problem stringlengths 10 5.15k | answer dict |
|---|---|
Let $S$ be a set of $6$ integers taken from $\{1,2,\dots,12\}$ with the property that if $a$ and $b$ are elements of $S$ with $a<b$, then $b$ is not a multiple of $a$. What is the least possible value of an element in $S$? | {
"answer": "4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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 $5^n<2^m<2^{m+2}<5^{n+1}$? | {
"answer": "279",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of the rectangular region is | {
"answer": ".088 m^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Square $PQRS$ lies in the first quadrant. Points $(3,0), (5,0), (7,0),$ and $(13,0)$ lie on lines $SP, RQ, PQ$, and $SR$, respectively. What is the sum of the coordinates of the center of the square $PQRS$? | {
"answer": "\\frac{32}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When simplified, the third term in the expansion of $(\frac{a}{\sqrt{x}} - \frac{\sqrt{x}}{a^2})^6$ is: | {
"answer": "\\frac{15}{x}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(n) = \frac{x_1 + x_2 + \cdots + x_n}{n}$, where $n$ is a positive integer. If $x_k = (-1)^k, k = 1, 2, \cdots, n$, the set of possible values of $f(n)$ is: | {
"answer": "$\\{0, -\\frac{1}{n}\\}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \le 50$ is $f_{50}(n) = 12?$ | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The roots of the equation $x^{2}-2x = 0$ can be obtained graphically by finding the abscissas of the points of intersection of each of the following pairs of equations except the pair:
[Note: Abscissas means x-coordinate.] | {
"answer": "$y = x$, $y = x-2$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A ray of light originates from point $A$ and travels in a plane, being reflected $n$ times between lines $AD$ and $CD$ before striking a point $B$ (which may be on $AD$ or $CD$) perpendicularly and retracing its path back to $A$ (At each point of reflection the light makes two equal angles as indicated in the adjoining figure. The figure shows the light path for $n=3$). If $\measuredangle CDA=8^\circ$, what is the largest value $n$ can have? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $y=x^2+px+q$, then if the least possible value of $y$ is zero $q$ is equal to: | {
"answer": "\\frac{p^2}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ways are there to place $3$ indistinguishable red chips, $3$ indistinguishable blue chips, and $3$ indistinguishable green chips in the squares of a $3 \times 3$ grid so that no two chips of the same color are directly adjacent to each other, either vertically or horizontally? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
George and Henry started a race from opposite ends of the pool. After a minute and a half, they passed each other in the center of the pool. If they lost no time in turning and maintained their respective speeds, how many minutes after starting did they pass each other the second time? | {
"answer": "4\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A watch loses $2\frac{1}{2}$ minutes per day. It is set right at $1$ P.M. on March 15. Let $n$ be the positive correction, in minutes, to be added to the time shown by the watch at a given time. When the watch shows $9$ A.M. on March 21, $n$ equals: | {
"answer": "14\\frac{14}{23}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ratio of $w$ to $x$ is $4:3$, of $y$ to $z$ is $3:2$ and of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y$? | {
"answer": "16:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A quadratic polynomial with real coefficients and leading coefficient $1$ is called $\emph{disrespectful}$ if the equation $p(p(x))=0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\tilde{p}(x)$ for which the sum of the roots is maximized. What is $\tilde{p}(1)$? | {
"answer": "\\frac{5}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with center $O$ has area $156\pi$. Triangle $ABC$ is equilateral, $\overline{BC}$ is a chord on the circle, $OA = 4\sqrt{3}$, and point $O$ is outside $\triangle ABC$. What is the side length of $\triangle ABC$? | {
"answer": "$6$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of two numbers is $S$. Suppose $3$ is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers? | {
"answer": "2S + 12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The rails on a railroad are $30$ feet long. As the train passes over the point where the rails are joined, there is an audible click.
