial Using Lagrange Interpolation:**
The polynomial $P(x)$ is defined... | 0 | 2 | <｜begin▁of▁sentence｜><｜User｜>Let $P(x)$ be a polynomial of degree $3n$ such that
\begin{align*} P(0) = P(3) = \dots = P(3n) &= 2, \\ P(1) = P(4) = \dots = P(3n+1-2) &= 1, \\ P(2) = P(5) = \dots = P(3n+2-2) &= 0. \end{align*}
Also, $P(3n+1) = 730$. Determine $n$. Please reason step by step, and put your final answer wit... | 4 |
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 $a=$ | \frac{\sqrt{2}}{2} | 1. **Define the function and apply it to $\sqrt{2}$:**
Given the function $f(x) = ax^2 - \sqrt{2}$, we first find $f(\sqrt{2})$:
\[
f(\sqrt{2}) = a(\sqrt{2})^2 - \sqrt{2} = 2a - \sqrt{2}.
\]
2. **Apply the function $f$ to the result of $f(\sqrt{2})$:**
Next, we need to find $f(f(\sqrt{2}))$:
\[
f(... | 1 | 0 | <｜begin▁of▁sentence｜><｜User｜>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 $a=$ Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
The graph, $G$ of $y=\log_{10}x$ is rotated $90^{\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. What is the equation for $G'$? | 10^{-x} | 1. **Understanding the rotation**: Rotating a point $(x, y)$ $90^\circ$ counterclockwise about the origin results in the point $(-y, x)$. This can be verified using rotation matrices:
\[
\begin{bmatrix}
\cos(90^\circ) & -\sin(90^\circ) \\
\sin(90^\circ) & \cos(90^\circ)
\end{bmatrix}
\begin{bmatrix}
... | 2 | 0 | <｜begin▁of▁sentence｜><｜User｜>The graph, $G$ of $y=\log_{10}x$ is rotated $90^{\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. What is the equation for $G'$? Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\frac{7}{17}$. What number was erased? | 7 | 1. **Identify the sum of the first $n$ positive integers**: The sum of the first $n$ positive integers is given by the formula for the sum of an arithmetic series:
\[
S = \frac{n(n+1)}{2}
\]
2. **Expression for the average after erasing one number**: If one number $x$ is erased, the sum of the remaining numbe... | 3 | 3 | <｜begin▁of▁sentence｜><｜User｜>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? Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜>... | 25 |
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 $(\cos 40^\circ,\sin 40^\circ)$, $(\cos 60^\circ,\sin 60^\circ)$, and $(\cos t^\circ,\sin t^\circ)$ is isosceles? | 380 | Let $A = (\cos 40^\circ, \sin 40^\circ)$, $B = (\cos 60^\circ, \sin 60^\circ)$, and $C = (\cos t^\circ, \sin t^\circ)$. We need to find the values of $t$ such that $\triangle ABC$ is isosceles. We consider three cases based on which sides of the triangle are equal.
**Case 1: $AB = AC$**
For $AB = AC$, $C$ must lie on... | 4 | 0 | <｜begin▁of▁sentence｜><｜User｜>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 $(\cos 40^\circ,\sin 40^\circ)$, $(\cos 60^\circ,\sin 60^\circ)$, and $(\cos t^\circ,\sin t^\circ)$ is isosceles? Please reason step by step, and put your fi... | 17 |
In the adjoining figure, points $B$ and $C$ lie on line segment $AD$, and $AB, BC$, and $CD$ are diameters of circle $O, N$, and $P$, respectively. Circles $O, N$, and $P$ all have radius $15$ and the line $AG$ is tangent to circle $P$ at $G$. If $AG$ intersects circle $N$ at points $E$ and $F$, then chord $EF$ has len... | 20 | 1. **Identify the Configuration**: Given that $AB$, $BC$, and $CD$ are diameters of circles $O$, $N$, and $P$ respectively, and all circles have a radius of $15$. Since $B$ and $C$ lie on $AD$, and each segment is a diameter, $AB = BC = CD = 30$. Therefore, $AD = AB + BC + CD = 30 + 30 + 30 = 90$.
2. **Position of $N$... | 5 | 0 | <｜begin▁of▁sentence｜><｜User｜>In the adjoining figure, points $B$ and $C$ lie on line segment $AD$, and $AB, BC$, and $CD$ are diameters of circle $O, N$, and $P$, respectively. Circles $O, N$, and $P$ all have radius $15$ and the line $AG$ is tangent to circle $P$ at $G$. If $AG$ intersects circle $N$ at points $E$ and... | 0 |
An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at $A$, $B$, and $C$ are $12$, $9$, and $10$ meter... | 17 | 1. **Establishing the Coordinate System and Points**:
Let's assume the side length of the hexagon is $6$ meters for simplicity. We place the hexagon in a 3D coordinate system with $A$ at the origin, i.e., $A = (0, 0, 0)$. The coordinates of $B$ and $C$ can be calculated based on the geometry of a regular hexagon:
... | 6 | 2 | <｜begin▁of▁sentence｜><｜User｜>An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at $A$, $B$, and $C$ ... | 31 |
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 $ABCD?$ | 360 | 1. **Assign Variables and Use Pythagorean Theorem in $\triangle ABC$:**
Let $AB = x$ and $BC = y$. Since $\angle ABC = 90^\circ$, by the Pythagorean theorem, we have:
\[
x^2 + y^2 = AC^2 = 20^2 = 400.
\]
2. **Calculate Area of $\triangle ACD$:**
Since $\angle ACD = 90^\circ$, the area of $\triangle ACD$... | 7 | 2 | <｜begin▁of▁sentence｜><｜User｜>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 $ABCD?$ Please reason step by step, and put your final answer within \boxed{}.<｜Assi... | 15 |
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 $k?$ | \frac{1+\sqrt{5}}{2} | 1. **Identify the Lines and Their Intersection**:
The lines given are $y = x$ and $y = 3x$. Both lines pass through the origin and form an acute angle there.
2. **Calculate the Slopes of the Lines**:
- The slope of the line $y = x$ is $1$.
- The slope of the line $y = 3x$ is $3$.
3. **Determine the Angle Be... | 8 | 0 | <｜begin▁of▁sentence｜><｜User｜>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 $k?$ Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
Consider all 1000-element subsets of the set $\{1, 2, 3, \dots , 2015\}$. 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 $p + q$. | 2016 |
To solve this problem, we need to find the arithmetic mean of the least elements of all 1000-element subsets of the set $\{1, 2, 3, \ldots, 2015\}$. We will use combinatorial arguments to derive the solution.
#### Step 1: Counting subsets with a fixed least element
Let $i$ be the least element of a 1000-element subse... | 9 | 0 | <｜begin▁of▁sentence｜><｜User｜>Consider all 1000-element subsets of the set $\{1, 2, 3, \dots , 2015\}$. 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 $p + q$. Please reason step by ste... | 0 |
If $x, 2x+2, 3x+3, \dots$ are in geometric progression, the fourth term is: | -13\frac{1}{2} | 1. **Identify the nature of the sequence**: Given that $x, 2x+2, 3x+3, \dots$ are in geometric progression, the ratio between consecutive terms must be constant. Let's denote this common ratio by $r$.
