problem
stringlengths 10
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A laser is placed at the point $(3,5)$. The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the $y$-axis, then hit and bounce off the $x$-axis, then hit the point $(7,5)$. What is the total distance the beam will travel along this path? | {
"answer": "10\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\log_{10}{m}= b-\log_{10}{n}$, then $m=$ | {
"answer": "\\frac{10^{b}}{n}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on pad 1. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad 0 it will be eaten by a patiently waiting snake. If the frog reaches pad 10 it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake? | {
"answer": "\\frac{63}{146}",
"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"
} |
Let $n$ be a positive integer greater than 4 such that the decimal representation of $n!$ ends in $k$ zeros and the decimal representation of $(2n)!$ ends in $3k$ zeros. Let $s$ denote the sum of the four least possible values of $n$. What is the sum of the digits of $s$? | {
"answer": "8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles lie outside regular hexagon $ABCDEF$. The first is tangent to $\overline{AB}$, and the second is tangent to $\overline{DE}$. Both are tangent to lines $BC$ and $FA$. What is the ratio of the area of the second circle to that of the first circle? | {
"answer": "81",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is equivalent to $\sqrt{\frac{x}{1-\frac{x-1}{x}}}$ when $x < 0$? | {
"answer": "-x",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a far-off land three fish can be traded for two loaves of bread and a loaf of bread can be traded for four bags of rice. How many bags of rice is one fish worth? | {
"answer": "2\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a finite sequence $S=(a_1,a_2,\ldots ,a_n)$ of $n$ real numbers, let $A(S)$ be the sequence
$\left(\frac{a_1+a_2}{2},\frac{a_2+a_3}{2},\ldots ,\frac{a_{n-1}+a_n}{2}\right)$
of $n-1$ real numbers. Define $A^1(S)=A(S)$ and, for each integer $m$, $2\le m\le n-1$, define $A^m(S)=A(A^{m-1}(S))$. Suppose $x>0$, and let $S=(1,x,x^2,\ldots ,x^{100})$. If $A^{100}(S)=\left(\frac{1}{2^{50}}\right)$, then what is $x$? | {
"answer": "\\sqrt{2}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside a right circular cone with base radius $5$ and height $12$ are three congruent spheres with radius $r$. Each sphere is tangent to the other two spheres and also tangent to the base and side of the cone. What is $r$? | {
"answer": "\\frac{90-40\\sqrt{3}}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle is inscribed in a triangle with side lengths $8, 13$, and $17$. Let the segments of the side of length $8$, made by a point of tangency, be $r$ and $s$, with $r<s$. What is the ratio $r:s$? | {
"answer": "1:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, $2,5,8,11,14$ is an arithmetic sequence with five terms, in which the first term is $2$ and the constant added is $3$. Each row and each column in this $5\times5$ array is an arithmetic sequence with five terms. The square in the center is labelled $X$ as shown. What is the value of $X$? | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Square $EFGH$ is inside the square $ABCD$ so that each side of $EFGH$ can be extended to pass through a vertex of $ABCD$. Square $ABCD$ has side length $\sqrt {50}$ and $BE = 1$. What is the area of the inner square $EFGH$? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $P$ equal the product of 3,659,893,456,789,325,678 and 342,973,489,379,256. The number of digits in $P$ is: | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points $A, B$ and $C$ on a circle of radius $r$ are situated so that $AB=AC, AB>r$, and the length of minor arc $BC$ is $r$. If angles are measured in radians, then $AB/BC=$ | {
"answer": "\\frac{1}{2}\\csc{\\frac{1}{4}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\frac{\frac{x}{4}}{2}=\frac{4}{\frac{x}{2}}$, then $x=$ | {
"answer": "\\pm 8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$\left(\frac{1}{4}\right)^{-\frac{1}{4}}=$ | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many numbers between $1$ and $2005$ are integer multiples of $3$ or $4$ but not $12$? | {
"answer": "1002",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where we keep track of the order of the summation. For example, $f(4)=6$ because $4$ can be written as $4$, $2+2$, $2+1+1$, $1+2+1$, $1+1+2$, and $1+1+1+1$. Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd. | {
"answer": "2016",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and $5x^2+kx+12=0$ has at least one integer solution for $x$. What is $N$? | {
"answer": "78",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\sin{2x}\sin{3x}=\cos{2x}\cos{3x}$, then one value for $x$ is | {
"answer": "18",
"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"
} |
Three-digit powers of $2$ and $5$ are used in this "cross-number" puzzle. What is the only possible digit for the outlined square?