The speed of the train in miles per hour is approximately the number of clicks heard in: | {
"answer": "20 seconds",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When the polynomial $x^3-2$ is divided by the polynomial $x^2-2$, the remainder is | {
"answer": "2x-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Marvin had a birthday on Tuesday, May 27 in the leap year $2008$. In what year will his birthday next fall on a Saturday? | {
"answer": "2017",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x, y$ and $2x + \frac{y}{2}$ are not zero, then
$\left( 2x + \frac{y}{2} \right)^{-1} \left[(2x)^{-1} + \left( \frac{y}{2} \right)^{-1} \right]$ equals | {
"answer": "\\frac{1}{xy}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
All of the triangles in the diagram below are similar to isosceles triangle $ABC$, in which $AB=AC$. Each of the $7$ smallest triangles has area $1,$ and $\triangle ABC$ has area $40$. What is the area of trapezoid $DBCE$? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?
[asy] unitsize(10mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqrt(2)); draw(A--B--C--D--E--F--G--H--cycle); draw(A--D); draw(B--G); draw(C--F); draw(E--H);[/asy] | {
"answer": "\\frac{\\sqrt{2} - 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jim starts with a positive integer $n$ and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with $n = 55$, then his sequence contains $5$ numbers:
$\begin{array}{ccccc} {}&{}&{}&{}&55\\ 55&-&7^2&=&6\\ 6&-&2^2&=&2\\ 2&-&1^2&=&1\\ 1&-&1^2&=&0\\ \end{array}$
Let $N$ be the smallest number for which Jim’s sequence has $8$ numbers. What is the units digit of $N$? | {
"answer": "3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A wooden cube has edges of length $3$ meters. Square holes, of side one meter, centered in each face are cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. The entire surface area including the inside, in square meters, is | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $c = \frac{2\pi}{11}.$ What is the value of
\[\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}?\] | {
"answer": "1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common endpoint.) | {
"answer": "1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A list of integers has mode 32 and mean 22. The smallest number in the list is 10. The median m of the list is a member of the list. If the list member m were replaced by m+10, the mean and median of the new list would be 24 and m+10, respectively. If m were instead replaced by m-8, the median of the new list would be m-4. What is m? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In one of the adjoining figures a square of side $2$ is dissected into four pieces so that $E$ and $F$ are the midpoints of opposite sides and $AG$ is perpendicular to $BF$. These four pieces can then be reassembled into a rectangle as shown in the second figure. The ratio of height to base, $XY / YZ$, in this rectangle is | {
"answer": "5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The clock in Sri's car, which is not accurate, gains time at a constant rate. One day as he begins shopping, he notes that his car clock and his watch (which is accurate) both say 12:00 noon. When he is done shopping, his watch says 12:30 and his car clock says 12:35. Later that day, Sri loses his watch. He looks at his car clock and it says 7:00. What is the actual time? | {
"answer": "6:00",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The value of $\frac{1}{16}a^0+\left (\frac{1}{16a} \right )^0- \left (64^{-\frac{1}{2}} \right )- (-32)^{-\frac{4}{5}}$ is: | {
"answer": "1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a certain positive integer $n$ less than $1000$, the decimal equivalent of $\frac{1}{n}$ is $0.\overline{abcdef}$, a repeating decimal of period of $6$, and the decimal equivalent of $\frac{1}{n+6}$ is $0.\overline{wxyz}$, a repeating decimal of period $4$. In which interval does $n$ lie? | {
"answer": "[201,400]",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the adjoining figure $ABCD$ is a square and $CMN$ is an equilateral triangle. If the area of $ABCD$ is one square inch, then the area of $CMN$ in square inches is | {
"answer": "2\\sqrt{3}-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For each integer $n\geq 2$, let $S_n$ be the sum of all products $jk$, where $j$ and $k$ are integers and $1\leq j<k\leq n$. What is the sum of the 10 least values of $n$ such that $S_n$ is divisible by $3$? | {
"answer": "197",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a square. Let $E, F, G$ and $H$ be the centers, respectively, of equilateral triangles with bases $\overline{AB}, \overline{BC}, \overline{CD},$ and $\overline{DA},$ each exterior to the square. What is the ratio of the area of square $EFGH$ to the area of square $ABCD$? | {
