problem stringlengths 10 5.15k | answer dict |
|---|---|
Let $T_1$ be a triangle with side lengths $2011, 2012,$ and $2013$. For $n \ge 1$, if $T_n = \triangle ABC$ and $D, E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ 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 $( T_n )$? | {
"answer": "\\frac{1509}{128}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A manufacturer built a machine which will address $500$ envelopes in $8$ minutes. He wishes to build another machine so that when both are operating together they will address $500$ envelopes in $2$ minutes. The equation used to find how many minutes $x$ it would require the second machine to address $500$ envelopes alone is: | {
"answer": "$\\frac{1}{8}+\\frac{1}{x}=\\frac{1}{2}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph relates the distance traveled [in miles] to the time elapsed [in hours] on a trip taken by an experimental airplane. During which hour was the average speed of this airplane the largest? | {
"answer": "second (1-2)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7, as shown. What is the area of the shaded quadrilateral? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x<-2$, then $|1-|1+x||$ equals | {
"answer": "-2-x",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$ | {
"answer": "117",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A store increased the original price of a shirt by a certain percent and then lowered the new price by the same amount. Given that the resulting price was $84\%$ of the original price, by what percent was the price increased and decreased? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $f(x)=4^x$ then $f(x+1)-f(x)$ equals: | {
"answer": "3f(x)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Tom has twelve slips of paper which he wants to put into five cups labeled $A$, $B$, $C$, $D$, $E$. He wants the sum of the numbers on the slips in each cup to be an integer. Furthermore, he wants the five integers to be consecutive and increasing from $A$ to $E$. The numbers on the papers are $2, 2, 2, 2.5, 2.5, 3, 3, 3, 3, 3.5, 4,$ and $4.5$. If a slip with $2$ goes into cup $E$ and a slip with $3$ goes into cup $B$, then the slip with $3.5$ must go into what cup? | {
"answer": "D",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in the middle, a path only allows moves from one letter to an adjacent (above, below, left, or right, but not diagonal) letter. One example of such a path is traced in the picture. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The amount $2.5$ is split into two nonnegative real numbers uniformly at random, for instance, into $2.143$ and $.357$, or into $\sqrt{3}$ and $2.5-\sqrt{3}$. Then each number is rounded to its nearest integer, for instance, $2$ and $0$ in the first case above, $2$ and $1$ in the second. What is the probability that the two integers sum to $3$? | {
"answer": "\\frac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $N$ be the positive integer $7777\ldots777$, a $313$-digit number where each digit is a $7$. Let $f(r)$ be the leading digit of the $r^{\text{th}}$ root of $N$. What is $f(2) + f(3) + f(4) + f(5)+ f(6)$? | {
"answer": "8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define the function $f_1$ on the positive integers by setting $f_1(1)=1$ and if $n=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is the prime factorization of $n>1$, then
\[f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}\cdots (p_k+1)^{e_k-1}.\]
For every $m\ge 2$, let $f_m(n)=f_1(f_{m-1}(n))$. For how many $N$s in the range $1\le N\le 400$ is the sequence $(f_1(N),f_2(N),f_3(N),\dots )$ unbounded?
Note: A sequence of positive numbers is unbounded if for every integer $B$, there is a member of the sequence greater than $B$. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A piece of paper containing six joined squares labeled as shown in the diagram is folded along the edges of the squares to form a cube. The label of the face opposite the face labeled $\text{X}$ is | {
"answer": "Y",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that the euro is worth 1.3 dollars. If Diana has 500 dollars and Etienne has 400 euros, by what percent is the value of Etienne's money greater that the value of Diana's money? | {
"answer": "4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of the shaded region $\text{BEDC}$ in parallelogram $\text{ABCD}$ is | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles that share the same center have radii $10$ meters and $20$ meters. An aardvark runs along the path shown, starting at $A$ and ending at $K$. How many meters does the aardvark run? | {
"answer": "20\\pi + 40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The horizontal and vertical distances between adjacent points equal 1 unit. What is the area of triangle $ABC$? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Tamara has three rows of two $6$-feet by $2$-feet flower beds in her garden. The beds are separated and also surrounded by $1$-foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet? | {
"answer": "78",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Handy Aaron helped a neighbor $1 \frac{1}{4}$ hours on Monday, $50$ minutes on Tuesday, from 8:20 to 10:45 on Wednesday morning, and a half-hour on Friday. He is paid $\textdollar 3$ per hour. How much did he earn for the week? | {
"answer": "\\$15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a_1, a_2, \dots, a_{2018}$ be a strictly increasing sequence of positive integers such that $a_1 + a_2 + \cdots + a_{2018} = 2018^{2018}$. What is the remainder when $a_1^3 + a_2^3 + \cdots + a_{2018}^3$ is divided by $6$? | {
"answer": "2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On each horizontal line in the figure below, the five large dots indicate the populations of cities $A, B, C, D$ and $E$ in the year indicated.
