problem stringlengths 10 5.15k | answer dict |
|---|---|
For how many integers $n$ between $1$ and $50$, inclusive, is $\frac{(n^2-1)!}{(n!)^n}$ an integer? | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a set $S$ consisting of two undefined elements "pib" and "maa", and the four postulates: $P_1$: Every pib is a collection of maas, $P_2$: Any two distinct pibs have one and only one maa in common, $P_3$: Every maa belongs to two and only two pibs, $P_4$: There are exactly four pibs. Consider the three theorems: $T_1$: There are exactly six maas, $T_2$: There are exactly three maas in each pib, $T_3$: For each maa there is exactly one other maa not in the same pid with it. The theorems which are deducible from the postulates are: | {
"answer": "all",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles? | {
"answer": "\\frac{\\sqrt{5}-1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The closed curve in the figure is made up of 9 congruent circular arcs each of length $\frac{2\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side 2. What is the area enclosed by the curve? | {
"answer": "\\pi + 6\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At the beginning of a trip, the mileage odometer read $56,200$ miles. The driver filled the gas tank with $6$ gallons of gasoline. During the trip, the driver filled his tank again with $12$ gallons of gasoline when the odometer read $56,560$. At the end of the trip, the driver filled his tank again with $20$ gallons of gasoline. The odometer read $57,060$. To the nearest tenth, what was the car's average miles-per-gallon for the entire trip? | {
"answer": "26.9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square with side length 8 is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles? | {
"answer": "4 \\times 8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Professor Gamble buys a lottery ticket, which requires that he pick six different integers from $1$ through $46$, inclusive. He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer. It so happens that the integers on the winning ticket have the same property— the sum of the base-ten logarithms is an integer. What is the probability that Professor Gamble holds the winning ticket? | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $t_n = \frac{n(n+1)}{2}$ be the $n$th triangular number. Find
\[\frac{1}{t_1} + \frac{1}{t_2} + \frac{1}{t_3} + ... + \frac{1}{t_{2002}}\] | {
"answer": "\\frac {4004}{2003}",
"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"
} |
If $p$ is a prime and both roots of $x^2+px-444p=0$ are integers, then | {
"answer": "31 < p \\le 41",
"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"
} |
There are $10$ horses, named Horse $1$, Horse $2$, . . . , Horse $10$. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time $0$ all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time $S > 0$, in minutes, at which all $10$ horses will again simultaneously be at the starting point is $S=2520$. Let $T > 0$ be the least time, in minutes, such that at least $5$ of the horses are again at the starting point. What is the sum of the digits of $T?$ | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In trapezoid $ABCD$, the sides $AB$ and $CD$ are equal. The perimeter of $ABCD$ is | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A non-zero digit is chosen in such a way that the probability of choosing digit $d$ is $\log_{10}{(d+1)}-\log_{10}{d}$. The probability that the digit $2$ is chosen is exactly $\frac{1}{2}$ the probability that the digit chosen is in the set | {
"answer": "{4, 5, 6, 7, 8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A dress originally priced at $80$ dollars was put on sale for $25\%$ off. If $10\%$ tax was added to the sale price, then the total selling price (in dollars) of the dress was | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points $R$, $S$ and $T$ are vertices of an equilateral triangle, and points $X$, $Y$ and $Z$ are midpoints of its sides. How many noncongruent triangles can be drawn using any three of these six points as vertices? | {
"answer": "4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of digits of all the numbers in the sequence $1,2,3,4,\cdots ,10000$. | {
"answer": "180001",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the dart board shown in the figure below, the outer circle has radius $6$ and the inner circle has radius $3$. Three radii divide each circle into three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to the area of the region. When two darts hit this board, the score is the sum of the point values of the regions hit. What is the probability that the score is odd? | {
"answer": "\\frac{35}{72}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A big $L$ is formed as shown. What is its area? | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Right triangle $ABC$ has leg lengths $AB=20$ and $BC=21$. Including $\overline{AB}$ and $\overline{BC}$, how many line segments with integer length can be drawn from vertex $B$ to a point on hypotenuse $\overline{AC}$? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jerry cuts a wedge from a 6-cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters? | {