"answer": "\\frac{2+\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At Jefferson Summer Camp, $60\%$ of the children play soccer, $30\%$ of the children swim, and $40\%$ of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer? | {
"answer": "51\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\overline{AB}$ be a diameter in a circle of radius $5\sqrt{2}.$ Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at a point $E$ such that $BE=2\sqrt{5}$ and $\angle AEC = 45^{\circ}.$ What is $CE^2+DE^2?$ | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$ has side lengths $AB = 11, BC=24$, and $CA = 20$. The bisector of $\angle{BAC}$ intersects $\overline{BC}$ in point $D$, and intersects the circumcircle of $\triangle{ABC}$ in point $E \ne A$. The circumcircle of $\triangle{BED}$ intersects the line $AB$ in points $B$ and $F \ne B$. What is $CF$? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A convex quadrilateral $ABCD$ with area $2002$ contains a point $P$ in its interior such that $PA = 24, PB = 32, PC = 28, PD = 45$. Find the perimeter of $ABCD$. | {
"answer": "4(36 + \\sqrt{113})",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Tyrone had $97$ marbles and Eric had $11$ marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a real number $a$, let $\lfloor a \rfloor$ denote the greatest integer less than or equal to $a$. Let $\mathcal{R}$ denote the region in the coordinate plane consisting of points $(x,y)$ such that $\lfloor x \rfloor ^2 + \lfloor y \rfloor ^2 = 25$. The region $\mathcal{R}$ is completely contained in a disk of radius $r$ (a disk is the union of a circle and its interior). The minimum value of $r$ can be written as $\frac {\sqrt {m}}{n}$, where $m$ and $n$ are integers and $m$ is not divisible by the square of any prime. Find $m + n$. | {
"answer": "69",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that \(\begin{array}{c} a \\ b \\ c \end{array}\) means $a+b-c$.
For example, \(\begin{array}{c} 5 \\ 4 \\ 6 \end{array}\) is $5+4-6 = 3$.
Then the sum \(\begin{array}{c} 3 \\ 2 \\ 5 \end{array}\) + \(\begin{array}{c} 4 \\ 1 \\ 6 \end{array}\) is | {
"answer": "1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two particles move along the edges of equilateral $\triangle ABC$ in the direction $A\Rightarrow B\Rightarrow C\Rightarrow A,$ starting simultaneously and moving at the same speed. One starts at $A$, and the other starts at the midpoint of $\overline{BC}$. The midpoint of the line segment joining the two particles traces out a path that encloses a region $R$. What is the ratio of the area of $R$ to the area of $\triangle ABC$? | {
"answer": "\\frac{1}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n$ denote the smallest positive integer that is divisible by both $4$ and $9,$ and whose base-$10$ representation consists of only $4$'s and $9$'s, with at least one of each. What are the last four digits of $n?$ | {
"answer": "4944",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A $9 \times 9 \times 9$ cube is composed of twenty-seven $3 \times 3 \times 3$ cubes. The big cube is ‘tunneled’ as follows: First, the six $3 \times 3 \times 3$ cubes which make up the center of each face as well as the center $3 \times 3 \times 3$ cube are removed. Second, each of the twenty remaining $3 \times 3 \times 3$ cubes is diminished in the same way. That is, the center facial unit cubes as well as each center cube are removed. The surface area of the final figure is: | {
"answer": "1056",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region ---- inside the hexagon but outside all of the semicircles? | {
"answer": "6\\sqrt{3} - 3\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The price of an article is cut $10 \%$. To restore it to its former value, the new price must be increased by: | {
"answer": "$11\\frac{1}{9} \\%$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads, and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0,1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $|x-y| > \frac{1}{2}$? | {
"answer": "\\frac{7}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid $A$ dollars and Bernardo had paid $B$ dollars, where $A < B$. How many dollars must LeRoy give to Bernardo so that they share the costs equally? | {
"answer": "\\frac{B - A}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the statements:
$\textbf{(1)}\ p\wedge \sim q\wedge r \qquad\textbf{(2)}\ \sim p\wedge \sim q\wedge r \qquad\textbf{(3)}\ p\wedge \sim q\wedge \sim r \qquad\textbf{(4)}\ \sim p\wedge q\wedge r$
where $p,q$, and $r$ are propositions. How many of these imply the truth of $(p\rightarrow q)\rightarrow r$? | {
"answer": "4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A scientist walking through a forest recorded as integers the heights of $5$ trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?