2. **Set up the equation for the common ratio**:
\[
\frac{2x+2}{x} = \frac{3x+3}{2x+2}
\]
This equation ar... | 10 | 0 | <｜begin▁of▁sentence｜><｜User｜>If $x, 2x+2, 3x+3, \dots$ are in geometric progression, the fourth term is: Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
At $2:15$ o'clock, the hour and minute hands of a clock form an angle of: | 22\frac {1}{2}^{\circ} | To find the angle between the hour and minute hands of a clock at a given time, we can use the formula:
\[
\text{Angle} = \left|\frac{60h - 11m}{2}\right|
\]
where $h$ is the hour and $m$ is the minute.
1. **Identify the values of $h$ and $m$:**
At $2:15$, $h = 2$ and $m = 15$.
2. **Substitute these values into th... | 11 | 0 | <｜begin▁of▁sentence｜><｜User｜>At $2:15$ o'clock, the hour and minute hands of a clock form an angle of: Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks at the ideal rate, what number of words would be an appropriate length for her speech? | 5650 | 1. **Convert Time to Minutes**:
- One-half hour is $\frac{1}{2} \times 60 = 30$ minutes.
- Three-quarters of an hour is $\frac{3}{4} \times 60 = 45$ minutes.
2. **Calculate the Range of Words**:
- At the ideal rate of 150 words per minute, the minimum number of words Julie should speak is:
\[
30 \t... | 12 | 0 | <｜begin▁of▁sentence｜><｜User｜>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, what number of words would be an appropriate length for her speech? Please reason step... | 0 |
If the following instructions are carried out by a computer, what value of \(X\) will be printed because of instruction \(5\)?
1. START \(X\) AT \(3\) AND \(S\) AT \(0\).
2. INCREASE THE VALUE OF \(X\) BY \(2\).
3. INCREASE THE VALUE OF \(S\) BY THE VALUE OF \(X\).
4. IF \(S\) IS AT LEAST \(10000\),
TH... | 23 | 1. **Initialization**: The program starts with $X = 3$ and $S = 0$.
2. **Loop Execution**:
- **Instruction 2**: Increase the value of $X$ by $2$.
- **Instruction 3**: Increase the value of $S$ by the current value of $X$.
- **Instruction 4**: Check if $S \geq 10000$. If true, go to instruction 5; otherwise, ... | 13 | 0 | <｜begin▁of▁sentence｜><｜User｜>If the following instructions are carried out by a computer, what 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 ... | 0 |
The sum of the greatest integer less than or equal to $x$ and the least integer greater than or equal to $x$ is $5$. The solution set for $x$ is | \{x \mid 2 < x < 3\} | 1. **Understanding the Problem:**
The problem asks us to find the set of values for $x$ such that the sum of the greatest integer less than or equal to $x$ (denoted $\lfloor x \rfloor$) and the least integer greater than or equal to $x$ (denoted $\lceil x \rceil$) equals 5.
2. **Analyzing the Floor and Ceiling Func... | 14 | 0 | <｜begin▁of▁sentence｜><｜User｜>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 Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
A powderman set a fuse for a blast to take place in $30$ seconds. He ran away at a rate of $8$ yards per second. Sound travels at the rate of $1080$ feet per second. When the powderman heard the blast, he had run approximately: | 245 yd. | 1. **Define the problem in terms of equations:**
- Let $p(t)$ be the distance in feet that the powderman is from the starting point at time $t$ seconds after the fuse is lit. Since he runs at $8$ yards per second, and there are $3$ feet in a yard, he runs $8 \times 3 = 24$ feet per second. Therefore, $p(t) = 24t$.
... | 15 | 2 | <｜begin▁of▁sentence｜><｜User｜>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: Please reason step by step, and put your final answer within \b... | 21 |
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 $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ... | 192 | 1. **Understanding the Problem**: We are given four circles with centers at $A$, $B$, $C$, and $D$. Points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and similarly for circles $C$ and $D$. The distances $AB$ and $CD$ are both 39, and the length of segm... | 16 | 2 | <｜begin▁of▁sentence｜><｜User｜>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 $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthe... | 10 |
Medians $AD$ and $CE$ of $\triangle ABC$ intersect in $M$. The midpoint of $AE$ is $N$.
Let the area of $\triangle MNE$ be $k$ times the area of $\triangle ABC$. Then $k$ equals: | \frac{1}{6} | 1. **Identify the relationships between the triangles**:
- Since $AD$ and $CE$ are medians, point $M$ is the centroid of $\triangle ABC$. The centroid divides each median into two segments, one of which is twice the length of the other, specifically, the segment connecting the centroid to the midpoint of a side is ... | 17 | 4 | <｜begin▁of▁sentence｜><｜User｜>Medians $AD$ and $CE$ of $\triangle ABC$ intersect in $M$. The midpoint of $AE$ is $N$.
Let the area of $\triangle MNE$ be $k$ times the area of $\triangle ABC$. Then $k$ equals: Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\frac34$ of the probability of rolling a sum of $10,$ and the probability of rolling a sum of $12$ is $\frac{... | 17 | 1. **Define Variables:**
Let the number of faces on the two dice be $a$ and $b$, respectively, with $a \geq b$. Assume each die has distinct integers from $1$ to the number of faces on that die.
2. **Analyze the Probability of Rolling a Sum of $7$:**
Since each die has at least $6$ faces, there are always $6$ wa... | 18 | 2 | <｜begin▁of▁sentence｜><｜User｜>Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\frac34$ of the probability of rolling a sum of $10,$ and the probability of rol... | 14 |
Two long cylindrical tubes of the same length but different diameters lie parallel to each other on a flat surface. The larger tube has radius $72$ and rolls along the surface toward the smaller tube, which has radius $24$. It rolls over the smaller tube and continues rolling along the flat surface until it comes to re... | 312 | 1. **Understanding the Problem:**
Two cylindrical tubes, one with radius 72 and the other with radius 24, are lying parallel on a flat surface. The larger tube rolls over the smaller tube and completes one full revolution, returning to the same point on its circumference, without slipping.
2. **Initial Setup:**
... | 19 | 0 | <｜begin▁of▁sentence｜><｜User｜>Two long cylindrical tubes of the same length but different diameters lie parallel to each other on a flat surface. The larger tube has radius $72$ and rolls along the surface toward the smaller tube, which has radius $24$. It rolls over the smaller tube and continues rolling along the flat... | 0 |
For all non-zero numbers $x$ and $y$ such that $x = 1/y$, $\left(x-\frac{1}{x}\right)\left(y+\frac{1}{y}\right)$ equals | x^2-y^2 | 1. **Substitute $y$ in terms of $x$:** Given $x = \frac{1}{y}$, we can rewrite $y$ as $y = \frac{1}{x}$.