\[\begin{array}{lcl} \textbf{ACROSS} & & \textbf{DOWN} \\ \textbf{2}.~ 2^m & & \textbf{1}.~ 5^n \end{array}\] | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$ and the line $x=5.1?$ | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Quadrilateral $ABCD$ is inscribed in circle $O$ and has side lengths $AB=3, BC=2, CD=6$, and $DA=8$. Let $X$ and $Y$ be points on $\overline{BD}$ such that $\frac{DX}{BD} = \frac{1}{4}$ and $\frac{BY}{BD} = \frac{11}{36}$.
Let $E$ be the intersection of line $AX$ and the line through $Y$ parallel to $\overline{AD}$. Let $F$ be the intersection of line $CX$ and the line through $E$ parallel to $\overline{AC}$. Let $G$ be the point on circle $O$ other than $C$ that lies on line $CX$. What is $XF\cdot XG$? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $a$, $b$ and $c$ are positive integers with $a+b+c=2006$, and $a!b!c!=m\cdot 10^n$, where $m$ and $n$ are integers and $m$ is not divisible by $10$. What is the smallest possible value of $n$? | {
"answer": "492",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x=t^{\frac{1}{t-1}}$ and $y=t^{\frac{t}{t-1}},t>0,t \ne 1$, a relation between $x$ and $y$ is: | {
"answer": "$y^x=x^y$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments.
How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?
some rotation around a point of line $\ell$
some translation in the direction parallel to line $\ell$
the reflection across line $\ell$
some reflection across a line perpendicular to line $\ell$ | {
"answer": "2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Liliane has $50\%$ more soda than Jacqueline, and Alice has $25\%$ more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alice have? | {
"answer": "$20\\%$",
"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"
} |
What is the correct ordering of the three numbers $\frac{5}{19}$, $\frac{7}{21}$, and $\frac{9}{23}$, in increasing order? | {
"answer": "\\frac{5}{19} < \\frac{7}{21} < \\frac{9}{23}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers on the faces of this cube are consecutive whole numbers. The sum of the two numbers on each of the three pairs of opposite faces are equal. The sum of the six numbers on this cube is | {
"answer": "81",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle with vertices $(6, 5)$, $(8, -3)$, and $(9, 1)$ is reflected about the line $x=8$ to create a second triangle. What is the area of the union of the two triangles? | {
"answer": "\\frac{32}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $B$ be a right rectangular prism (box) with edges lengths $1,$ $3,$ and $4$, together with its interior. For real $r\geq0$, let $S(r)$ be the set of points in $3$-dimensional space that lie within a distance $r$ of some point in $B$. The volume of $S(r)$ can be expressed as $ar^{3} + br^{2} + cr +d$, where $a,$ $b,$ $c,$ and $d$ are positive real numbers. What is $\frac{bc}{ad}?$ | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen? | {
"answer": "170",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside square $ABCD$ with side $s$, quarter-circle arcs with radii $s$ and centers at $A$ and $B$ are drawn. These arcs intersect at a point $X$ inside the square. How far is $X$ from the side of $CD$? | {
"answer": "\\frac{1}{2} s(2-\\sqrt{3})",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ratio of the areas of two concentric circles is $1: 3$. If the radius of the smaller is $r$, then the difference between the radii is best approximated by: | {
"answer": "0.73r",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sides of a regular polygon of $n$ sides, $n>4$, are extended to form a star. The number of degrees at each point of the star is: | {
"answer": "\\frac{(n-2)180}{n}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ways are there to paint each of the integers $2, 3, \cdots , 9$ either red, green, or blue so that each number has a different color from each of its proper divisors? | {
"answer": "432",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One of the factors of $x^4+4$ is: | {
"answer": "$x^2-2x+2$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product
\[n = f_1\cdot f_2\cdots f_k,\]where $k\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number $6$ can be written as $6$, $2\cdot 3$, and $3\cdot2$, so $D(6) = 3$. What is $D(96)$? | {
"answer": "112",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
John scores 93 on this year's AHSME. Had the old scoring system still been in effect, he would score only 84 for the same answers.