"answer": "603",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two distinct numbers $a$ and $b$ are chosen randomly from the set $\{2, 2^2, 2^3, ..., 2^{25}\}$. What is the probability that $\log_a b$ is an integer? | {
"answer": "\\frac{31}{300}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Regular polygons with $5, 6, 7,$ and $8$ sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect? | {
"answer": "68",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $n$ is a multiple of $4$, the sum $s=1+2i+3i^2+\cdots+(n+1)i^n$, where $i=\sqrt{-1}$, equals: | {
"answer": "\\frac{1}{2}(n+2-ni)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn't there, the two helpers painted only 24% of the house and quit at 2:12 PM. On Wednesday Paula worked by herself and finished the house by working until 7:12 P.M. How long, in minutes, was each day's lunch break? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Pauline Bunyan can shovel snow at the rate of $20$ cubic yards for the first hour, $19$ cubic yards for the second, $18$ for the third, etc., always shoveling one cubic yard less per hour than the previous hour. If her driveway is $4$ yards wide, $10$ yards long, and covered with snow $3$ yards deep, then the number of hours it will take her to shovel it clean is closest to | {
"answer": "7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What percent of the blend is pear juice? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube? | {
"answer": "5\\sqrt{2} - 7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is $\frac{1}{2}$ foot from the top face. The second cut is $\frac{1}{3}$ foot below the first cut, and the third cut is $\frac{1}{17}$ foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end. What is the total surface area of this solid in square feet? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Of the following complex numbers $z$, which one has the property that $z^5$ has the greatest real part? | {
"answer": "-\\sqrt{3} + i",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $m$ and $b$ are real numbers and $mb>0$, then the line whose equation is $y=mx+b$ cannot contain the point | {
"answer": "(1997,0)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A housewife saved $2.50 in buying a dress on sale. If she spent $25 for the dress, she saved about: | {
"answer": "9 \\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When one ounce of water is added to a mixture of acid and water, the new mixture is $20\%$ acid. When one ounce of acid is added to the new mixture, the result is $33\frac13\%$ acid. The percentage of acid in the original mixture is | {
"answer": "25\\%",
"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"
} |
A mixture of $30$ liters of paint is $25\%$ red tint, $30\%$ yellow tint and $45\%$ water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$ has $AB=2 \cdot AC$. Let $D$ and $E$ be on $\overline{AB}$ and $\overline{BC}$, respectively, such that $\angle BAE = \angle ACD$. Let $F$ be the intersection of segments $AE$ and $CD$, and suppose that $\triangle CFE$ is equilateral. What is $\angle ACB$? | {
"answer": "90^{\\circ}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sofia ran $5$ laps around the $400$-meter track at her school. For each lap, she ran the first $100$ meters at an average speed of $4$ meters per second and the remaining $300$ meters at an average speed of $5$ meters per second. How much time did Sofia take running the $5$ laps? | {
"answer": "7 minutes and 5 seconds",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each half of this figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. When the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. There are 2 red-white pairs. How many white pairs coincide? | {
"answer": "5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n?$ | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of two numbers is $S$. Suppose $3$ is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers? | {
"answer": "2S + 12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once? | {
"answer": "\\frac{1}{729}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of $\triangle ABC$? | {
"answer": "\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Values for $A, B, C,$ and $D$ are to be selected from $\{1, 2, 3, 4, 5, 6\}$ without replacement (i.e. no two letters have the same value). How many ways are there to make such choices so that the two curves $y=Ax^2+B$ and $y=Cx^2+D$ intersect? (The order in which the curves are listed does not matter; for example, the choices $A=3, B=2, C=4, D=1$ is considered the same as the choices $A=4, B=1, C=3, D=2.$) | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many positive integers $n \le 1000$ is$\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor$not divisible by $3$? | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $F=\frac{6x^2+16x+3m}{6}$ be the square of an expression which is linear in $x$. Then $m$ has a particular value between: | {
"answer": "3 and 4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two positive numbers $a$, $b$ such that $a<b$. Let $A.M.$ be their arithmetic mean and let $G.M.$ be their positive geometric mean. Then $A.M.$ minus $G.M.$ is always less than: | {
"answer": "\\frac{(b-a)^2}{8a}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the right triangle shown the sum of the distances $BM$ and $MA$ is equal to the sum of the distances $BC$ and $CA$.