\begin{tabular}{|c|c|} \hline Tree 1 & meters \\ Tree 2 & 11 meters \\ Tree 3 & meters \\ Tree 4 & meters \\ Tree 5 & meters \\ \hline Average height & .2 meters \\ \hline \end{tabular}
| {
"answer": "24.2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $1 more than a pink pill, and Al's pills cost a total of $546 for the two weeks. How much does one green pill cost? | {
"answer": "$19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The simplest form of $1 - \frac{1}{1 + \frac{a}{1 - a}}$ is: | {
"answer": "a",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
[asy]
draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle, black+linewidth(.75));
draw((0,-1)--(0,1), black+linewidth(.75));
draw((-1,0)--(1,0), black+linewidth(.75));
draw((-1,-1/sqrt(3))--(1,1/sqrt(3)), black+linewidth(.75));
draw((-1,1/sqrt(3))--(1,-1/sqrt(3)), black+linewidth(.75));
draw((-1/sqrt(3),-1)--(1/sqrt(3),1), black+linewidth(.75));
draw((1/sqrt(3),-1)--(-1/sqrt(3),1), black+linewidth(.75));
[/asy]
Amy painted a dartboard over a square clock face using the "hour positions" as boundaries. If $t$ is the area of one of the eight triangular regions such as that between 12 o'clock and 1 o'clock, and $q$ is the area of one of the four corner quadrilaterals such as that between 1 o'clock and 2 o'clock, then $\frac{q}{t}=$ | {
"answer": "2\\sqrt{3}-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows the miles traveled by bikers Alberto and Bjorn. After four hours, about how many more miles has Alberto biked than Bjorn? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7.$ Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice? | {
"answer": "\\frac{7}{36}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number $(2^{48}-1)$ is exactly divisible by two numbers between $60$ and $70$. These numbers are | {
"answer": "63,65",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $B$ lies on line segment $\overline{AC}$ with $AB=16$ and $BC=4$. Points $D$ and $E$ lie on the same side of line $AC$ forming equilateral triangles $\triangle ABD$ and $\triangle BCE$. Let $M$ be the midpoint of $\overline{AE}$, and $N$ be the midpoint of $\overline{CD}$. The area of $\triangle BMN$ is $x$. Find $x^2$. | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers $1, 2, 3, 4, 5$ are to be arranged in a circle. An arrangement is $\textit{bad}$ if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there? | {
"answer": "2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $a$ is $150\%$ of $b$. What percent of $a$ is $3b$? | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A plastic snap-together cube has a protruding snap on one side and receptacle holes on the other five sides as shown. What is the smallest number of these cubes that can be snapped together so that only receptacle holes are showing?