How many questions does he leave unanswered? | {
"answer": "9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to
[asy]
unitsize(3mm); defaultpen(linewidth(0.8pt));
path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0);
path p2=(0,1)--(1,1)--(1,0);
path p3=(2,0)--(2,1)--(3,1);
path p4=(3,2)--(2,2)--(2,3);
path p5=(1,3)--(1,2)--(0,2);
path p6=(1,1)--(2,2);
path p7=(2,1)--(1,2);
path[] p=p1^^p2^^p3^^p4^^p5^^p6^^p7;
for(int i=0; i<3; ++i) {
for(int j=0; j<3; ++j) {
draw(shift(3*i,3*j)*p);
}
}
[/asy] | {
"answer": "56",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The average cost of a long-distance call in the USA in $1985$ was
$41$ cents per minute, and the average cost of a long-distance
call in the USA in $2005$ was $7$ cents per minute. Find the
approximate percent decrease in the cost per minute of a long-
distance call. | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a rectangle and let $\overline{DM}$ be a segment perpendicular to the plane of $ABCD$. Suppose that $\overline{DM}$ has integer length, and the lengths of $\overline{MA}, \overline{MC},$ and $\overline{MB}$ are consecutive odd positive integers (in this order). What is the volume of pyramid $MABCD?$ | {
"answer": "24\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the adjoining plane figure, sides $AF$ and $CD$ are parallel, as are sides $AB$ and $EF$,
and sides $BC$ and $ED$. Each side has length $1$. Also, $\angle FAB = \angle BCD = 60^\circ$.
The area of the figure is | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are shown. What is the area of the fourth rectangle? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $|x-\log y|=x+\log y$ where $x$ and $\log y$ are real, then | {
"answer": "x(y-1)=0",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x_{k+1} = x_k + \frac{1}{2}$ for $k=1, 2, \dots, n-1$ and $x_1=1$, find $x_1 + x_2 + \dots + x_n$. | {
"answer": "\\frac{n^2+3n}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define $x\otimes y=x^3-y$. What is $h\otimes (h\otimes h)$? | {
"answer": "h",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $9$ trapezoids, let $x$ be the angle measure in degrees of the larger interior angle of the trapezoid. What is $x$?
[asy] unitsize(4mm); defaultpen(linewidth(.8pt)); int i; real r=5, R=6; path t=r*dir(0)--r*dir(20)--R*dir(20)--R*dir(0); for(i=0; i<9; ++i) { draw(rotate(20*i)*t); } draw((-r,0)--(R+1,0)); draw((-R,0)--(-R-1,0)); [/asy] | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number $S$. What is the smallest possible value for the sum of the digits of $S$? | {
"answer": "4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Indicate in which one of the following equations $y$ is neither directly nor inversely proportional to $x$: | {
"answer": "$3x + y = 10$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime? | {
"answer": "\\frac{7}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Circles with centers $A$, $B$, and $C$ each have radius $r$, where $1 < r < 2$. The distance between each pair of centers is $2$. If $B'$ is the point of intersection of circle $A$ and circle $C$ which is outside circle $B$, and if $C'$ is the point of intersection of circle $A$ and circle $B$ which is outside circle $C$, then length $B'C'$ equals | {
"answer": "1+\\sqrt{3(r^2-1)}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A positive number is mistakenly divided by $6$ instead of being multiplied by $6.$ Based on the correct answer, the error thus committed, to the nearest percent, is | {
"answer": "97",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The domain of the function $f(x)=\log_{\frac{1}{2}}(\log_4(\log_{\frac{1}{4}}(\log_{16}(\log_{\frac{1}{16}}x))))$ is an interval of length $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | {
"answer": "271",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the $1992^{\text{nd}}$ letter in this sequence?
