problem stringlengths 10 5.15k | answer dict |
|---|---|
The $120$ permutations of $AHSME$ are arranged in dictionary order as if each were an ordinary five-letter word.
The last letter of the $86$th word in this list is: | {
"answer": "E",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two equilateral triangles are contained in a square whose side length is $2\sqrt{3}$. The bases of these triangles are the opposite sides of the square, and their intersection is a rhombus. What is the area of the rhombus? | {
"answer": "8\\sqrt{3} - 12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From time $t=0$ to time $t=1$ a population increased by $i\%$, and from time $t=1$ to time $t=2$ the population increased by $j\%$. Therefore, from time $t=0$ to time $t=2$ the population increased by | {
"answer": "$\\left(i+j+\\frac{ij}{100}\\right)\\%$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $a<b<c<d<e$ are consecutive positive integers such that $b+c+d$ is a perfect square and $a+b+c+d+e$ is a perfect cube, what is the smallest possible value of $c$? | {
"answer": "182",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ann made a $3$-step staircase using $18$ toothpicks as shown in the figure. How many toothpicks does she need to add to complete a $5$-step staircase?
[asy]
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path h = ellipse((0.5,0),0.45,0.015), v = ellipse((0,0.5),0.015,0.45);
for(int i=0;i<=2;i=i+1) {
for(int j=0;j<=3-i;j=j+1) {
filldraw(shift((i,j))*h,black);
filldraw(shift((j,i))*v,black);
}
}
[/asy] | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color? | {
"answer": "\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The larger root minus the smaller root of the equation $(7+4\sqrt{3})x^2+(2+\sqrt{3})x-2=0$ is | {
"answer": "6-3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the median of the following list of $4040$ numbers?
\[1, 2, 3, \ldots, 2020, 1^2, 2^2, 3^2, \ldots, 2020^2\] | {
"answer": "1976.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N = 749$, Bernardo writes the numbers $10444$ and $3245$, and LeRoy obtains the sum $S = 13689$. For how many choices of $N$ are the two rightmost digits of $S$, in order, the same as those of $2N$? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cubical cake with edge length $2$ inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where $M$ is the midpoint of a top edge. The piece whose top is triangle $B$ contains $c$ cubic inches of cake and $s$ square inches of icing. What is $c+s$? | {
"answer": "\\frac{32}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1+11+21+31+41)+(9+19+29+39+49)= | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$?
$\begin{array}[t]{r} ABBCB \\ + \\ BCADA \\ \hline DBDDD \end{array}$ | {
"answer": "7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The factors of $x^4+64$ are: | {
"answer": "(x^2-4x+8)(x^2+4x+8)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac ab,$ where $a$ and $b$ are relatively prime positive integers. What is $a+b?$ | {
"answer": "85",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ways can a student schedule $3$ mathematics courses -- algebra, geometry, and number theory -- in a $6$-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other $3$ periods is of no concern here.) | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $L(m)$ be the $x$ coordinate of the left end point of the intersection of the graphs of $y=x^2-6$ and $y=m$, where $-6<m<6$. Let $r=[L(-m)-L(m)]/m$. Then, as $m$ is made arbitrarily close to zero, the value of $r$ is: | {
"answer": "\\frac{1}{\\sqrt{6}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many four-digit integers $abcd$, with $a \neq 0$, have the property that the three two-digit integers $ab<bc<cd$ form an increasing arithmetic sequence? One such number is $4692$, where $a=4$, $b=6$, $c=9$, and $d=2$. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid A dollars and Bernardo had paid B dollars, where $A < B.$ How many dollars must LeRoy give to Bernardo so that they share the costs equally? | {
"answer": "\\frac{B-A}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangle with a diagonal of length $x$ is twice as long as it is wide. What is the area of the rectangle? | {
"answer": "\\frac{2}{5}x^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In this diagram the center of the circle is $O$, the radius is $a$ inches, chord $EF$ is parallel to chord $CD$. $O$,$G$,$H$,$J$ are collinear, and $G$ is the midpoint of $CD$. Let $K$ (sq. in.) represent the area of trapezoid $CDFE$ and let $R$ (sq. in.) represent the area of rectangle $ELMF.$ Then, as $CD$ and $EF$ are translated upward so that $OG$ increases toward the value $a$, while $JH$ always equals $HG$, the ratio $K:R$ becomes arbitrarily close to: | {
"answer": "\\frac{1}{\\sqrt{2}}+\\frac{1}{2}",
"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{2}{243}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $V = gt + V_0$ and $S = \frac{1}{2}gt^2 + V_0t$, then $t$ equals: | {
"answer": "\\frac{2S}{V+V_0}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
To complete the grid below, each of the digits 1 through 4 must occur once
in each row and once in each column. What number will occupy the lower
right-hand square?
