problem stringlengths 10 5.15k | answer dict |
|---|---|
A list of five positive integers has mean $12$ and range $18$. The mode and median are both $8$. How many different values are possible for the second largest element of the list? | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ordered triples of integers $(a,b,c)$ satisfy $|a+b|+c = 19$ and $ab+|c| = 97$? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $(c,0)$ to $(3,3)$, divides the entire region into two regions of equal area. What is $c$? | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that $\triangle{ABC}$ is an equilateral triangle of side length $s$, with the property that there is a unique point $P$ inside the triangle such that $AP=1$, $BP=\sqrt{3}$, and $CP=2$. What is $s$? | {
"answer": "\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two cubical dice each have removable numbers $1$ through $6$. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probability that the sum is $7$? | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\angle DBA = \angle DCB$ and $\angle ADB = \angle CBD$. Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two largest angles of $ABCD$? | {
"answer": "240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Successive discounts of $10\%$ and $20\%$ are equivalent to a single discount of: | {
"answer": "28\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $r$ be the number that results when both the base and the exponent of $a^b$ are tripled, where $a,b>0$. If $r$ equals the product of $a^b$ and $x^b$ where $x>0$, then $x=$ | {
"answer": "27a^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed $37$th and $64$th, respectively. How many schools are in the city? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An organization has $30$ employees, $20$ of whom have a brand A computer while the other $10$ have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other if their computers are directly connected by a cable or by relaying messages through a series of connected computers. Initially, no computer is connected to any other. A technician arbitrarily selects one computer of each brand and installs a cable between them, provided there is not already a cable between that pair. The technician stops once every employee can communicate with each other. What is the maximum possible number of cables used? | {
"answer": "191",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x, y$, and $y-\frac{1}{x}$ are not $0$, then $\frac{x-\frac{1}{y}}{y-\frac{1}{x}}$ equals | {
"answer": "\\frac{x}{y}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is $ax^2+bx+c, a \neq 0,$ and the roots are $r$ and $s,$ then the requirement is that $\{a,b,c\}=\{r,s\}$.) | {
"answer": "4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each of the points $A,B,C,D,E,$ and $F$ in the figure below represents a different digit from $1$ to $6.$ Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is $47.$ What is the digit represented by $B?$ | {
"answer": "5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$ | {
"answer": "\\pi + 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is now between 10:00 and 11:00 o'clock, and six minutes from now, the minute hand of a watch will be exactly opposite the place where the hour hand was three minutes ago. What is the exact time now? | {
"answer": "10:15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When $(a-b)^n,n\ge2,ab\ne0$, is expanded by the binomial theorem, it is found that when $a=kb$, where $k$ is a positive integer, the sum of the second and third terms is zero. Then $n$ equals: | {
"answer": "2k+1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A particle moves through the first quadrant as follows. During the first minute it moves from the origin to $(1,0)$. Thereafter, it continues to follow the directions indicated in the figure, going back and forth between the positive x and y axes, moving one unit of distance parallel to an axis in each minute. At which point will the particle be after exactly 1989 minutes?
[asy] import graph; Label f; f.p=fontsize(6); xaxis(0,3.5,Ticks(f, 1.0)); yaxis(0,4.5,Ticks(f, 1.0)); draw((0,0)--(1,0)--(1,1)--(0,1)--(0,2)--(2,2)--(2,0)--(3,0)--(3,3)--(0,3)--(0,4)--(1.5,4),blue+linewidth(2)); arrow((2,4),dir(180),blue); [/asy] | {
"answer": "(44,35)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the same line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture? | {
"answer": "\\frac{3}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles of radius 1 are to be constructed as follows. The center of circle $A$ is chosen uniformly and at random from the line segment joining $(0,0)$ and $(2,0)$. The center of circle $B$ is chosen uniformly and at random, and independently of the first choice, from the line segment joining $(0,1)$ to $(2,1)$. What is the probability that circles $A$ and $B$ intersect? | {
"answer": "\\frac {4 \\sqrt {3} - 3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\triangle PQR$ with $\overline{RS}$ bisecting $\angle R$, $PQ$ extended to $D$ and $\angle n$ a right angle, then: | {
"answer": "\\frac{1}{2}(\\angle p + \\angle q)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four distinct points are arranged on a plane so that the segments connecting them have lengths $a$, $a$, $a$, $a$, $2a$, and $b$. What is the ratio of $b$ to $a$? | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $T_1$ be a triangle with side lengths $2011$, $2012$, and $2013$. For $n \geq 1$, if $T_n = \Delta ABC$ and $D, E$, and $F$ are the points of tangency of the incircle of $\Delta ABC$ to the sides $AB$, $BC$, and $AC$, respectively, then $T_{n+1}$ is a triangle with side lengths $AD, BE$, and $CF$, if it exists. What is the perimeter of the last triangle in the sequence $\left(T_n\right)$? | {
"answer": "\\frac{1509}{128}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle with integral sides has perimeter $8$. The area of the triangle is | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\angle \text{CBD}$ is a right angle, then this protractor indicates that the measure of $\angle \text{ABC}$ is approximately | {
"answer": "20^{\\circ}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the digit $1$ is placed after a two digit number whose tens' digit is $t$, and units' digit is $u$, the new number is: | {
"answer": "100t+10u+1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of the shaded region. | {
"answer": "6\\dfrac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A box contains $5$ chips, numbered $1$, $2$, $3$, $4$, and $5$. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds $4$. What is the probability that $3$ draws are required? | {
"answer": "\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Team A and team B play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team B wins the second game and team A wins the series, what is the probability that team B wins the first game? | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Through a point $P$ inside the $\triangle ABC$ a line is drawn parallel to the base $AB$, dividing the triangle into two equal areas.