\[\text{ABCDEDCBAABCDEDCBAABCDEDCBAABCDEDC}\cdots\] | {
"answer": "C",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each of the $12$ edges of a cube is labeled $0$ or $1$. Two labelings are considered different even if one can be obtained from the other by a sequence of one or more rotations and/or reflections. For how many such labelings is the sum of the labels on the edges of each of the $6$ faces of the cube equal to $2$? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of a circle is doubled when its radius $r$ is increased by $n$. Then $r$ equals: | {
"answer": "n(\\sqrt{2} + 1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Liquid $X$ does not mix with water. Unless obstructed, it spreads out on the surface of water to form a circular film $0.1$cm thick. A rectangular box measuring $6$cm by $3$cm by $12$cm is filled with liquid $X$. Its contents are poured onto a large body of water. What will be the radius, in centimeters, of the resulting circular film? | {
"answer": "\\sqrt{\\frac{2160}{\\pi}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $a=\frac{1}{2}$ and $(a+1)(b+1)=2$ then the radian measure of $\arctan a + \arctan b$ equals | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A painting $18$" X $24$" is to be placed into a wooden frame with the longer dimension vertical. The wood at the top and bottom is twice as wide as the wood on the sides. If the frame area equals that of the painting itself, the ratio of the smaller to the larger dimension of the framed painting is: | {
"answer": "2:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, $AB=AC$ and $\measuredangle A=80^\circ$. If points $D, E$, and $F$ lie on sides $BC, AC$ and $AB$, respectively, and $CE=CD$ and $BF=BD$, then $\measuredangle EDF$ equals | {
"answer": "50^\\circ",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points.
Suppose $PQ=6$ and $RS=8,$ and let $d$ denote the length of $\overline{BD},$ the longer diagonal of $ABCD.$ Then $d^2$ can be written in the form $m+n\sqrt p,$ where $m,n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m+n+p?$ | {
"answer": "81",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures
Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.
$\circ$ Art's cookies are trapezoids:
$\circ$ Roger's cookies are rectangles:
$\circ$ Paul's cookies are parallelograms:
$\circ$ Trisha's cookies are triangles:
How many cookies will be in one batch of Trisha's cookies? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular octahedron has side length $1$. A plane parallel to two of its opposite faces cuts the octahedron into the two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area $\frac {a\sqrt {b}}{c}$, where $a$, $b$, and $c$ are positive integers, $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of any prime. What is $a + b + c$? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The times between $7$ and $8$ o'clock, correct to the nearest minute, when the hands of a clock will form an angle of $84^{\circ}$ are: | {
"answer": "7: 23 and 7: 53",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\frac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\frac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N) < \frac{321}{400}$? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The probability that event $A$ occurs is $\frac{3}{4}$; the probability that event B occurs is $\frac{2}{3}$.
Let $p$ be the probability that both $A$ and $B$ occur. The smallest interval necessarily containing $p$ is the interval | {
"answer": "[\\frac{5}{12},\\frac{2}{3}]",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a number $N, N \ne 0$, diminished by four times its reciprocal, equals a given real constant $R$, then, for this given $R$, the sum of all such possible values of $N$ is | {
"answer": "R",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\triangle{ABC}$? | {
"answer": "$3\\sqrt{3}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\angle 1 + \angle 2 = 180^\circ
\angle 3 = \angle 4
Find \angle 4. | {
"answer": "35^\\circ",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Functions $f$ and $g$ are quadratic, $g(x) = - f(100 - x)$, and the graph of $g$ contains the vertex of the graph of $f$. The four $x$-intercepts on the two graphs have $x$-coordinates $x_1$, $x_2$, $x_3$, and $x_4$, in increasing order, and $x_3 - x_2 = 150$. Then $x_4 - x_1 = m + n\sqrt p$, where $m$, $n$, and $p$ are positive integers, and $p$ is not divisible by the square of any prime. What is $m + n + p$? | {
"answer": "752",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mr. Lopez has a choice of two routes to get to work. Route A is $6$ miles long, and his average speed along this route is $30$ miles per hour. Route B is $5$ miles long, and his average speed along this route is $40$ miles per hour, except for a $\frac{1}{2}$-mile stretch in a school zone where his average speed is $20$ miles per hour. By how many minutes is Route B quicker than Route A? | {
"answer": "3 \\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What fraction of the large $12$ by $18$ rectangular region is shaded? | {
"answer": "\\frac{1}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points $A$ and $B$ are on a circle of radius $5$ and $AB = 6$. Point $C$ is the midpoint of the minor arc $AB$. What is the length of the line segment $AC$? | {
"answer": "\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Trapezoid $ABCD$ has $\overline{AB}\parallel\overline{CD}, BC=CD=43$, and $\overline{AD}\perp\overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP=11$, the length of $AD$ can be written in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m+n$? | {
"answer": "194",
"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"
} |
An "$n$-pointed star" is formed as follows: the sides of a convex polygon are numbered consecutively $1,2,\cdots ,k,\cdots,n,\text{ }n\ge 5$; for all $n$ values of $k$, sides $k$ and $k+2$ are non-parallel, sides $n+1$ and $n+2$ being respectively identical with sides $1$ and $2$; prolong the $n$ pairs of sides numbered $k$ and $k+2$ until they meet. (A figure is shown for the case $n=5$).