\[\begin{tabular}{|c|c|c|c|}\hline 1 & & 2 &\ \hline 2 & 3 & &\ \hline & &&4\ \hline & &&\ \hline\end{tabular}\] | {
"answer": "1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $a = \log_8 225$ and $b = \log_2 15$, then $a$, in terms of $b$, is: | {
"answer": "\\frac{2b}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of each of the four congruent L-shaped regions of this 100-inch by 100-inch square is 3/16 of the total area. How many inches long is the side of the center square? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the volume of tetrahedron $ABCD$ with edge lengths $AB = 2$, $AC = 3$, $AD = 4$, $BC = \sqrt{13}$, $BD = 2\sqrt{5}$, and $CD = 5$? | {
"answer": "4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$ has a right angle at $B$, $AB=1$, and $BC=2$. The bisector of $\angle BAC$ meets $\overline{BC}$ at $D$. What is $BD$? | {
"answer": "\\frac{\\sqrt{5} - 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the following six statements:
(1) All women are good drivers
(2) Some women are good drivers
(3) No men are good drivers
(4) All men are bad drivers
(5) At least one man is a bad driver
(6) All men are good drivers.
The statement that negates statement (6) is: | {
"answer": "(5)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A farmer bought $749$ sheep. He sold $700$ of them for the price paid for the $749$ sheep. The remaining $49$ sheep were sold at the same price per head as the other $700$. Based on the cost, the percent gain on the entire transaction is: | {
"answer": "7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let $R$ be the region formed by the union of the square and all the triangles, and $S$ be the smallest convex polygon that contains $R$. What is the area of the region that is inside $S$ but outside $R$? | {
"answer": "1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cassie leaves Escanaba at 8:30 AM heading for Marquette on her bike. She bikes at a uniform rate of 12 miles per hour. Brian leaves Marquette at 9:00 AM heading for Escanaba on his bike. He bikes at a uniform rate of 16 miles per hour. They both bike on the same 62-mile route between Escanaba and Marquette. At what time in the morning do they meet? | {
"answer": "11:00",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible? | {
"answer": "3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to 43, what is the probability that this number will be divisible by 11? | {
"answer": "\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Abby, Bernardo, Carl, and Debra play a game in which each of them starts with four coins. The game consists of four rounds. In each round, four balls are placed in an urn---one green, one red, and two white. The players each draw a ball at random without replacement. Whoever gets the green ball gives one coin to whoever gets the red ball. What is the probability that, at the end of the fourth round, each of the players has four coins? | {
"answer": "\\frac{5}{192}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
As shown in the figure below, a regular dodecahedron (the polyhedron consisting of $12$ congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom face. How many ways are there to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and moves are not permitted from the bottom ring to the top ring? | {
"answer": "810",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circular disk is divided by $2n$ equally spaced radii ($n>0$) and one secant line. The maximum number of non-overlapping areas into which the disk can be divided is | {
"answer": "3n+1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x = \sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}$, then: | {
"answer": "1 < x < 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rectangles $R_1$ and $R_2,$ and squares $S_1, S_2,$ and $S_3,$ shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of $S_2$ in units? | {
"answer": "651",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
John ordered $4$ pairs of black socks and some additional pairs of blue socks. The price of the black socks per pair was twice that of the blue. When the order was filled, it was found that the number of pairs of the two colors had been interchanged. This increased the bill by $50\%$. The ratio of the number of pairs of black socks to the number of pairs of blue socks in the original order was: | {