If the altitude to $AB$ has a length of $1$, then the distance from $P$ to $AB$ is: | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions? | {
"answer": "9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number halfway between $\frac{1}{8}$ and $\frac{1}{10}$ is | {
"answer": "\\frac{1}{9}",
"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"
} |
When simplified and expressed with negative exponents, the expression $(x + y)^{ - 1}(x^{ - 1} + y^{ - 1})$ is equal to: | {
"answer": "x^{ - 1}y^{ - 1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Convex polygons $P_1$ and $P_2$ are drawn in the same plane with $n_1$ and $n_2$ sides, respectively, $n_1\le n_2$. If $P_1$ and $P_2$ do not have any line segment in common, then the maximum number of intersections of $P_1$ and $P_2$ is: | {
"answer": "n_1n_2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the area of the shaded figure shown below? | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The perimeter of the polygon shown is | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the four equations:
$\textbf{(1)}\ 3y-2x=12 \qquad\textbf{(2)}\ -2x-3y=10 \qquad\textbf{(3)}\ 3y+2x=12 \qquad\textbf{(4)}\ 2y+3x=10$
The pair representing the perpendicular lines is: | {
"answer": "\\text{(1) and (4)}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On an auto trip, the distance read from the instrument panel was $450$ miles. With snow tires on for the return trip over the same route, the reading was $440$ miles. Find, to the nearest hundredth of an inch, the increase in radius of the wheels if the original radius was 15 inches. | {
"answer": ".34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coordinates is it true that $|4x+3y| \leq 1000$? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with center $O$ is tangent to the coordinate axes and to the hypotenuse of the $30^\circ$-$60^\circ$-$90^\circ$ triangle $ABC$ as shown, where $AB=1$. To the nearest hundredth, what is the radius of the circle? | {
"answer": "2.37",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Carmen takes a long bike ride on a hilly highway. The graph indicates the miles traveled during the time of her ride. What is Carmen's average speed for her entire ride in miles per hour? | {
"answer": "5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three red beads, two white beads, and one blue bead are placed in line in random order. What is the probability that no two neighboring beads are the same color? | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On February 13 The Oshkosh Northwester listed the length of daylight as 10 hours and 24 minutes, the sunrise was $6:57\textsc{am}$, and the sunset as $8:15\textsc{pm}$. The length of daylight and sunrise were correct, but the sunset was wrong. When did the sun really set? | {
"answer": "$5:21\\textsc{pm}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cryptographic code is designed as follows. The first time a letter appears in a given message it is replaced by the letter that is $1$ place to its right in the alphabet (asumming that the letter $A$ is one place to the right of the letter $Z$). The second time this same letter appears in the given message, it is replaced by the letter that is $1+2$ places to the right, the third time it is replaced by the letter that is $1+2+3$ places to the right, and so on. For example, with this code the word "banana" becomes "cbodqg". What letter will replace the last letter $s$ in the message "Lee's sis is a Mississippi miss, Chriss!?" | {
"answer": "s",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters? | {
"answer": "400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
All $20$ diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the boundary) do two or more diagonals intersect? | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCDE$ be an equiangular convex pentagon of perimeter $1$. The pairwise intersections of the lines that extend the sides of the pentagon determine a five-pointed star polygon. Let $s$ be the perimeter of this star. What is the difference between the maximum and the minimum possible values of $s$? | {
"answer": "0",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the units digit of the decimal expansion of $\left(15 + \sqrt{220}\right)^{19} + \left(15 + \sqrt{220}\right)^{82}$. | {
"answer": "9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For what real values of $K$ does $x = K^2 (x-1)(x-2)$ have real roots? | {
"answer": "all",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Bernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds $50$ to it and passes the result to Bernardo. The winner is the last person who produces a number less than $1000$. Let $N$ be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of $N$? | {
"answer": "7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number $a=\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying
\[\lfloor x \rfloor \cdot \{x\} = a \cdot x^2\]is $420$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\}=x- \lfloor x \rfloor$ denotes the fractional part of $x$. What is $p+q$? | {
"answer": "929",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four cubes with edge lengths $1$, $2$, $3$, and $4$ are stacked as shown. What is the length of the portion of $\overline{XY}$ contained in the cube with edge length $3$? | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A large urn contains $100$ balls, of which $36 \%$ are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be $72 \%$? (No red balls are to be removed.) | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, $3$, $23578$, and $987620$ are monotonous, but $88$, $7434$, and $23557$ are not. How many monotonous positive integers are there? | {