Which city had the greatest percentage increase in population from $1970$ to $1980$? | {
"answer": "C",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $a, b, c$ are real numbers such that $a^2 + 2b = 7$, $b^2 + 4c = -7$, and $c^2 + 6a = -14$, find $a^2 + b^2 + c^2$. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $4^x - 4^{x - 1} = 24$, then $(2x)^x$ equals: | {
"answer": "25\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?
[asy]
size(110);
pair A, B, C, D, E, F;
A = (0,0);
B = (1,0);
C = (2,0);
D = rotate(60, A)*B;
E = B + D;
F = rotate(60, A)*C;
draw(Circle(A, 0.5));
draw(Circle(B, 0.5));
draw(Circle(C, 0.5));
draw(Circle(D, 0.5));
draw(Circle(E, 0.5));
draw(Circle(F, 0.5));
[/asy] | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A privateer discovers a merchantman $10$ miles to leeward at 11:45 a.m. and with a good breeze bears down upon her at $11$ mph, while the merchantman can only make $8$ mph in her attempt to escape. After a two hour chase, the top sail of the privateer is carried away; she can now make only $17$ miles while the merchantman makes $15$. The privateer will overtake the merchantman at: | {
"answer": "$5\\text{:}30\\text{ p.m.}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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 $ABCD?$ | {
"answer": "360",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In rectangle $ABCD$, $DC = 2 \cdot CB$ and points $E$ and $F$ lie on $\overline{AB}$ so that $\overline{ED}$ and $\overline{FD}$ trisect $\angle ADC$ as shown. What is the ratio of the area of $\triangle DEF$ to the area of rectangle $ABCD$? | {
"answer": "\\frac{3\\sqrt{3}}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point $(x, y)$ is to be chosen in the coordinate plane so that it is equally distant from the x-axis, the y-axis, and the line $x+y=2$. Then $x$ is | {
"answer": "1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The values of $y$ which will satisfy the equations $2x^{2}+6x+5y+1=0$, $2x+y+3=0$ may be found by solving: | {
"answer": "$y^{2}+10y-7=0$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the set of all equations $x^3 + a_2x^2 + a_1x + a_0 = 0$, where $a_2$, $a_1$, $a_0$ are real constants and $|a_i| < 2$ for $i = 0,1,2$. Let $r$ be the largest positive real number which satisfies at least one of these equations. Then | {
"answer": "\\frac{5}{2} < r < 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
2^{-(2k+1)}-2^{-(2k-1)}+2^{-2k} is equal to | {
"answer": "-2^{-(2k+1)}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x_1$ and $x_2$ be such that $x_1 \not= x_2$ and $3x_i^2-hx_i=b$, $i=1, 2$. Then $x_1+x_2$ equals | {
"answer": "-\\frac{h}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23, -21, -17,$ and $-15$, and $Q(P(x))$ has zeros at $x=-59,-57,-51$ and $-49$. What is the sum of the minimum values of $P(x)$ and $Q(x)$? | {
"answer": "-100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $r$ and $s$ are the roots of $x^2-px+q=0$, then $r^2+s^2$ equals: | {
"answer": "p^2-2q",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, $30 = 6 \times 5$. What is the missing number in the top row? | {
"answer": "4",
"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"
} |
Find $i + 2i^2 +3i^3 + ... + 2002i^{2002}.$ | {
"answer": "-1001 + 1000i",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cassandra sets her watch to the correct time at noon. At the actual time of 1:00 PM, she notices that her watch reads 12:57 and 36 seconds. Assuming that her watch loses time at a constant rate, what will be the actual time when her watch first reads 10:00 PM? | {
"answer": "10:25 PM",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Assume the adjoining chart shows the $1980$ U.S. population, in millions, for each region by ethnic group. To the nearest percent, what percent of the U.S. Black population lived in the South?