"answer": "1:4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ryan got $80\%$ of the problems correct on a $25$-problem test, $90\%$ on a $40$-problem test, and $70\%$ on a $10$-problem test. What percent of all the problems did Ryan answer correctly? | {
"answer": "84",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Distinct points $P$, $Q$, $R$, $S$ lie on the circle $x^{2}+y^{2}=25$ and have integer coordinates. The distances $PQ$ and $RS$ are irrational numbers. What is the greatest possible value of the ratio $\frac{PQ}{RS}$? | {
"answer": "7",
"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"
} |
A $3 \times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90^{\circ}$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability the grid is now entirely black? | {
"answer": "\\frac{49}{512}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An artist has $14$ cubes, each with an edge of $1$ meter. She stands them on the ground to form a sculpture as shown. She then paints the exposed surface of the sculpture. How many square meters does she paint? | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five runners, $P$, $Q$, $R$, $S$, $T$, have a race, and $P$ beats $Q$, $P$ beats $R$, $Q$ beats $S$, and $T$ finishes after $P$ and before $Q$. Who could NOT have finished third in the race? | {
"answer": "P and S",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A baseball league consists of two four-team divisions. Each team plays every other team in its division $N$ games. Each team plays every team in the other division $M$ games with $N>2M$ and $M>4$. Each team plays a $76$ game schedule. How many games does a team play within its own division? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $2:1$. The ratio of the rectangle's length to its width is $2:1$. What percent of the rectangle's area is inside the square? | {
"answer": "12.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The difference of the roots of $x^2-7x-9=0$ is: | {
"answer": "\\sqrt{85}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle has a radius of $\log_{10}{(a^2)}$ and a circumference of $\log_{10}{(b^4)}$. What is $\log_{a}{b}$? | {
"answer": "\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A barn with a roof is rectangular in shape, $10$ yd. wide, $13$ yd. long and $5$ yd. high. It is to be painted inside and outside, and on the ceiling, but not on the roof or floor. The total number of sq. yd. to be painted is: | {
"answer": "490",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A six digit number (base 10) is squarish if it satisfies the following conditions:
(i) none of its digits are zero;
(ii) it is a perfect square; and
(iii) the first of two digits, the middle two digits and the last two digits of the number are all perfect squares when considered as two digit numbers.
How many squarish numbers are there? | {
"answer": "2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a whole number $n$ is not prime, then the whole number $n-2$ is not prime. A value of $n$ which shows this statement to be false is | {
"answer": "9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a plane, four circles with radii $1,3,5,$ and $7$ are tangent to line $\ell$ at the same point $A,$ but they may be on either side of $\ell$. Region $S$ consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region $S$? | {
"answer": "65\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two equal parallel chords are drawn $8$ inches apart in a circle of radius $8$ inches. The area of that part of the circle that lies between the chords is: | {
"answer": "$32\\sqrt{3}+21\\frac{1}{3}\\pi$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Small lights are hung on a string $6$ inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of $2$ red lights followed by $3$ green lights. How many feet separate the 3rd red light and the 21st red light? | {
"answer": "22.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n$ be a $5$-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by $100$. For how many values of $n$ is $q+r$ divisible by $11$? | {
"answer": "9000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cube of edge $3$ cm is cut into $N$ smaller cubes, not all the same size. If the edge of each of the smaller cubes is a whole number of centimeters, then $N=$ | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle of radius $2$, there are many line segments of length $2$ that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments. | {