2. **Substitute and simplify the expression:** We start by substituting $y = \frac{1}{x}$ into the expression $\left(x-\frac{1}{x}\right)\left(y+\frac{1}{y}\right)$:
\[
\left(x - \frac{1}{x}\right)\left(\fra... | 20 | 0 | <｜begin▁of▁sentence｜><｜User｜>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 Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
The values of $k$ for which the equation $2x^2-kx+x+8=0$ will have real and equal roots are: | 9 and -7 | To find the values of $k$ for which the quadratic equation $2x^2 - kx + x + 8 = 0$ has real and equal roots, we first simplify the equation:
\[ 2x^2 - kx + x + 8 = 0 \]
\[ 2x^2 + (1-k)x + 8 = 0 \]
For a quadratic equation $ax^2 + bx + c = 0$, the condition for the roots to be real and equal is given by the discriminan... | 21 | 0 | <｜begin▁of▁sentence｜><｜User｜>The values of $k$ for which the equation $2x^2-kx+x+8=0$ will have real and equal roots are: Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
A line that passes through the origin intersects both the line $x = 1$ and the line $y=1+ \frac{\sqrt{3}}{3} x$. The three lines create an equilateral triangle. What is the perimeter of the triangle? | 3 + 2\sqrt{3} | 1. **Identify the slopes of the lines**:
- The line $y = 1 + \frac{\sqrt{3}}{3}x$ has a slope of $\frac{\sqrt{3}}{3}$.
- Since the triangle formed is equilateral, and one side is vertical ($x=1$), the line passing through the origin that forms another side of the triangle must have a slope that is the negative r... | 22 | 0 | <｜begin▁of▁sentence｜><｜User｜>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? Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\... | 2 |
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? | 12 | 1. **Define the problem in terms of algebra**: Let the two-digit number Mary thought of be $x$. According to the problem, she performs the following operations on $x$:
- Multiplies by $3$: $3x$
- Adds $11$: $3x + 11$
- Switches the digits of the result.
2. **Analyze the range of the final number**: The switch... | 23 | 4 | <｜begin▁of▁sentence｜><｜User｜>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? Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><... | 32 |
Trapezoid $ABCD$ has $\overline{AB} \parallel \overline{CD}, BC=CD=43$, and $\overline{AD} \perp \overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP=11$, the length of $AD$ can be written in the form $m\sqrt{n}$... | 194 | 1. **Identify the properties of the trapezoid**: Given that $ABCD$ is a trapezoid with $\overline{AB}\parallel\overline{CD}$ and $BC=CD=43$. Also, $\overline{AD}\perp\overline{BD}$, which implies that $\triangle ABD$ is a right triangle.
2. **Examine the diagonals and intersection**: The diagonals $\overline{AC}$ and ... | 24 | 0 | <｜begin▁of▁sentence｜><｜User｜>Trapezoid $ABCD$ has $\overline{AB} \parallel \overline{CD}, BC=CD=43$, and $\overline{AD} \perp \overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP=11$, the length of $AD$ can be wr... | 5 |
If $q_1(x)$ and $r_1$ are the quotient and remainder, respectively, when the polynomial $x^8$ is divided by $x + \frac{1}{2}$, and if $q_2(x)$ and $r_2$ are the quotient and remainder, respectively, when $q_1(x)$ is divided by $x + \frac{1}{2}$, then $r_2$ equals | -\frac{1}{16} | 1. **Divide \(x^8\) by \(x + \frac{1}{2}\):**
We use synthetic division to divide \(x^8\) by \(x + \frac{1}{2}\). Setting \(x + \frac{1}{2} = 0\) gives \(x = -\frac{1}{2}\). We substitute \(x = -\frac{1}{2}\) into \(x^8\) to find the remainder:
\[
\left(-\frac{1}{2}\right)^8 = \frac{1}{256}
\]
Thus, the ... | 25 | 2 | <｜begin▁of▁sentence｜><｜User｜>If $q_1(x)$ and $r_1$ are the quotient and remainder, respectively, when the polynomial $x^8$ is divided by $x + \frac{1}{2}$, and if $q_2(x)$ and $r_2$ are the quotient and remainder, respectively, when $q_1(x)$ is divided by $x + \frac{1}{2}$, then $r_2$ equals Please reason step by step,... | 29 |
In the cube $ABCDEFGH$ with opposite vertices $C$ and $E,$ $J$ and $I$ are the midpoints of segments $\overline{FB}$ and $\overline{HD},$ respectively. Let $R$ be the ratio of the area of the cross-section $EJCI$ to the area of one of the faces of the cube. What is $R^2?$ | \frac{9}{4} | 1. **Assign Coordinates to Vertices:**
Let's consider a cube $ABCDEFGH$ with side length $a$. We can assign coordinates to each vertex assuming $A$ as the origin $(0,0,0)$, and aligning the cube along the axes:
- $A = (0,0,0)$
- $B = (a,0,0)$
- $C = (a,0,a)$
- $D = (0,0,a)$
- $E = (0,a,0)$
- $F = (... | 26 | 0 | <｜begin▁of▁sentence｜><｜User｜>In the cube $ABCDEFGH$ with opposite vertices $C$ and $E,$ $J$ and $I$ are the midpoints of segments $\overline{FB}$ and $\overline{HD},$ respectively. Let $R$ be the ratio of the area of the cross-section $EJCI$ to the area of one of the faces of the cube. What is $R^2?$ Please reason step... | 0 |
Quadrilateral $ABCD$ has $AB = BC = CD$, $m\angle ABC = 70^\circ$ and $m\angle BCD = 170^\circ$. What is the degree measure of $\angle BAD$? | 85 | 1. **Assign the unknown and draw necessary diagonals**: Let $\angle BAD = x$. Draw diagonals $BD$ and $AC$. Let $I$ be the intersection of diagonals $BD$ and $AC$.