\begin{tabular}{|c|cccc|} \hline & NE & MW & South & West \\ \hline White & 42 & 52 & 57 & 35 \\ Black & 5 & 5 & 15 & 2 \\ Asian & 1 & 1 & 1 & 3 \\ Other & 1 & 1 & 2 & 4 \\ \hline \end{tabular} | {
"answer": "56\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $a$ and $b$ are positive numbers such that $a^b=b^a$ and $b=9a$, then the value of $a$ is | {
"answer": "\\sqrt[4]{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3$ cm and $6$ cm. Into each cone is dropped a spherical marble of radius $1$ cm, which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of the rise of the liquid level in the narrow cone to the rise of the liquid level in the wide cone? | {
"answer": "4:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number $2013$ has the property that its units digit is the sum of its other digits, that is $2+0+1=3$. How many integers less than $2013$ but greater than $1000$ have this property? | {
"answer": "46",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular hexagon and an equilateral triangle have equal areas. What is the ratio of the length of a side of the triangle to the length of a side of the hexagon? | {
"answer": "\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the $xy$-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at $(0,0)$, $(0,3)$, $(3,3)$, $(3,1)$, $(5,1)$ and $(5,0)$. The slope of the line through the origin that divides the area of this region exactly in half is | {
"answer": "\\frac{7}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle of radius $10$ inches has its center at the vertex $C$ of an equilateral triangle $ABC$ and passes through the other two vertices. The side $AC$ extended through $C$ intersects the circle at $D$. The number of degrees of angle $ADB$ is: | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$-axis or $y$-axis. Let $A = (-3, 2)$ and $B = (3, -2)$. Consider all possible paths of the bug from $A$ to $B$ of length at most $20$. How many points with integer coordinates lie on at least one of these paths? | {
"answer": "195",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A round table has radius $4$. Six rectangular place mats are placed on the table. Each place mat has width $1$ and length $x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $x$? | {
"answer": "$\\frac{3\\sqrt{7}-\\sqrt{3}}{2}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The price of an article was increased $p\%$. Later the new price was decreased $p\%$. If the last price was one dollar, the original price was: | {
"answer": "\\frac{10000}{10000-p^2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A semicircle is inscribed in an isosceles triangle with base 16 and height 15 so that the diameter of the semicircle is contained in the base of the triangle. What is the radius of the semicircle? | {
"answer": "\\frac{120}{17}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the correct order of the fractions $\frac{15}{11}, \frac{19}{15},$ and $\frac{17}{13},$ from least to greatest? | {
"answer": "\\frac{19}{15}<\\frac{17}{13}<\\frac{15}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled? | {
"answer": "900",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the perimeter of a rectangle is $p$ and its diagonal is $d$, the difference between the length and width of the rectangle is: | {
"answer": "\\frac {\\sqrt {8d^2 - p^2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x) = \frac{x+1}{x-1}$. Then for $x^2 \neq 1$, $f(-x)$ is | {
"answer": "\\frac{1}{f(x)}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with a circumscribed and an inscribed square centered at the origin of a rectangular coordinate system with positive $x$ and $y$ axes is shown in each figure I to IV below.
The inequalities
\(|x|+|y| \leq \sqrt{2(x^{2}+y^{2})} \leq 2\mbox{Max}(|x|, |y|)\)
are represented geometrically* by the figure numbered
* An inequality of the form $f(x, y) \leq g(x, y)$, for all $x$ and $y$ is represented geometrically by a figure showing the containment
$\{\mbox{The set of points }(x, y)\mbox{ such that }g(x, y) \leq a\} \subset\\
\{\mbox{The set of points }(x, y)\mbox{ such that }f(x, y) \leq a\}$
for a typical real number $a$. | {
"answer": "II",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the set of $x$-values satisfying the inequality $|\frac{5-x}{3}|<2$. [The symbol $|a|$ means $+a$ if $a$ is positive,
$-a$ if $a$ is negative,$0$ if $a$ is zero. The notation $1<a<2$ means that a can have any value between $1$ and $2$, excluding $1$ and $2$. ] | {
"answer": "-1 < x < 11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ABCD$ is a rectangle (see the accompanying diagram) with $P$ any point on $\overline{AB}$. $\overline{PS} \perp \overline{BD}$ and $\overline{PR} \perp \overline{AC}$. $\overline{AF} \perp \overline{BD}$ and $\overline{PQ} \perp \overline{AF}$. Then $PR + PS$ is equal to: | {
"answer": "$AF$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements.