[asy] draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); draw(circle((2,2),1)); draw((4,0)--(6,1)--(6,5)--(4,4)); draw((6,5)--(2,5)--(0,4)); draw(ellipse((5,2.5),0.5,1)); fill(ellipse((3,4.5),1,0.25),black); fill((2,4.5)--(2,5.25)--(4,5.25)--(4,4.5)--cycle,black); fill(ellipse((3,5.25),1,0.25),black); [/asy] | {
"answer": "4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $M$ is $30 \%$ of $Q$, $Q$ is $20 \%$ of $P$, and $N$ is $50 \%$ of $P$, then $\frac {M}{N} =$ | {
"answer": "\\frac {3}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible? | {
"answer": "729",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the non-convex quadrilateral $ABCD$ shown below, $\angle BCD$ is a right angle, $AB=12$, $BC=4$, $CD=3$, and $AD=13$. What is the area of quadrilateral $ABCD$? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For $x$ real, the inequality $1 \le |x-2| \le 7$ is equivalent to | {
"answer": "$-5 \\le x \\le 1$ or $3 \\le x \\le 9$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Star lists the whole numbers $1$ through $30$ once. Emilio copies Star's numbers, replacing each occurrence of the digit $2$ by the digit $1$. Star adds her numbers and Emilio adds his numbers. How much larger is Star's sum than Emilio's? | {
"answer": "103",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Before Ashley started a three-hour drive, her car's odometer reading was 29792, a palindrome. (A palindrome is a number that reads the same way from left to right as it does from right to left). At her destination, the odometer reading was another palindrome. If Ashley never exceeded the speed limit of 75 miles per hour, what was her greatest possible average speed? | {
"answer": "70 \\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When a positive integer $x$ is divided by a positive integer $y$, the quotient is $u$ and the remainder is $v$, where $u$ and $v$ are integers.
What is the remainder when $x+2uy$ is divided by $y$? | {
"answer": "v",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In rectangle $ABCD$, $AB=1$, $BC=2$, and points $E$, $F$, and $G$ are midpoints of $\overline{BC}$, $\overline{CD}$, and $\overline{AD}$, respectively. Point $H$ is the midpoint of $\overline{GE}$. What is the area of the shaded region? | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The expression $a^3-a^{-3}$ equals: | {
"answer": "\\left(a-\\frac{1}{a}\\right)\\left(a^2+1+\\frac{1}{a^2}\\right)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Segment $BD$ and $AE$ intersect at $C$, as shown, $AB=BC=CD=CE$, and $\angle A = \frac{5}{2} \angle B$. What is the degree measure of $\angle D$? | {
"answer": "52.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (0,4), (4,4),$ and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square? | {
"answer": "\\frac{5}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Debra flips a fair coin repeatedly, keeping track of how many heads and how many tails she has seen in total, until she gets either two heads in a row or two tails in a row, at which point she stops flipping. What is the probability that she gets two heads in a row but she sees a second tail before she sees a second head？ | {
"answer": "\\frac{1}{24}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The vertices of a quadrilateral lie on the graph of $y=\ln{x}$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln{\frac{91}{90}}$. What is the $x$-coordinate of the leftmost vertex? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The first three terms of a geometric progression are $\sqrt 3$, $\sqrt[3]3$, and $\sqrt[6]3$. What is the fourth term? | {
"answer": "1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the expression $\frac{x + 1}{x - 1}$ each $x$ is replaced by $\frac{x + 1}{x - 1}$. The resulting expression, evaluated for $x = \frac{1}{2}$, equals: | {
"answer": "3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x$ is a number satisfying the equation $\sqrt[3]{x+9}-\sqrt[3]{x-9}=3$, then $x^2$ is between: | {
"answer": "75 \\text{ and } 85",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangle with diagonal length $x$ is twice as long as it is wide. What is the area of the rectangle? | {
"answer": "\\frac{2}{5} x^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The table below displays the grade distribution of the $30$ students in a mathematics class on the last two tests. For example, exactly one student received a 'D' on Test 1 and a 'C' on Test 2. What percent of the students received the same grade on both tests? | {
"answer": "40\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a particular game, each of $4$ players rolls a standard $6$-sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again and this process will continue until one player wins. Hugo is one of the players in this game. What is the probability that Hugo's first roll was a $5,$ given that he won the game? | {