2. **Analyze the isosceles triangles**: Since $AB = BC = CD$, triangles $\triangle ABC$ and $\triangle BCD$ are isosceles. Therefore, $\angle DBC = \angle ... | 27 | 0 | <｜begin▁of▁sentence｜><｜User｜>Quadrilateral $ABCD$ has $AB = BC = CD$, $m\angle ABC = 70^\circ$ and $m\angle BCD = 170^\circ$. What is the degree measure of $\angle BAD$? Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 12 |
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
\[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?\] | \frac{7}{16} | 1. **Express $(2+i)$ in polar form**:
We start by expressing the complex number $2+i$ in polar form. We calculate the modulus and the argument of $2+i$:
\[ |2+i| = \sqrt{2^2 + 1^2} = \sqrt{5}, \]
\[ \theta = \arctan\left(\frac{1}{2}\right). \]
Therefore, we can write:
\[ 2+i = \sqrt{5} \left(\cos \theta... | 28 | 5 | <｜begin▁of▁sentence｜><｜User｜>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
\[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?\] Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 26 |
If $\frac{1}{x} - \frac{1}{y} = \frac{1}{z}$, then $z$ equals: | \frac{xy}{y - x} | 1. Start with the given equation:
\[
\frac{1}{x} - \frac{1}{y} = \frac{1}{z}
\]
2. Find a common denominator for the left-hand side:
\[
\frac{y}{xy} - \frac{x}{xy} = \frac{1}{z}
\]
3. Combine the fractions:
\[
\frac{y-x}{xy} = \frac{1}{z}
\]
4. To isolate $z$, take the reciprocal of both s... | 29 | 0 | <｜begin▁of▁sentence｜><｜User｜>If $\frac{1}{x} - \frac{1}{y} = \frac{1}{z}$, then $z$ equals: Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
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 and Carlos had was $7$ dollars, between Car... | 10 | Let's denote the amount of money each person has as follows:
- Ashley: $A$
- Betty: $B$
- Carlos: $C$
- Dick: $D$
- Elgin: $E$
From the problem, we have the following equations based on the absolute differences:
1. $|A - B| = 19$
2. $|B - C| = 7$
3. $|C - D| = 5$
4. $|D - E| = 4$
5. $|E - A| = 11$
Additionally, we kn... | 30 | 0 | <｜begin▁of▁sentence｜><｜User｜>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 and Carlos had... | 30 |
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 | 4K | 1. **Draw the triangle and identify key components**: In $\triangle ABC$, where $\angle C = 90^\circ$, draw altitude $CH$ from $C$ to hypotenuse $AB$ and median $CM$ from $C$ to the midpoint $M$ of $AB$. Since $CM$ trisects the right angle, $\angle MCB = 30^\circ$ and $\angle MCH = 15^\circ$.
2. **Properties of median... | 31 | 0 | <｜begin▁of▁sentence｜><｜User｜>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 Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
In this diagram, not drawn to scale, Figures $I$ and $III$ are equilateral triangular regions with respective areas of $32\sqrt{3}$ and $8\sqrt{3}$ square inches. Figure $II$ is a square region with area $32$ square inches. Let the length of segment $AD$ be decreased by $12\frac{1}{2}$% of itself, while the lengths of ... | 25 | To solve this problem, we first need to determine the side lengths of the equilateral triangles and the square based on the given areas.
1. **Calculate the side length of the equilateral triangles:**
- For an equilateral triangle, the area $A$ is given by $A = \frac{\sqrt{3}}{4} s^2$, where $s$ is the side length.
... | 32 | 1 | <｜begin▁of▁sentence｜><｜User｜>In this diagram, not drawn to scale, Figures $I$ and $III$ are equilateral triangular regions with respective areas of $32\sqrt{3}$ and $8\sqrt{3}$ square inches. Figure $II$ is a square region with area $32$ square inches. Let the length of segment $AD$ be decreased by $12\frac{1}{2}$% of ... | 0 |
In square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$, respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at right angles at $R$, with $BR = 6$ and $PR = 7$. What is the area of the square? | 117 | 1. **Identify Similar Triangles**: Notice that $\triangle CRB \sim \triangle BAP$ by AA similarity (both have a right angle and share angle $BRP$).
2. **Set Up Ratio of Sides**: From the similarity, we have the ratio of corresponding sides:
\[
\frac{CB}{CR} = \frac{PB}{AB}
\]
Since $CB = AB = s$ (side leng... | 33 | 0 | <｜begin▁of▁sentence｜><｜User｜>In square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$, respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at right angles at $R$, with $BR = 6$ and $PR = 7$. What is the area of the square? Please reason step by step, and put your final answer wit... | 22 |
Sides $\overline{AB}$ and $\overline{AC}$ of equilateral triangle $ABC$ are tangent to a circle at points $B$ and $C$ respectively. What fraction of the area of $\triangle ABC$ lies outside the circle? | \frac{4}{3}-\frac{4\sqrt{3}\pi}{27} | 1. **Identify the Geometry and Relationships**:
- Let the radius of the circle be $r$, and its center be $O$.
- Since $\overline{AB}$ and $\overline{AC}$ are tangent to the circle at points $B$ and $C$, respectively, $\angle OBA = \angle OCA = 90^\circ$.
- Triangle $OBC$ is isosceles with $\overline{OB} = \ove... | 34 | 0 | <｜begin▁of▁sentence｜><｜User｜>Sides $\overline{AB}$ and $\overline{AC}$ of equilateral triangle $ABC$ are tangent to a circle at points $B$ and $C$ respectively. What fraction of the area of $\triangle ABC$ lies outside the circle? Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think... | 0 |
In $\triangle PQR$, $PR=15$, $QR=20$, and $PQ=25$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, and points $E$ and $F$ lie on $\overline{PR}$, with $PA=QB=QC=RD=RE=PF=5$. Find the area of hexagon $ABCDEF$. | 150 | 1. **Identify the type of triangle $PQR$**:
Given $PR=15$, $QR=20$, and $PQ=25$, we check if $\triangle PQR$ is a right triangle using the Pythagorean theorem:
\[
PR^2 + QR^2 = 15^2 + 20^2 = 225 + 400 = 625 = 25^2 = PQ^2.
\]
Since $PR^2 + QR^2 = PQ^2$, $\triangle PQR$ is a right triangle with $PQ$ as the... | 35 | 1 | <｜begin▁of▁sentence｜><｜User｜>In $\triangle PQR$, $PR=15$, $QR=20$, and $PQ=25$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, and points $E$ and $F$ lie on $\overline{PR}$, with $PA=QB=QC=RD=RE=PF=5$. Find the area of hexagon $ABCDEF$. Please reason step by step, and put your fin... | 0 |
Adams plans a profit of $10$ % on the selling price of an article and his expenses are $15$ % of sales. The rate of markup on an article that sells for $ $5.00$ is: | 33\frac {1}{3}\% | 1. **Define Variables:**
Let $C$ be the cost of the product and $S$ be the selling price. According to the problem, Adams plans a profit of 10% on the selling price, and his expenses are 15% of the selling price.
2. **Express the Relationships:**
The profit is 10% of the selling price, so the profit amount is $0... | 36 | 0 | <｜begin▁of▁sentence｜><｜User｜>Adams plans a profit of $10$ % on the selling price of an article and his expenses are $15$ % of sales. The rate of markup on an article that sells for $ $5.00$ is: Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$? | 19 | To solve this problem, we need to consider the possible number of intersection points formed by four distinct lines in a plane. Each pair of lines can intersect at most once, and the maximum number of intersection points is determined by the number of ways to choose 2 lines from 4, which is given by the binomial coeffi... | 37 | 1 | <｜begin▁of▁sentence｜><｜User｜>For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$? Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 23 |
Let $f(t)=\frac{t}{1-t}$, $t \not= 1$. If $y=f(x)$, then $x$ can be expressed as | -f(-y) | 1. Given the function $f(t) = \frac{t}{1-t}$, we know that $y = f(x)$, which implies:
\[
y = \frac{x}{1-x}
\]
2. We rearrange the equation to solve for $x$ in terms of $y$:
\[
y(1-x) = x \implies y - yx = x \implies y = x + yx \implies y = x(1+y)
\]
\[
x = \frac{y}{1+y}
\]
3. We need to fin... | 38 | 0 | <｜begin▁of▁sentence｜><｜User｜>Let $f(t)=\frac{t}{1-t}$, $t \not= 1$. If $y=f(x)$, then $x$ can be expressed as Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
If $x \geq 0$, then $\sqrt{x\sqrt{x\sqrt{x}}} =$ | $\sqrt[8]{x^7}$ | 1. **Understanding the expression**: We start with the expression $\sqrt{x\sqrt{x\sqrt{x}}}$. This involves nested square roots, which can be simplified using the property that $\sqrt{y} = y^{\frac{1}{2}}$ for all $y \geq 0$.