"answer": "1524",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The square of $5-\sqrt{y^2-25}$ is: | {
"answer": "y^2-10\\sqrt{y^2-25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set of $n$ numbers; $n > 1$, of which one is $1 - \frac {1}{n}$ and all the others are $1$. The arithmetic mean of the $n$ numbers is: | {
"answer": "1 - \\frac{1}{n^2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A dilation of the plane—that is, a size transformation with a positive scale factor—sends the circle of radius $2$ centered at $A(2,2)$ to the circle of radius $3$ centered at $A’(5,6)$. What distance does the origin $O(0,0)$, move under this transformation? | {
"answer": "\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $d(n)$ denote the number of positive integers that divide $n$, including $1$ and $n$. For example, $d(1)=1,d(2)=2,$ and $d(12)=6$. (This function is known as the divisor function.) Let\[f(n)=\frac{d(n)}{\sqrt [3]n}.\]There is a unique positive integer $N$ such that $f(N)>f(n)$ for all positive integers $n\ne N$. What is the sum of the digits of $N?$ | {
"answer": "9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A particle is placed on the parabola $y = x^2- x -6$ at a point $P$ whose $y$-coordinate is $6$. It is allowed to roll along the parabola until it reaches the nearest point $Q$ whose $y$-coordinate is $-6$. The horizontal distance traveled by the particle (the numerical value of the difference in the $x$-coordinates of $P$ and $Q$) is: | {
"answer": "4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circular grass plot 12 feet in diameter is cut by a straight gravel path 3 feet wide, one edge of which passes through the center of the plot. The number of square feet in the remaining grass area is | {
"answer": "30\\pi - 9\\sqrt3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ms.Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of $50$ units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles? | {
"answer": "128",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
After school, Maya and Naomi headed to the beach, $6$ miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi's and Maya's average speeds? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the given figure, hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\triangle KBC$? | {
"answer": "$12$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be an isosceles trapezoid having parallel bases $\overline{AB}$ and $\overline{CD}$ with $AB>CD.$ Line segments from a point inside $ABCD$ to the vertices divide the trapezoid into four triangles whose areas are $2, 3, 4,$ and $5$ starting with the triangle with base $\overline{CD}$ and moving clockwise as shown in the diagram below. What is the ratio $\frac{AB}{CD}?$ | {
"answer": "2+\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are $10$ people standing equally spaced around a circle. Each person knows exactly $3$ of the other $9$ people: the $2$ people standing next to her or him, as well as the person directly across the circle. How many ways are there for the $10$ people to split up into $5$ pairs so that the members of each pair know each other? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra \$2.50 to cover her portion of the total bill. What was the total bill? | {
"answer": "$120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\log_{10} (x^2-3x+6)=1$, the value of $x$ is: | {
"answer": "4 or -1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Square pyramid $ABCDE$ has base $ABCD$, which measures $3$ cm on a side, and altitude $AE$ perpendicular to the base, which measures $6$ cm. Point $P$ lies on $BE$, one third of the way from $B$ to $E$; point $Q$ lies on $DE$, one third of the way from $D$ to $E$; and point $R$ lies on $CE$, two thirds of the way from $C$ to $E$. What is the area, in square centimeters, of $\triangle{PQR}$? | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular hexagon of side length $1$ is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. What is the area of the region bounded by these $6$ reflected arcs? | {
"answer": "3\\sqrt{3}-\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Which one of the following points is not on the graph of $y=\dfrac{x}{x+1}$? | {
"answer": "(-1,1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circular table has 60 chairs around it. There are $N$ people seated at this table in such a way that the next person seated must sit next to someone. What is the smallest possible value for $N$? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $a > 1$, then the sum of the real solutions of
$\sqrt{a - \sqrt{a + x}} = x$ | {
"answer": "\\frac{\\sqrt{4a- 3} - 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A group of $12$ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^{\text{th}}$ pirate to take a share takes $\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the $12^{\text{th}}$ pirate receive? | {
"answer": "1925",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the expression $(\underline{\qquad}\times\underline{\qquad})+(\underline{\qquad}\times\underline{\qquad})$ each blank is to be filled in with one of the digits $1,2,3,$ or $4,$ with each digit being used once. How many different values can be obtained? | {