Brian: "Mike and I are different species."
Chris: "LeRoy is a frog."
LeRoy: "Chris is a frog."
Mike: "Of the four of us, at least two are toads."
How many of these amphibians are frogs? | {
"answer": "3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up? | {
"answer": "2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $m \ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $(a_1, a_2, a_3, a_4)$ of distinct integers with $1 \le a_i \le m$ for all $i$ such that $m$ divides $a_1+a_2+a_3+a_4$. There is a polynomial
\[q(x) = c_3x^3+c_2x^2+c_1x+c_0\]such that $D(m) = q(m)$ for all odd integers $m\ge 5$. What is $c_1?$ | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $O$ be an interior point of triangle $ABC$, and let $s_1=OA+OB+OC$. If $s_2=AB+BC+CA$, then | {
"answer": "$s_2\\ge 2s_1,s_1 \\le s_2$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For each positive integer $n$, let $S(n)$ be the number of sequences of length $n$ consisting solely of the letters $A$ and $B$, with no more than three $A$s in a row and no more than three $B$s in a row. What is the remainder when $S(2015)$ is divided by $12$? | {
"answer": "8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $f(n)=\tfrac{1}{3} n(n+1)(n+2)$, then $f(r)-f(r-1)$ equals: | {
"answer": "r(r+1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Estimate the time it takes to send $60$ blocks of data over a communications channel if each block consists of $512$ "chunks" and the channel can transmit $120$ chunks per second. | {
"answer": "240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A merchant buys goods at $25\%$ off the list price. He desires to mark the goods so that he can give a discount of $20\%$ on the marked price and still clear a profit of $25\%$ on the selling price. What percent of the list price must he mark the goods? | {
"answer": "125\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A straight line passing through the point $(0,4)$ is perpendicular to the line $x-3y-7=0$. Its equation is: | {
"answer": "y+3x-4=0",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three identical rectangles are put together to form rectangle $ABCD$, as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is 5 feet, what is the area in square feet of rectangle $ABCD$? | {
"answer": "150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a circle of radius $5$ units, $CD$ and $AB$ are perpendicular diameters. A chord $CH$ cutting $AB$ at $K$ is $8$ units long. The diameter $AB$ is divided into two segments whose dimensions are: | {
"answer": "2,8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $Q(z)$ and $R(z)$ be the unique polynomials such that $z^{2021}+1=(z^2+z+1)Q(z)+R(z)$ and the degree of $R$ is less than $2.$ What is $R(z)?$ | {
"answer": "-z",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x)=\sum_{k=2}^{10}(\lfloor kx \rfloor -k \lfloor x \rfloor)$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to $r$. How many distinct values does $f(x)$ assume for $x \ge 0$? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the figure, $\angle A$, $\angle B$, and $\angle C$ are right angles. If $\angle AEB = 40^\circ$ and $\angle BED = \angle BDE$, then $\angle CDE =$ | {
"answer": "95^\\circ",
"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": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rectangle $ABCD$ and right triangle $DCE$ have the same area. They are joined to form a trapezoid. What is $DE$? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three $\text{A's}$, three $\text{B's}$, and three $\text{C's}$ are placed in the nine spaces so that each row and column contains one of each letter. If $\text{A}$ is placed in the upper left corner, how many arrangements are possible? | {
"answer": "4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are $2$ boys for every $3$ girls in Ms. Johnson's math class. If there are $30$ students in her class, what percent of them are boys? | {
"answer": "40\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Trapezoid $ABCD$ has $AD||BC$, $BD = 1$, $\angle DBA = 23^{\circ}$, and $\angle BDC = 46^{\circ}$. The ratio $BC: AD$ is $9: 5$. What is $CD$? | {
"answer": "\\frac{4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the figure, $ABCD$ is a square of side length $1$. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$? | {
"answer": "2-\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Square $ABCD$ has sides of length 3. Segments $CM$ and $CN$ divide the square's area into three equal parts. How long is segment $CM$? | {
"answer": "\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $M$ is the midpoint of side $BC$, $AN$ bisects $\angle BAC$, and $BN \perp AN$. If sides $AB$ and $AC$ have lengths $14$ and $19$, respectively, then find $MN$. | {
"answer": "\\frac{5}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The adjacent map is part of a city: the small rectangles are blocks, and the paths in between are streets.