2. **Simplifying the innermost square root**: Begin by simplifying the innermost square root:... | 39 | 0 | <｜begin▁of▁sentence｜><｜User｜>If $x \geq 0$, then $\sqrt{x\sqrt{x\sqrt{x}}} =$ Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
There are $120$ seats in a row. What is the fewest number of seats that must be occupied so the next person to be seated must sit next to someone? | 40 | To solve this problem, we need to find the minimum number of people that must be seated such that any additional person must sit next to someone already seated. We aim to maximize the number of empty seats between seated people under this constraint.
1. **Understanding the Pattern**:
- If we place a person in a se... | 40 | 1 | <｜begin▁of▁sentence｜><｜User｜>There are $120$ seats in a row. What is the fewest number of seats that must be occupied so the next person to be seated must sit next to someone? Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 28 |
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of $6$ meters, and it takes her $36$ seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in me... | \frac{\pi}{3} |
Let's denote Keiko's speed as $s$ meters per second. The track consists of two straight sections and two semicircular ends. The width of the track is $6$ meters, which means the difference in radius between the inner and outer semicircles is $6$ meters.
#### Step 1: Define the dimensions of the track
Let $b$ be the r... | 41 | 0 | <｜begin▁of▁sentence｜><｜User｜>Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of $6$ meters, and it takes her $36$ seconds longer to walk around the outside edge of the track than around the inside edge... | 0 |
Semicircles $POQ$ and $ROS$ pass through the center $O$. What is the ratio of the combined areas of the two semicircles to the area of circle $O$? | \frac{1}{2} | 1. **Identify the radii of the semicircles and the circle**:
- Assume the radius of each semicircle $POQ$ and $ROS$ is $r$. Since these semicircles pass through the center $O$ of the circle, and each semicircle's diameter is a chord of the circle that passes through $O$, the radius of the circle $O$ must be $r$ as ... | 42 | 1 | <｜begin▁of▁sentence｜><｜User｜>Semicircles $POQ$ and $ROS$ pass through the center $O$. What is the ratio of the combined areas of the two semicircles to the area of circle $O$? Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 30 |
An automobile travels $a/6$ feet in $r$ seconds. If this rate is maintained for $3$ minutes, how many yards does it travel in $3$ minutes? | \frac{10a}{r} | 1. **Identify the rate of travel**: The automobile travels $\frac{a}{6}$ feet in $r$ seconds. Thus, the rate of travel is:
\[
\text{Rate} = \frac{\frac{a}{6} \text{ feet}}{r \text{ seconds}}
\]
2. **Convert the rate to yards per second**: Since there are 3 feet in a yard, we convert feet to yards:
\[
\t... | 43 | 0 | <｜begin▁of▁sentence｜><｜User｜>An automobile travels $a/6$ feet in $r$ seconds. If this rate is maintained for $3$ minutes, how many yards does it travel in $3$ minutes? Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
For real numbers $a$ and $b$, define $a \diamond b = \sqrt{a^2 + b^2}$. What is the value of $(5 \diamond 12) \diamond ((-12) \diamond (-5))$? | 13\sqrt{2} | 1. **Calculate $5 \diamond 12$:**
\[
5 \diamond 12 = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13
\]
2. **Calculate $(-12) \diamond (-5)$:**
\[
(-12) \diamond (-5) = \sqrt{(-12)^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13
\]
3. **Apply the operation $\diamond$ to the results from steps ... | 44 | 0 | <｜begin▁of▁sentence｜><｜User｜>For real numbers $a$ and $b$, define $a \diamond b = \sqrt{a^2 + b^2}$. What is the value of $(5 \diamond 12) \diamond ((-12) \diamond (-5))$? Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than $2019$? | 22 | To find the greatest possible sum of the digits in the base-seven representation of a positive integer less than $2019$, we first need to determine the maximum number of digits that a base-seven number can have without exceeding $2019$ in decimal.
1. **Convert $2019$ to base-seven:**
- The largest power of $7$ less... | 45 | 0 | <｜begin▁of▁sentence｜><｜User｜>What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than $2019$? Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 10 |
The measures of the interior angles of a convex polygon of $n$ sides are in arithmetic progression. If the common difference is $5^{\circ}$ and the largest angle is $160^{\circ}$, then $n$ equals: | 16 | 1. **Identify the formula for the sum of interior angles of a polygon**: The sum of the interior angles of a polygon with $n$ sides is given by the formula:
\[
S = 180^\circ (n-2)
\]
2. **Set up the arithmetic sequence**: Given that the angles are in arithmetic progression with a common difference of $5^\circ... | 46 | 1 | <｜begin▁of▁sentence｜><｜User｜>The measures of the interior angles of a convex polygon of $n$ sides are in arithmetic progression. If the common difference is $5^{\circ}$ and the largest angle is $160^{\circ}$, then $n$ equals: Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
The fraction $\frac{a^{-4}-b^{-4}}{a^{-2}-b^{-2}}$ is equal to: | a^{-2}+b^{-2} | 1. **Rewrite the numerator using the difference of squares formula**:
The difference of squares formula states that $x^2 - y^2 = (x+y)(x-y)$. Applying this to $a^{-4} - b^{-4}$, where $x = a^{-2}$ and $y = b^{-2}$, we get:
\[
a^{-4} - b^{-4} = (a^{-2} - b^{-2})(a^{-2} + b^{-2})
\]
2. **Substitute the rewr... | 47 | 0 | <｜begin▁of▁sentence｜><｜User｜>The fraction $\frac{a^{-4}-b^{-4}}{a^{-2}-b^{-2}}$ is equal to: Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the side... | 2(w+h)^2 | 1. **Understanding the Problem Setup**: We have a square sheet of wrapping paper and a box with a square base of side $w$ and height $h$. The box is placed such that its base vertices lie on the midlines of the wrapping paper. The wrapping paper is folded up to meet at a point $A$ at the center of the top of the box.
... | 48 | 0 | <｜begin▁of▁sentence｜><｜User｜>A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are ... | 7 |
Of the following expressions the one equal to $\frac{a^{-1}b^{-1}}{a^{-3} - b^{-3}}$ is: | \frac{a^2b^2}{b^3 - a^3} | 1. Start by rewriting the given expression using the properties of exponents:
\[
\frac{a^{-1}b^{-1}}{a^{-3} - b^{-3}} = \frac{\frac{1}{ab}}{\frac{1}{a^3} - \frac{1}{b^3}}
\]
2. To simplify the expression, multiply the numerator and the denominator by $a^3b^3$ (the least common multiple of $a^3$ and $b^3$):
... | 49 | 0 | <｜begin▁of▁sentence｜><｜User｜>Of the following expressions the one equal to $\frac{a^{-1}b^{-1}}{a^{-3} - b^{-3}}$ is: Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
Frieda the frog begins a sequence of hops on a $3 \times 3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite edge... | \frac{25}{32} | To solve this problem, we will calculate the probability that Frieda reaches a corner square within four hops, starting from the center of a $3 \times 3$ grid. We will use a state-based approach to model Frieda's possible positions and transitions.