If $MB = x, CB = h$, and $CA = d$, then $x$ equals: | {
"answer": "\\frac{hd}{2h+d}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle of radius 2 is centered at $A$. An equilateral triangle with side 4 has a vertex at $A$. What is the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle? | {
"answer": "$4(\\pi - \\sqrt{3})$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Real numbers $x$ and $y$ satisfy $x + y = 4$ and $x \cdot y = -2$. What is the value of
\[x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y?\] | {
"answer": "440",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Samuel's birthday cake is in the form of a $4 \times 4 \times 4$ inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into $64$ smaller cubes, each measuring $1 \times 1 \times 1$ inch, as shown below. How many of the small pieces will have icing on exactly two sides? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The bar graph shows the grades in a mathematics class for the last grading period. If A, B, C, and D are satisfactory grades, what fraction of the grades shown in the graph are satisfactory? | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On this monthly calendar, the date behind one of the letters is added to the date behind $\text{C}$. If this sum equals the sum of the dates behind $\text{A}$ and $\text{B}$, then the letter is | {
"answer": "P",
"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"
} |
What is the area enclosed by the geoboard quadrilateral below?
[asy] unitsize(3mm); defaultpen(linewidth(.8pt)); dotfactor=2; for(int a=0; a<=10; ++a) for(int b=0; b<=10; ++b) { dot((a,b)); }; draw((4,0)--(0,5)--(3,4)--(10,10)--cycle); [/asy] | {
"answer": "22\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Part of an \(n\)-pointed regular star is shown. It is a simple closed polygon in which all \(2n\) edges are congruent, angles \(A_1,A_2,\cdots,A_n\) are congruent, and angles \(B_1,B_2,\cdots,B_n\) are congruent. If the acute angle at \(A_1\) is \(10^\circ\) less than the acute angle at \(B_1\), then \(n=\) | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four of the eight vertices of a cube are the vertices of a regular tetrahedron. Find the ratio of the surface area of the cube to the surface area of the tetrahedron. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Squares $ABCD$ and $EFGH$ are congruent, $AB=10$, and $G$ is the center of square $ABCD$. The area of the region in the plane covered by these squares is | {
"answer": "175",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two counterfeit coins of equal weight are mixed with $8$ identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the $10$ coins. A second pair is selected at random without replacement from the remaining $8$ coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all $4$ selected coins are genuine? | {
"answer": "\\frac{15}{19}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the xy-plane, what is the length of the shortest path from $(0,0)$ to $(12,16)$ that does not go inside the circle $(x-6)^{2}+(y-8)^{2}= 25$? | {
"answer": "10\\sqrt{3}+\\frac{5\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram above shows several numbers in the complex plane. The circle is the unit circle centered at the origin.
One of these numbers is the reciprocal of $F$. Which one? | {
"answer": "C",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each of the $20$ balls is tossed independently and at random into one of the $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ balls. What is $\frac{p}{q}$? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7.$ Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice? | {
"answer": "\\frac{7}{36}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the pattern in the diagram continues, what fraction of eighth triangle would be shaded?