Each morning, a student walks from intersection $A$ to intersection $B$, always walking along streets shown,
and always going east or south. For variety, at each intersection where he has a choice, he chooses with
probability $\frac{1}{2}$ whether to go east or south. Find the probability that through any given morning, he goes through $C$. | {
"answer": "\\frac{21}{32}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$\overline{AB}$ is a diameter of a circle. Tangents $\overline{AD}$ and $\overline{BC}$ are drawn so that $\overline{AC}$ and $\overline{BD}$ intersect in a point on the circle. If $\overline{AD}=a$ and $\overline{BC}=b$, $a \not= b$, the diameter of the circle is: | {
"answer": "\\sqrt{ab}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eighth row and eighth column) and places other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest number and the least number that appear in the second row from the top? | {
"answer": "367",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $x$ and $y$ are inversely proportional and positive. If $x$ increases by $p\%$, then $y$ decreases by | {
"answer": "\\frac{100p}{100+p}\\%$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx + 2$ passes through no lattice point with $0 < x \leq 100$ for all $m$ such that $\frac{1}{2} < m < a$. What is the maximum possible value of $a$? | {
"answer": "\\frac{50}{99}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A man can commute either by train or by bus. If he goes to work on the train in the morning, he comes home on the bus in the afternoon; and if he comes home in the afternoon on the train, he took the bus in the morning. During a total of $x$ working days, the man took the bus to work in the morning $8$ times, came home by bus in the afternoon $15$ times, and commuted by train (either morning or afternoon) $9$ times. Find $x$. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The arithmetic mean (average) of the first $n$ positive integers is: | {
"answer": "\\frac{n+1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles is this last portion of her run? | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x$ is real and $4y^2+4xy+x+6=0$, then the complete set of values of $x$ for which $y$ is real, is: | {
"answer": "$x \\le -2$ or $x \\ge 3$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Charles has $5q + 1$ quarters and Richard has $q + 5$ quarters. The difference in their money in dimes is: | {
"answer": "10(q - 1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An equilateral triangle is drawn with a side of length $a$. A new equilateral triangle is formed by joining the midpoints of the sides of the first one. Then a third equilateral triangle is formed by joining the midpoints of the sides of the second; and so on forever. The limit of the sum of the perimeters of all the triangles thus drawn is: | {
"answer": "6a",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x$ is such that $\frac{1}{x}<2$ and $\frac{1}{x}>-3$, then: | {
"answer": "x>\\frac{1}{2} \\text{ or } x<-\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $a = -2$, the largest number in the set $\{ -3a, 4a, \frac{24}{a}, a^2, 1\}$ is | {
"answer": "-3a",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle? | {
"answer": "\\frac{\\sqrt{5}-1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let
\[T=\frac{1}{3-\sqrt{8}}-\frac{1}{\sqrt{8}-\sqrt{7}}+\frac{1}{\sqrt{7}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{5}-2}.\]
Then | {
"answer": "T>2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n ?$ | {
"answer": "471",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is , $X$ in centimeters? | {
"answer": "5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ordered triples $(x,y,z)$ of positive integers satisfy $\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600 \text{ and lcm}(y,z)=900$? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden (back, bottom, between). The total number of dots NOT visible in this view is | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A box contains $2$ pennies, $4$ nickels, and $6$ dimes. Six coins are drawn without replacement, with each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least $50$ cents? | {
"answer": "\\frac{127}{924}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Elmer's new car gives $50\%$ percent better fuel efficiency, measured in kilometers per liter, than his old car. However, his new car uses diesel fuel, which is $20\%$ more expensive per liter than the gasoline his old car used. By what percent will Elmer save money if he uses his new car instead of his old car for a long trip? | {
"answer": "20\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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