#### Definitions:
- **State**: Represents Frieda's position on the gri... | 50 | 0 | <｜begin▁of▁sentence｜><｜User｜>Frieda the frog begins a sequence of hops on a $3 \times 3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" a... | 18 |
How many integers $n \geq 2$ are there such that whenever $z_1, z_2, \dots, z_n$ are complex numbers such that
\[|z_1| = |z_2| = \dots = |z_n| = 1 \text{ and } z_1 + z_2 + \dots + z_n = 0,\]
then the numbers $z_1, z_2, \dots, z_n$ are equally spaced on the unit circle in the complex plane? | 2 | 1. **Case $n=2$:**
Consider $z_1$ and $z_2$ on the unit circle such that $|z_1| = |z_2| = 1$ and $z_1 + z_2 = 0$. This implies $z_2 = -z_1$. Since both $z_1$ and $-z_1$ lie on the unit circle and are diametrically opposite, they are equally spaced on the unit circle.
2. **Case $n=3$:**
Without loss of genera... | 51 | 5 | <｜begin▁of▁sentence｜><｜User｜>How many integers $n \geq 2$ are there such that whenever $z_1, z_2, \dots, z_n$ are complex numbers such that
\[|z_1| = |z_2| = \dots = |z_n| = 1 \text{ and } z_1 + z_2 + \dots + z_n = 0,\]
then the numbers $z_1, z_2, \dots, z_n$ are equally spaced on the unit circle in the complex p... | 27 |
If $x$ cows give $x+1$ cans of milk in $x+2$ days, how many days will it take $x+3$ cows to give $x+5$ cans of milk? | \frac{x(x+2)(x+5)}{(x+1)(x+3)} | 1. **Understanding the problem**: We are given that $x$ cows produce $x+1$ cans of milk in $x+2$ days. We need to find out how many days it will take for $x+3$ cows to produce $x+5$ cans of milk.
2. **Calculate the daily milk production per cow**:
- The daily production per cow can be calculated by dividing the to... | 52 | 0 | <｜begin▁of▁sentence｜><｜User｜>If $x$ cows give $x+1$ cans of milk in $x+2$ days, how many days will it take $x+3$ cows to give $x+5$ cans of milk? Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
A $6$-inch and $18$-inch diameter poles are placed together and bound together with wire.
The length of the shortest wire that will go around them is: | 12\sqrt{3}+14\pi | 1. **Understanding the Setup**: We have two poles with diameters of $6$ inches and $18$ inches, respectively. This means the radii are $3$ inches and $9$ inches. When bound together, the wire will wrap around the outer edges of both poles.
2. **Visualizing the Geometry**: The poles are placed side by side. The wire wi... | 53 | 0 | <｜begin▁of▁sentence｜><｜User｜>A $6$-inch and $18$-inch diameter poles are placed together and bound together with wire.
The length of the shortest wire that will go around them is: Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 2 |
Odell and Kershaw run for $30$ minutes on a circular track. Odell runs clockwise at $250 m/min$ and uses the inner lane with a radius of $50$ meters. Kershaw runs counterclockwise at $300 m/min$ and uses the outer lane with a radius of $60$ meters, starting on the same radial line as Odell. How many times after the sta... | 47 | 1. **Determine the Circumference of Each Track**:
- Odell's track radius = $50$ meters, so the circumference is $C_O = 2\pi \times 50 = 100\pi$ meters.
- Kershaw's track radius = $60$ meters, so the circumference is $C_K = 2\pi \times 60 = 120\pi$ meters.
2. **Calculate the Speed in Terms of Radians per Minute**... | 54 | 2 | <｜begin▁of▁sentence｜><｜User｜>Odell and Kershaw run for $30$ minutes on a circular track. Odell runs clockwise at $250 m/min$ and uses the inner lane with a radius of $50$ meters. Kershaw runs counterclockwise at $300 m/min$ and uses the outer lane with a radius of $60$ meters, starting on the same radial line as Odell.... | 32 |
650 students were surveyed about their pasta preferences. The choices were lasagna, manicotti, ravioli and spaghetti. The results of the survey are displayed in the bar graph. What is the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti? | \frac{5}{2} | 1. **Identify the number of students who preferred each pasta type:** According to the problem, the number of students who preferred spaghetti is 250 and the number of students who preferred manicotti is 100.
2. **Set up the ratio:** The ratio of the number of students who preferred spaghetti to the number of students... | 55 | 1 | <｜begin▁of▁sentence｜><｜User｜>650 students were surveyed about their pasta preferences. The choices were lasagna, manicotti, ravioli and spaghetti. The results of the survey are displayed in the bar graph. What is the ratio of the number of students who preferred spaghetti to the number of students who preferred manicot... | 0 |
In a magic triangle, each of the six whole numbers $10-15$ is placed in one of the circles so that the sum, $S$, of the three numbers on each side of the triangle is the same. The largest possible value for $S$ is
[asy] draw(circle((0,0),1)); draw(dir(60)--6*dir(60)); draw(circle(7*dir(60),1)); draw(8*dir(60)--13*dir(... | 39 | 1. **Assign Variables to Circles**: Let the numbers in the circles be $a$, $b$, $c$, $d$, $e$, and $f$ starting from the top circle and moving clockwise.
2. **Set Up Equations for Each Side of the Triangle**:
- The sum of the numbers on the first side is $S = a + b + c$.
- The sum of the numbers on the second si... | 56 | 5 | <｜begin▁of▁sentence｜><｜User｜>In a magic triangle, each of the six whole numbers $10-15$ is placed in one of the circles so that the sum, $S$, of the three numbers on each side of the triangle is the same. The largest possible value for $S$ is
[asy] draw(circle((0,0),1)); draw(dir(60)--6*dir(60)); draw(circle(7*dir(60)... | 29 |
In parallelogram $ABCD$, $\overline{DE}$ is the altitude to the base $\overline{AB}$ and $\overline{DF}$ is the altitude to the base $\overline{BC}$. [Note: Both pictures represent the same parallelogram.] If $DC=12$, $EB=4$, and $DE=6$, then $DF=$ | 6.4 | To solve for $DF$, we first need to find the length of $AB$ using the information given about $EB$ and the properties of a parallelogram.
1. **Identify the length of $AB$**:
Since $ABCD$ is a parallelogram, opposite sides are equal, so $AB = DC = 12$.