"answer": "\\frac{41}{144}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Lines in the $xy$-plane are drawn through the point $(3,4)$ and the trisection points of the line segment joining the points $(-4,5)$ and $(5,-1)$. One of these lines has the equation | {
"answer": "x-4y+13=0",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ways can a student schedule $3$ mathematics courses -- algebra, geometry, and number theory -- in a $6$-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other $3$ periods is of no concern here.) | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than $5$ steps left). Suppose Dash takes $19$ fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an arcade game, the "monster" is the shaded sector of a circle of radius $1$ cm, as shown in the figure. The missing piece (the mouth) has central angle $60^\circ$. What is the perimeter of the monster in cm? | {
"answer": "\\frac{5}{3}\\pi + 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A man has part of $4500 invested at 4% and the rest at 6%. If his annual return on each investment is the same, the average rate of interest which he realizes of the $4500 is: | {
"answer": "4.8\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The negation of the proposition "For all pairs of real numbers $a,b$, if $a=0$, then $ab=0$" is: There are real numbers $a,b$ such that | {
"answer": "$a=0$ and $ab \\ne 0$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$ has $\angle BAC = 60^{\circ}$, $\angle CBA \leq 90^{\circ}$, $BC=1$, and $AC \geq AB$. Let $H$, $I$, and $O$ be the orthocenter, incenter, and circumcenter of $\triangle ABC$, respectively. Assume that the area of pentagon $BCOIH$ is the maximum possible. What is $\angle CBA$? | {
"answer": "80^{\\circ}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the line $y = \frac{3}{4}x + 6$ and a line $L$ parallel to the given line and $4$ units from it. A possible equation for $L$ is: | {
"answer": "y =\\frac{3}{4}x+1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
[asy]
draw((-7,0)--(7,0),black+linewidth(.75));
draw((-3*sqrt(3),0)--(-2*sqrt(3),3)--(-sqrt(3),0)--(0,3)--(sqrt(3),0)--(2*sqrt(3),3)--(3*sqrt(3),0),black+linewidth(.75));
draw((-2*sqrt(3),0)--(-1*sqrt(3),3)--(0,0)--(sqrt(3),3)--(2*sqrt(3),0),black+linewidth(.75));
[/asy]
Five equilateral triangles, each with side $2\sqrt{3}$, are arranged so they are all on the same side of a line containing one side of each vertex. Along this line, the midpoint of the base of one triangle is a vertex of the next. The area of the region of the plane that is covered by the union of the five triangular regions is | {
"answer": "12\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $f(x)=\log \left(\frac{1+x}{1-x}\right)$ for $-1<x<1$, then $f\left(\frac{3x+x^3}{1+3x^2}\right)$ in terms of $f(x)$ is | {
"answer": "3f(x)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of this figure is $100\text{ cm}^2$. Its perimeter is
[asy] draw((0,2)--(2,2)--(2,1)--(3,1)--(3,0)--(1,0)--(1,1)--(0,1)--cycle,linewidth(1)); draw((1,2)--(1,1)--(2,1)--(2,0),dashed); [/asy]
[figure consists of four identical squares] | {
"answer": "50 cm",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the figure, polygons $A$, $E$, and $F$ are isosceles right triangles; $B$, $C$, and $D$ are squares with sides of length $1$; and $G$ is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is | {
"answer": "5/6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$[x-(y-z)] - [(x-y) - z] = $ | {
"answer": "2z",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The limit of the sum of an infinite number of terms in a geometric progression is $\frac {a}{1 - r}$ where $a$ denotes the first term and $- 1 < r < 1$ denotes the common ratio. The limit of the sum of their squares is: | {
"answer": "\\frac {a^2}{1 - r^2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A house and store were sold for $12,000 each. The house was sold at a loss of 20% of the cost, and the store at a gain of 20% of the cost. The entire transaction resulted in: | {
"answer": "-$1000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a room containing $N$ people, $N > 3$, at least one person has not shaken hands with everyone else in the room.
What is the maximum number of people in the room that could have shaken hands with everyone else? | {
"answer": "$N-1$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Quadrilateral $ABCD$ is a trapezoid, $AD = 15$, $AB = 50$, $BC = 20$, and the altitude is $12$. What is the area of the trapezoid? | {
"answer": "750",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The solution set of $6x^2+5x<4$ is the set of all values of $x$ such that | {
"answer": "-\\frac{4}{3}<x<\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The closed curve in the figure is made up of 9 congruent circular arcs each of length $\frac{2\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side 2. What is the area enclosed by the curve? | {
"answer": "\\pi + 6\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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