"answer": "\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Equilateral $\triangle ABC$ has side length $2$, $M$ is the midpoint of $\overline{AC}$, and $C$ is the midpoint of $\overline{BD}$. What is the area of $\triangle CDM$? | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graphs of $y=\log_3 x$, $y=\log_x 3$, $y=\log_{\frac{1}{3}} x$, and $y=\log_x \dfrac{1}{3}$ are plotted on the same set of axes. How many points in the plane with positive $x$-coordinates lie on two or more of the graphs? | {
"answer": "3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When you simplify $\left[ \sqrt [3]{\sqrt [6]{a^9}} \right]^4\left[ \sqrt [6]{\sqrt [3]{a^9}} \right]^4$, the result is: | {
"answer": "a^4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the adjoining figure triangle $ABC$ is such that $AB = 4$ and $AC = 8$. IF $M$ is the midpoint of $BC$ and $AM = 3$, what is the length of $BC$? | {
"answer": "2\\sqrt{31}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of the shaded region. | {
"answer": "6\\dfrac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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"
} |
The vertical axis indicates the number of employees, but the scale was accidentally omitted from this graph. What percent of the employees at the Gauss company have worked there for $5$ years or more? | {
"answer": "30 \\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x,y>0$, $\log_y(x)+\log_x(y)=\frac{10}{3}$ and $xy=144$, then $\frac{x+y}{2}=$ | {
"answer": "13\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mia is "helping" her mom pick up $30$ toys that are strewn on the floor. Mia’s mom manages to put $3$ toys into the toy box every $30$ seconds, but each time immediately after those $30$ seconds have elapsed, Mia takes $2$ toys out of the box. How much time, in minutes, will it take Mia and her mom to put all $30$ toys into the box for the first time? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can the sequence $1,2,3,4,5$ be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$ shown in the figure, $AB=7$, $BC=8$, $CA=9$, and $\overline{AH}$ is an altitude. Points $D$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $\overline{BD}$ and $\overline{CE}$ are angle bisectors, intersecting $\overline{AH}$ at $Q$ and $P$, respectively. What is $PQ$? | {
"answer": "\\frac{8}{15}\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Anita attends a baseball game in Atlanta and estimates that there are 50,000 fans in attendance. Bob attends a baseball game in Boston and estimates that there are 60,000 fans in attendance. A league official who knows the actual numbers attending the two games note that:
i. The actual attendance in Atlanta is within $10 \%$ of Anita's estimate.
ii. Bob's estimate is within $10 \%$ of the actual attendance in Boston.
To the nearest 1,000, the largest possible difference between the numbers attending the two games is | {
"answer": "22000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a convex quadrilateral with $BC=2$ and $CD=6.$ Suppose that the centroids of $\triangle ABC, \triangle BCD,$ and $\triangle ACD$ form the vertices of an equilateral triangle. What is the maximum possible value of the area of $ABCD$? | {
"answer": "12+10\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $N = 34 \cdot 34 \cdot 63 \cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$? | {
"answer": "1 : 14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A $3 \times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90^{\circ}$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability the grid is now entirely black? | {
"answer": "\\frac{49}{512}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The fraction $\frac{\sqrt{a^2+x^2}-\frac{x^2-a^2}{\sqrt{a^2+x^2}}}{a^2+x^2}$ reduces to: | {
"answer": "\\frac{2a^2}{(a^2+x^2)^{\\frac{3}{2}}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six. (The probability of obtaining a six on any toss is $\frac{1}{6}$, independent of the outcome of any other toss.) | {
"answer": "\\frac{36}{91}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A straight one-mile stretch of highway, 40 feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at 5 miles per hour, how many hours will it take to cover the one-mile stretch?