2. **Determine the full length of $AE$**:
Since $EB = 4$ and... | 57 | 0 | <｜begin▁of▁sentence｜><｜User｜>In parallelogram $ABCD$, $\overline{DE}$ is the altitude to the base $\overline{AB}$ and $\overline{DF}$ is the altitude to the base $\overline{BC}$. [Note: Both pictures represent the same parallelogram.] If $DC=12$, $EB=4$, and $DE=6$, then $DF=$ Please reason step by step, and put your ... | 0 |
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions? | 28 | We need to consider different seating arrangements for Alice, as her position affects the seating of the others due to her restrictions with Bob and Carla.
1. **Alice sits in the center chair (3rd position):**
- The 2nd and 4th chairs must be occupied by Derek and Eric in either order because Alice cannot sit next ... | 58 | 0 | <｜begin▁of▁sentence｜><｜User｜>Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions? Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 16 |
If $P_1P_2P_3P_4P_5P_6$ is a regular hexagon whose apothem (distance from the center to midpoint of a side) is $2$, and $Q_i$ is the midpoint of side $P_iP_{i+1}$ for $i=1,2,3,4$, then the area of quadrilateral $Q_1Q_2Q_3Q_4$ is | 4\sqrt{3} | 1. **Identify the Geometry of the Hexagon and Midpoints:**
Given a regular hexagon $P_1P_2P_3P_4P_5P_6$, each side of the hexagon is equal, and each internal angle is $120^\circ$. The apothem, which is the distance from the center to the midpoint of any side, is given as $2$.
2. **Calculate the Side Length of the H... | 59 | 1 | <｜begin▁of▁sentence｜><｜User｜>If $P_1P_2P_3P_4P_5P_6$ is a regular hexagon whose apothem (distance from the center to midpoint of a side) is $2$, and $Q_i$ is the midpoint of side $P_iP_{i+1}$ for $i=1,2,3,4$, then the area of quadrilateral $Q_1Q_2Q_3Q_4$ is Please reason step by step, and put your final answer within \... | 0 |
Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between 0 and 1, and Bob announce... | \frac{13}{24} | Let $a$, $b$, and $c$ be the numbers that Alice, Bob, and Carol choose, respectively. Alice chooses $a$ uniformly from $[0,1]$, Bob chooses $b$ uniformly from $[\frac{1}{2}, \frac{2}{3}]$, and Carol aims to choose $c$ optimally.
Carol wins if her number $c$ is between the numbers chosen by Alice and Bob. We analyze th... | 60 | 3 | <｜begin▁of▁sentence｜><｜User｜>Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers betw... | 28 |
The sum of the real values of $x$ satisfying the equality $|x+2|=2|x-2|$ is: | 6\frac{2}{3} | To solve the equation $|x+2| = 2|x-2|$, we need to consider the different cases for the absolute values based on the sign of the expressions inside them.
#### Case 1: $x + 2 \geq 0$ and $x - 2 \geq 0$
This implies $x \geq 2$. The equation becomes:
\[ x+2 = 2(x-2) \]
\[ x+2 = 2x - 4 \]
\[ 2 + 4 = 2x - x \]
\[ 6 = x \]
... | 61 | 0 | <｜begin▁of▁sentence｜><｜User｜>The sum of the real values of $x$ satisfying the equality $|x+2|=2|x-2|$ is: Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
Let $ABCD$ be an isosceles trapezoid with $\overline{BC} \parallel \overline{AD}$ and $AB=CD$. Points $X$ and $Y$ lie on diagonal $\overline{AC}$ with $X$ between $A$ and $Y$. Suppose $\angle AXD = \angle BYC = 90^\circ$, $AX = 3$, $XY = 1$, and $YC = 2$. What is the area of $ABCD?$ | $3\sqrt{35}$ | 1. **Setting up the coordinate system**: Place $X$ at the origin $(0,0)$, align $AC$ along the $x$-axis, and $DX$ along the $y$-axis. This gives us:
- $X = (0,0)$
- $A = (3,0)$ (since $AX = 3$)
- $Y = (-1,0)$ (since $XY = 1$)
- $C = (-3,0)$ (since $YC = 2$)
2. **Locating points $B$ and $D$**: Let $BY = u$ ... | 62 | 0 | <｜begin▁of▁sentence｜><｜User｜>Let $ABCD$ be an isosceles trapezoid with $\overline{BC} \parallel \overline{AD}$ and $AB=CD$. Points $X$ and $Y$ lie on diagonal $\overline{AC}$ with $X$ between $A$ and $Y$. Suppose $\angle AXD = \angle BYC = 90^\circ$, $AX = 3$, $XY = 1$, and $YC = 2$. What is the area of $ABCD?$ Please ... | 29 |
The number of revolutions of a wheel, with fixed center and with an outside diameter of $6$ feet, required to cause a point on the rim to go one mile is: | \frac{880}{\pi} | 1. **Identify the radius of the wheel**: Given the diameter of the wheel is $6$ feet, the radius $r$ is half of the diameter:
\[
r = \frac{6}{2} = 3 \text{ feet}
\]
2. **Calculate the circumference of the wheel**: The circumference $C$ of a circle is given by the formula $C = 2\pi r$. Substituting the radius ... | 63 | 0 | <｜begin▁of▁sentence｜><｜User｜>The number of revolutions of a wheel, with fixed center and with an outside diameter of $6$ feet, required to cause a point on the rim to go one mile is: Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and $5x^2+kx+12=0$ has at least one integer solution for $x$. What is $N$? | 78 | 1. **Identify the integer solution condition**:
The quadratic equation given is:
\[
5x^2 + kx + 12 = 0
\]
For this equation to have an integer solution, let's denote that integer by $-n$. Thus, the equation can be factored as:
\[
(5x + \frac{12}{n})(x + n) = 0
\]
This factorization is valid... | 128 | 0 | <｜begin▁of▁sentence｜><｜User｜>There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and $5x^2+kx+12=0$ has at least one integer solution for $x$. What is $N$? Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 1 |
If $\sin{2x}\sin{3x}=\cos{2x}\cos{3x}$, then one value for $x$ is | 18 | 1. **Start with the given equation:**
\[
\sin{2x}\sin{3x} = \cos{2x}\cos{3x}
\]
2. **Divide both sides by $\cos{2x}\cos{3x}$:**
\[
\frac{\sin{2x}\sin{3x}}{\cos{2x}\cos{3x}} = 1
\]
Assuming $\cos{2x} \neq 0$ and $\cos{3x} \neq 0$, we can simplify this to:
\[
\tan{2x}\tan{3x} = 1
\]
3. **R... | 129 | 0 | <｜begin▁of▁sentence｜><｜User｜>If $\sin{2x}\sin{3x}=\cos{2x}\cos{3x}$, then one value for $x$ is Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
A number of linked rings, each $1$ cm thick, are hanging on a peg. The top ring has an outside diameter of $20$ cm. The outside diameter of each of the outer rings is $1$ cm less than that of the ring above it. The bottom ring has an outside diameter of $3$ cm. What is the distance, in cm, from the top of the top ring ... | 173 | 1. **Identify the sequence of inside diameters**: The outside diameter of the top ring is $20$ cm, and each subsequent ring has an outside diameter $1$ cm less than the ring above it. Since each ring is $1$ cm thick, the inside diameter of each ring is $2$ cm less than its outside diameter. Therefore, the inside diamet... | 130 | 0 | <｜begin▁of▁sentence｜><｜User｜>A number of linked rings, each $1$ cm thick, are hanging on a peg. The top ring has an outside diameter of $20$ cm. The outside diameter of each of the outer rings is $1$ cm less than that of the ring above it. The bottom ring has an outside diameter of $3$ cm. What is the distance, in cm, ... | 0 |
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$ and the line $x=5.1?$ | 50 | 1. **Understanding the Problem:**
We need to find the number of squares with integer coordinates for vertices, sides parallel to the axes, and lying entirely within the region bounded by:
- $y = \pi x$
- $y = -0.1$
- $x = 5.1$
2. **Visualizing the Region:**
The region is bounded by a line with a slope o... | 131 | 2 | <｜begin▁of▁sentence｜><｜User｜>How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$ and the line $x=5.1?$ Please reason step by step, and put your final answer within \boxed{}.<｜Assistant... | 12 |
Quadrilateral $ABCD$ is inscribed in circle $O$ and has side lengths $AB=3, BC=2, CD=6$, and $DA=8$. Let $X$ and $Y$ be points on $\overline{BD}$ such that $\frac{DX}{BD} = \frac{1}{4}$ and $\frac{BY}{BD} = \frac{11}{36}$.