[asy] unitsize(10); draw((0,0)--(12,0)--(6,6sqrt(3))--cycle); draw((15,0)--(27,0)--(21,6sqrt(3))--cycle); fill((21,0)--(18,3sqrt(3))--(24,3sqrt(3))--cycle,black); draw((30,0)--(42,0)--(36,6sqrt(3))--cycle); fill((34,0)--(32,2sqrt(3))--(36,2sqrt(3))--cycle,black); fill((38,0)--(36,2sqrt(3))--(40,2sqrt(3))--cycle,black); fill((36,2sqrt(3))--(34,4sqrt(3))--(38,4sqrt(3))--cycle,black); draw((45,0)--(57,0)--(51,6sqrt(3))--cycle); fill((48,0)--(46.5,1.5sqrt(3))--(49.5,1.5sqrt(3))--cycle,black); fill((51,0)--(49.5,1.5sqrt(3))--(52.5,1.5sqrt(3))--cycle,black); fill((54,0)--(52.5,1.5sqrt(3))--(55.5,1.5sqrt(3))--cycle,black); fill((49.5,1.5sqrt(3))--(48,3sqrt(3))--(51,3sqrt(3))--cycle,black); fill((52.5,1.5sqrt(3))--(51,3sqrt(3))--(54,3sqrt(3))--cycle,black); fill((51,3sqrt(3))--(49.5,4.5sqrt(3))--(52.5,4.5sqrt(3))--cycle,black); [/asy] | {
"answer": "\\frac{7}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cube with $3$-inch edges is to be constructed from $27$ smaller cubes with $1$-inch edges. Twenty-one of the cubes are colored red and $6$ are colored white. If the $3$-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white? | {
"answer": "\\frac{5}{54}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a convex quadrilateral with $AB = CD = 10$, $BC = 14$, and $AD = 2\sqrt{65}$. Assume that the diagonals of $ABCD$ intersect at point $P$, and that the sum of the areas of triangles $APB$ and $CPD$ equals the sum of the areas of triangles $BPC$ and $APD$. Find the area of quadrilateral $ABCD$. | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the figure, the length of side $AB$ of square $ABCD$ is $\sqrt{50}$ and $BE=1$. What is the area of the inner square $EFGH$? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task? | {
"answer": "3:30 PM",
"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"
} |
Figures $I$, $II$, and $III$ are squares. The perimeter of $I$ is $12$ and the perimeter of $II$ is $24$. The perimeter of $III$ is | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $OAB$ has $O=(0,0)$, $B=(5,0)$, and $A$ in the first quadrant. In addition, $\angle ABO=90^\circ$ and $\angle AOB=30^\circ$. Suppose that $OA$ is rotated $90^\circ$ counterclockwise about $O$. What are the coordinates of the image of $A$? | {
"answer": "$\\left( - \\frac {5}{3}\\sqrt {3},5\\right)$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a neglibile height above the equator, then, among the following choices, the best estimate of the number of hours of flight is: | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A coin is altered so that the probability that it lands on heads is less than $\frac{1}{2}$ and when the coin is flipped four times, the probability of an equal number of heads and tails is $\frac{1}{6}$. What is the probability that the coin lands on heads? | {
"answer": "\\frac{3-\\sqrt{3}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The figure is constructed from $11$ line segments, each of which has length $2$. The area of pentagon $ABCDE$ can be written as $\sqrt{m} + \sqrt{n}$, where $m$ and $n$ are positive integers. What is $m + n ?$ | {
"answer": "23",
"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"
} |
Erin the ant starts at a given corner of a cube and crawls along exactly 7 edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point. How many paths are there meeting these conditions? | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two boys start moving from the same point A on a circular track but in opposite directions. Their speeds are 5 ft. per second and 9 ft. per second. If they start at the same time and finish when they first meet at the point A again, then the number of times they meet, excluding the start and finish, is | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. What is the smallest possible value of the length of the side of the original square? | {
"answer": "4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM the machine has completed one third of the day's job. At what time will the doughnut machine complete the job? | {
"answer": "4:30 PM",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ be the square one of whose diagonals has endpoints $(1/10,7/10)$ and $(-1/10,-7/10)$. A point $v=(x,y)$ is chosen uniformly at random over all pairs of real numbers $x$ and $y$ such that $0 \le x \le 2012$ and $0\le y\le 2012$. Let $T(v)$ be a translated copy of $S$ centered at $v$. What is the probability that the square region determined by $T(v)$ contains exactly two points with integer coefficients in its interior? | {
"answer": "\\frac{4}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are exactly $77,000$ ordered quadruplets $(a, b, c, d)$ such that $\gcd(a, b, c, d) = 77$ and $\operatorname{lcm}(a, b, c, d) = n$. What is the smallest possible value for $n$? | {
"answer": "27,720",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\circ$. The enclosed area of the hexagon is $6\sqrt{3}$. What is the perimeter of the hexagon? | {
"answer": "12\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome $n$ is chosen uniformly at random. What is the probability that $\frac{n}{11}$ is also a palindrome? | {
"answer": "\\frac{11}{30}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Figure $OPQR$ is a square. Point $O$ is the origin, and point $Q$ has coordinates (2,2). What are the coordinates for $T$ so that the area of triangle $PQT$ equals the area of square $OPQR$? | {
"answer": "(-2,0)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6$ numbers obtained. What is the probability that the product is divisible by $4$? | {
"answer": "\\frac{63}{64}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A permutation $(a_1,a_2,a_3,a_4,a_5)$ of $(1,2,3,4,5)$ is heavy-tailed if $a_1 + a_2 < a_4 + a_5$. What is the number of heavy-tailed permutations? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two distinct lines pass through the center of three concentric circles of radii 3, 2, and 1. The area of the shaded region in the diagram is $\frac{8}{13}$ of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: $\pi$ radians is $180$ degrees.)