[asy]
size(10cm); pathpen=black; pointpen=black; D(arc((-2,0),1,300,360)); D(arc((0,0),1,0,180)); D(arc((2,0),1,180,360)); D(arc((4,0),1,0,180)); D(arc((6,0),1,180,240)); D((-1.5,-1)--(5.5,-1));
[/asy]
Note: 1 mile = 5280 feet | {
"answer": "\\frac{\\pi}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool, one at the rate of $3$ feet per second, the other at $2$ feet per second. They swim back and forth for $12$ minutes. Allowing no loss of times at the turns, find the number of times they pass each other. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose a square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed - a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle? | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area in square units of the region enclosed by parallelogram $ABCD$ is | {
"answer": "8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$ be a positive number. Consider the set $S$ of all points whose rectangular coordinates $(x, y)$ satisfy all of the following conditions:
\begin{enumerate}
\item $\frac{a}{2} \le x \le 2a$
\item $\frac{a}{2} \le y \le 2a$
\item $x+y \ge a$
\item $x+a \ge y$
\item $y+a \ge x$
\end{enumerate}
The boundary of set $S$ is a polygon with | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A man has $10,000 to invest. He invests $4000 at 5% and $3500 at 4%. In order to have a yearly income of $500, he must invest the remainder at: | {
"answer": "6.4\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Which pair of numbers does NOT have a product equal to $36$? | {
"answer": "{\\frac{1}{2},-72}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the length of a diagonal of a square is $a + b$, then the area of the square is: | {
"answer": "\\frac{1}{2}(a+b)^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A telephone number has the form \text{ABC-DEF-GHIJ}, where each letter represents
a different digit. The digits in each part of the number are in decreasing
order; that is, $A > B > C$, $D > E > F$, and $G > H > I > J$. Furthermore,
$D$, $E$, and $F$ are consecutive even digits; $G$, $H$, $I$, and $J$ are consecutive odd
digits; and $A + B + C = 9$. Find $A$. | {
"answer": "8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three equally spaced parallel lines intersect a circle, creating three chords of lengths 38, 38, and 34. What is the distance between two adjacent parallel lines? | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $\triangle ABC$ in the figure has area $10$. Points $D, E$ and $F$, all distinct from $A, B$ and $C$,
are on sides $AB, BC$ and $CA$ respectively, and $AD = 2, DB = 3$. If triangle $\triangle ABE$ and quadrilateral $DBEF$
have equal areas, then that area is | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sphere with center $O$ has radius $6$. A triangle with sides of length $15, 15,$ and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle? | {
"answer": "2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For every $n$ the sum of $n$ terms of an arithmetic progression is $2n + 3n^2$. The $r$th term is: | {
"answer": "6r - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Figure $ABCD$ is a trapezoid with $AB \parallel DC$, $AB=5$, $BC=3\sqrt{2}$, $\angle BCD=45^\circ$, and $\angle CDA=60^\circ$. The length of $DC$ is | {
"answer": "8 + \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular field is 300 feet wide and 400 feet long. Random sampling indicates that there are, on the average, three ants per square inch through out the field. [12 inches = 1 foot.] Of the following, the number that most closely approximates the number of ants in the field is | {
"answer": "50000000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular box has a total surface area of 94 square inches. The sum of the lengths of all its edges is 48 inches. What is the sum of the lengths in inches of all of its interior diagonals? | {
"answer": "20\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\frac{3}{2}$ and center $(0,\frac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to $(3,0)$. What is the area of the shark's fin falcata? | {
"answer": "\\frac{9\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $R$ be a rectangle. How many circles in the plane of $R$ have a diameter both of whose endpoints are vertices of $R$? | {
"answer": "5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number $2013$ is expressed in the form $2013 = \frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$,where $a_1 \ge a_2 \ge \cdots \ge a_m$ and $b_1 \ge b_2 \ge \cdots \ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. What is $|a_1 - b_1|$? | {
"answer": "2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A pair of standard $6$-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference? | {
"answer": "\\frac{1}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A shop advertises everything is "half price in today's sale." In addition, a coupon gives a 20% discount on sale prices. Using the coupon, the price today represents what percentage off the original price? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is $11/4$. To the nearest whole percent, what percent of its games did the team lose? | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jeff rotates spinners $P$, $Q$ and $R$ and adds the resulting numbers. What is the probability that his sum is an odd number? | {
"answer": "1/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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