Let $S$ be the degree-sum of the interior angles at the $n$ points of the star; then $S$ equals: | {
"answer": "180(n-4)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\angle AFE$? | {
"answer": "170",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cube with side length $1$ is sliced by a plane that passes through two diagonally opposite vertices $A$ and $C$ and the midpoints $B$ and $D$ of two opposite edges not containing $A$ or $C$, as shown. What is the area of quadrilateral $ABCD$? | {
"answer": "\\frac{\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\tan{\alpha}$ and $\tan{\beta}$ are the roots of $x^2 - px + q = 0$, and $\cot{\alpha}$ and $\cot{\beta}$ are the roots of $x^2 - rx + s = 0$, then $rs$ is necessarily | {
"answer": "\\frac{p}{q^2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given right $\triangle ABC$ with legs $BC=3, AC=4$. Find the length of the shorter angle trisector from $C$ to the hypotenuse: | {
"answer": "\\frac{12\\sqrt{3}-9}{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Hexadecimal (base-16) numbers are written using numeric digits $0$ through $9$ as well as the letters $A$ through $F$ to represent $10$ through $15$. Among the first $1000$ positive integers, there are $n$ whose hexadecimal representation contains only numeric digits. What is the sum of the digits of $n$? | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integers $N$ between $1$ and $1990$ is the improper fraction $\frac{N^2+7}{N+4}$ not in lowest terms? | {
"answer": "86",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$. Diagonals $AC$ and $BD$ intersect at $E$, $AC = 14$, and $\triangle AED$ and $\triangle BEC$ have equal areas. What is $AE$? | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The expression $\frac{2021}{2020} - \frac{2020}{2021}$ is equal to the fraction $\frac{p}{q}$ in which $p$ and $q$ are positive integers whose greatest common divisor is 1. What is $p?$ | {
"answer": "2021",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The radius of the first circle is $1$ inch, that of the second $\frac{1}{2}$ inch, that of the third $\frac{1}{4}$ inch and so on indefinitely. The sum of the areas of the circles is: | {
"answer": "\\frac{4\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Michael walks at the rate of $5$ feet per second on a long straight path. Trash pails are located every $200$ feet along the path. A garbage truck traveling at $10$ feet per second in the same direction as Michael stops for $30$ seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet? | {
"answer": "5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Makarla attended two meetings during her $9$-hour work day. The first meeting took $45$ minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A month with $31$ days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month? | {
"answer": "2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a tournament there are six teams that play each other twice. A team earns $3$ points for a win, $1$ point for a draw, and $0$ points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$? | {
"answer": "75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\frac{x}{x-1} = \frac{y^2 + 2y - 1}{y^2 + 2y - 2},$ then $x$ equals | {
"answer": "y^2 + 2y - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A paint brush is swept along both diagonals of a square to produce the symmetric painted area, as shown. Half the area of the square is painted. What is the ratio of the side length of the square to the brush width? | {
"answer": "2\\sqrt{2}+2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sequence of numbers is defined by $D_0=0,D_1=0,D_2=1$ and $D_n=D_{n-1}+D_{n-3}$ for $n\ge 3$. What are the parities (evenness or oddness) of the triple of numbers $(D_{2021},D_{2022},D_{2023})$, where $E$ denotes even and $O$ denotes odd? | {
"answer": "(E,O,E)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The lengths of two line segments are $a$ units and $b$ units respectively. Then the correct relation between them is: | {
"answer": "\\frac{a+b}{2} \\geq \\sqrt{ab}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Subsets and Splits