[asy] size(85); fill((-30,0)..(-24,18)--(0,0)--(-24,-18)..cycle,gray(0.7)); fill((30,0)..(24,18)--(0,0)--(24,-18)..cycle,gray(0.7)); fill((-20,0)..(0,20)--(0,-20)..cycle,white); fill((20,0)..(0,20)--(0,-20)..cycle,white); fill((0,20)..(-16,12)--(0,0)--(16,12)..cycle,gray(0.7)); fill((0,-20)..(-16,-12)--(0,0)--(16,-12)..cycle,gray(0.7)); fill((0,10)..(-10,0)--(10,0)..cycle,white); fill((0,-10)..(-10,0)--(10,0)..cycle,white); fill((-10,0)..(-8,6)--(0,0)--(-8,-6)..cycle,gray(0.7)); fill((10,0)..(8,6)--(0,0)--(8,-6)..cycle,gray(0.7)); draw(Circle((0,0),10),linewidth(0.7)); draw(Circle((0,0),20),linewidth(0.7)); draw(Circle((0,0),30),linewidth(0.7)); draw((-28,-21)--(28,21),linewidth(0.7)); draw((-28,21)--(28,-21),linewidth(0.7));[/asy] | {
"answer": "\\frac{\\pi}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the figure, $\overline{AB}$ and $\overline{CD}$ are diameters of the circle with center $O$, $\overline{AB} \perp \overline{CD}$, and chord $\overline{DF}$ intersects $\overline{AB}$ at $E$. If $DE = 6$ and $EF = 2$, then the area of the circle is | {
"answer": "24 \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In right triangle $ABC$ the hypotenuse $\overline{AB}=5$ and leg $\overline{AC}=3$. The bisector of angle $A$ meets the opposite side in $A_1$. A second right triangle $PQR$ is then constructed with hypotenuse $\overline{PQ}=A_1B$ and leg $\overline{PR}=A_1C$. If the bisector of angle $P$ meets the opposite side in $P_1$, the length of $PP_1$ is: | {
"answer": "\\frac{3\\sqrt{5}}{4}",
"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"
} |
Carlos took $70\%$ of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left? | {
"answer": "20\\%",
"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"
} |
The graph shows the distribution of the number of children in the families of the students in Ms. Jordan's English class. The median number of children in the family for this distribution is | {
"answer": "4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $C_1$ and $C_2$ be circles of radius 1 that are in the same plane and tangent to each other. How many circles of radius 3 are in this plane and tangent to both $C_1$ and $C_2$? | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Seven cubes, whose volumes are $1$, $8$, $27$, $64$, $125$, $216$, and $343$ cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units? | {
"answer": "749",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a geometric sequence of real numbers, the sum of the first $2$ terms is $7$, and the sum of the first $6$ terms is $91$. The sum of the first $4$ terms is | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The symbol $R_k$ stands for an integer whose base-ten representation is a sequence of $k$ ones. For example, $R_3=111, R_5=11111$, etc. When $R_{24}$ is divided by $R_4$, the quotient $Q=R_{24}/R_4$ is an integer whose base-ten representation is a sequence containing only ones and zeroes. The number of zeros in $Q$ is: | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A person starting with $64$ and making $6$ bets, wins three times and loses three times, the wins and losses occurring in random order. The chance for a win is equal to the chance for a loss. If each wager is for half the money remaining at the time of the bet, then the final result is: | {
"answer": "$37$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rhombus $ABCD$ has side length $2$ and $\angle B = 120^\circ$. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$? | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Squares $ABCD$, $EFGH$, and $GHIJ$ are equal in area. Points $C$ and $D$ are the midpoints of sides $IH$ and $HE$, respectively. What is the ratio of the area of the shaded pentagon $AJICB$ to the sum of the areas of the three squares? | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number obtained from the last two nonzero digits of $90!$ is equal to $n$. What is $n$? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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