Let $E$ be the intersection of line $AX$ and the line through $Y$ parallel to $\overline{AD}$. Le... | 17 | 1. **Identify Similar Triangles and Use Parallel Lines**:
- Since $AC \parallel EF$, triangles $\triangle ACX$ and $\triangle EFX$ are similar. Therefore, $\frac{XF}{XC} = \frac{XE}{XA}$.
- Since $YE \parallel AD$, triangles $\triangle EYX$ and $\triangle ADX$ are similar. Therefore, $\frac{XD}{XY} = \frac{XA}{XE... | 132 | 1 | <｜begin▁of▁sentence｜><｜User｜>Quadrilateral $ABCD$ is inscribed in circle $O$ and has side lengths $AB=3, BC=2, CD=6$, and $DA=8$. Let $X$ and $Y$ be points on $\overline{BD}$ such that $\frac{DX}{BD} = \frac{1}{4}$ and $\frac{BY}{BD} = \frac{11}{36}$.
Let $E$ be the intersection of line $AX$ and the line through $Y$ pa... | 0 |
Suppose $a$, $b$ and $c$ are positive integers with $a+b+c=2006$, and $a!b!c!=m\cdot 10^n$, where $m$ and $n$ are integers and $m$ is not divisible by $10$. What is the smallest possible value of $n$? | 492 | 1. **Understanding the Problem**: We need to find the smallest possible value of $n$ in the expression $a!b!c! = m \cdot 10^n$, where $a+b+c = 2006$ and $m$ is not divisible by $10$. This means we need to minimize the number of factors of $10$ in $a!b!c!$, which is equivalent to minimizing the number of factors of $5$ ... | 133 | 2 | <｜begin▁of▁sentence｜><｜User｜>Suppose $a$, $b$ and $c$ are positive integers with $a+b+c=2006$, and $a!b!c!=m\cdot 10^n$, where $m$ and $n$ are integers and $m$ is not divisible by $10$. What is the smallest possible value of $n$? Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>... | 1 |
If $x=t^{\frac{1}{t-1}}$ and $y=t^{\frac{t}{t-1}},t>0,t \ne 1$, a relation between $x$ and $y$ is: | $y^x=x^y$ | 1. **Define the expressions for $x$ and $y$:**
Given $x = t^{1/(t-1)}$ and $y = t^{t/(t-1)}$ where $t > 0$ and $t \neq 1$.
2. **Substitute $x$ and $y$ into the expressions in option (C):**
We need to verify if $y^x = x^y$ holds true.
3. **Calculate $x^y$:**
\[
x^y = \left(t^{\frac{1}{t-1}}\right)^{t^{\fra... | 134 | 1 | <｜begin▁of▁sentence｜><｜User｜>If $x=t^{\frac{1}{t-1}}$ and $y=t^{\frac{t}{t-1}},t>0,t \ne 1$, a relation between $x$ and $y$ is: Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 26 |
Liliane has $50\%$ more soda than Jacqueline, and Alice has $25\%$ more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alice have? | $20\%$ | Let's denote the amount of soda Jacqueline has as $J$. According to the problem, Liliane has $50\%$ more soda than Jacqueline, and Alice has $25\%$ more soda than Jacqueline. We can express the amounts of soda Liliane and Alice have in terms of $J$.
1. **Calculate the amount of soda Liliane has:**
Liliane has $50\%... | 135 | 0 | <｜begin▁of▁sentence｜><｜User｜>Liliane has $50\%$ more soda than Jacqueline, and Alice has $25\%$ more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alice have? Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 28 |
The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and $5^n<2^m<2^{m+2}<5^{n+1}$? | 279 | To solve this problem, we need to understand the relationship between the powers of $5$ and $2$ and how they are distributed between $5^n$ and $5^{n+1}$.
1. **Understanding the relationship between $5^n$ and $2^m$:**
We know that $5^{867}$ is between $2^{2013}$ and $2^{2014}$. This gives us a way to compare the gro... | 136 | 0 | <｜begin▁of▁sentence｜><｜User｜>The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and $5^n<2^m<2^{m+2}<5^{n+1}$? Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 5 |
What is the correct ordering of the three numbers $\frac{5}{19}$, $\frac{7}{21}$, and $\frac{9}{23}$, in increasing order? | \frac{5}{19} < \frac{7}{21} < \frac{9}{23} | To find the correct ordering of the fractions $\frac{5}{19}$, $\frac{7}{21}$, and $\frac{9}{23}$, we can compare each pair of fractions.
1. **Simplify $\frac{7}{21}$:**
\[\frac{7}{21} = \frac{1}{3}\]
2. **Compare $\frac{5}{19}$ and $\frac{7}{21}$:**
To compare $\frac{5}{19}$ and $\frac{1}{3}$, we find a common ... | 137 | 0 | <｜begin▁of▁sentence｜><｜User｜>What is the correct ordering of the three numbers $\frac{5}{19}$, $\frac{7}{21}$, and $\frac{9}{23}$, in increasing order? Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 3 |
The numbers on the faces of this cube are consecutive whole numbers. The sum of the two numbers on each of the three pairs of opposite faces are equal. The sum of the six numbers on this cube is | 81 | 1. **Identify the properties of the cube**: The problem states that the numbers on the faces of the cube are consecutive whole numbers and the sum of the numbers on each pair of opposite faces is equal.
2. **Determine possible sets of numbers**: Since the numbers are consecutive and there are six faces on a cube, we c... | 138 | 0 | <｜begin▁of▁sentence｜><｜User｜>The numbers on the faces of this cube are consecutive whole numbers. The sum of the two numbers on each of the three pairs of opposite faces are equal. The sum of the six numbers on this cube is Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>\n | 0 |
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