"answer": "4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the small country of Mathland, all automobile license plates have four symbols. The first must be a vowel (A, E, I, O, or U), the second and third must be two different letters among the 21 non-vowels, and the fourth must be a digit (0 through 9). If the symbols are chosen at random subject to these conditions, what is the probability that the plate will read "AMC8"? | {
"answer": "\\frac{1}{21,000}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers $1,2,\dots,9$ are randomly placed into the $9$ squares of a $3 \times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd? | {
"answer": "\\frac{1}{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the same line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture? | {
"answer": "\\frac{3}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $AB=6$, $AC=8$, $BC=10$, and $D$ is the midpoint of $\overline{BC}$. What is the sum of the radii of the circles inscribed in $\triangle ADB$ and $\triangle ADC$? | {
"answer": "3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many solutions does the equation $\tan(2x)=\cos(\frac{x}{2})$ have on the interval $[0,2\pi]?$ | {
"answer": "5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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 to the bottom of the bottom ring? | {
"answer": "173",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $R$ be a set of nine distinct integers. Six of the elements are $2$, $3$, $4$, $6$, $9$, and $14$. What is the number of possible values of the median of $R$? | {
"answer": "7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A$? | {
"answer": "385",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In regular hexagon $ABCDEF$, points $W$, $X$, $Y$, and $Z$ are chosen on sides $\overline{BC}$, $\overline{CD}$, $\overline{EF}$, and $\overline{FA}$ respectively, so lines $AB$, $ZW$, $YX$, and $ED$ are parallel and equally spaced. What is the ratio of the area of hexagon $WCXYFZ$ to the area of hexagon $ABCDEF$? | {
"answer": "\\frac{11}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Candy sales from the Boosters Club from January through April are shown. What were the average sales per month in dollars? | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Angle $ABC$ of $\triangle ABC$ is a right angle. The sides of $\triangle ABC$ are the diameters of semicircles as shown. The area of the semicircle on $\overline{AB}$ equals $8\pi$, and the arc of the semicircle on $\overline{AC}$ has length $8.5\pi$. What is the radius of the semicircle on $\overline{BC}$? | {
"answer": "7.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The base-ten representation for $19!$ is $121,6T5,100,40M,832,H00$, where $T$, $M$, and $H$ denote digits that are not given. What is $T+M+H$? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cars A and B travel the same distance. Car A travels half that distance at $u$ miles per hour and half at $v$ miles per hour. Car B travels half the time at $u$ miles per hour and half at $v$ miles per hour. The average speed of Car A is $x$ miles per hour and that of Car B is $y$ miles per hour. Then we always have | {
"answer": "$x \\leq y$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In quadrilateral $ABCD$ with diagonals $AC$ and $BD$, intersecting at $O$, $BO=4$, $OD = 6$, $AO=8$, $OC=3$, and $AB=6$. The length of $AD$ is: | {
"answer": "{\\sqrt{166}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A chord which is the perpendicular bisector of a radius of length 12 in a circle, has length | {
"answer": "12\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
To $m$ ounces of a $m\%$ solution of acid, $x$ ounces of water are added to yield a $(m-10)\%$ solution. If $m>25$, then $x$ is | {
"answer": "\\frac{10m}{m-10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An open box is constructed by starting with a rectangular sheet of metal 10 in. by 14 in. and cutting a square of side $x$ inches from each corner. The resulting projections are folded up and the seams welded. The volume of the resulting box is: | {
"answer": "140x - 48x^2 + 4x^3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$\frac{9}{7 \times 53} =$ | {
"answer": "$\\frac{0.9}{0.7 \\times 53}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many sets of two or more consecutive positive integers have a sum of $15$? | {
"answer": "2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of counter examples to the statement: | {
"answer": "2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$K$ takes $30$ minutes less time than $M$ to travel a distance of $30$ miles. $K$ travels $\frac {1}{3}$ mile per hour faster than $M$. If $x$ is $K$'s rate of speed in miles per hours, then $K$'s time for the distance is: | {
"answer": "\\frac{30}{x}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The fraction \(\frac{1}{99^2}=0.\overline{b_{n-1}b_{n-2}\ldots b_2b_1b_0},\) where $n$ is the length of the period of the repeating decimal expansion. What is the sum $b_0+b_1+\cdots+b_{n-1}$? | {
"answer": "883",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The pie charts below indicate the percent of students who prefer golf, bowling, or tennis at East Junior High School and West Middle School. The total number of students at East is 2000 and at West, 2500. In the two schools combined, the percent of students who prefer tennis is | {
"answer": "32\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A farmer's rectangular field is partitioned into a $2$ by $2$ grid of $4$ rectangular sections. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field? | {
"answer": "84",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.