problem stringlengths 10 5.15k | answer dict |
|---|---|
At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the $150 regular price did he save? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The largest divisor of $2,014,000,000$ is itself. What is its fifth-largest divisor? | {
"answer": "125,875,000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The total area of all the faces of a rectangular solid is $22\text{cm}^2$, and the total length of all its edges is $24\text{cm}$. Then the length in cm of any one of its interior diagonals is | {
"answer": "\\sqrt{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Six squares are colored, front and back, (R = red, B = blue, O = orange, Y = yellow, G = green, and W = white). They are hinged together as shown, then folded to form a cube. The face opposite the white face is | {
"answer": "B",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the bisectors of the exterior angles at $B$ and $C$ of triangle $ABC$ meet at $D$. Then, if all measurements are in degrees, angle $BDC$ equals: | {
"answer": "\\frac{1}{2}(180-A)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $A>B>0$ and A is $x$% greater than $B$. What is $x$? | {
"answer": "100\\left(\\frac{A-B}{B}\\right)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles, expressed in degrees, is: | {
"answer": "540",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A solid cube of side length $1$ is removed from each corner of a solid cube of side length $3$. How many edges does the remaining solid have? | {
"answer": "84",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units?
[asy] import three; defaultpen(linewidth(0.8)); real r=0.5; currentprojection=orthographic(1,1/2,1/4); draw(unitcube, white, thick(), nolight); draw(shift(1,0,0)*unitcube, white, thick(), nolight); draw(shift(1,-1,0)*unitcube, white, thick(), nolight); draw(shift(1,0,-1)*unitcube, white, thick(), nolight); draw(shift(2,0,0)*unitcube, white, thick(), nolight); draw(shift(1,1,0)*unitcube, white, thick(), nolight); draw(shift(1,0,1)*unitcube, white, thick(), nolight);[/asy] | {
"answer": "\\frac{7}{30}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x \ne 0$ or $4$ and $y \ne 0$ or $6$, then $\frac{2}{x} + \frac{3}{y} = \frac{1}{2}$ is equivalent to | {
"answer": "\\frac{4y}{y-6}=x",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
To control her blood pressure, Jill's grandmother takes one half of a pill every other day. If one supply of medicine contains $60$ pills, then the supply of medicine would last approximately | {
"answer": "8\\text{ months}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the radius of a circle is increased by $1$ unit, the ratio of the new circumference to the new diameter is: | {
"answer": "\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many of the twelve pentominoes pictured below have at least one line of reflectional symmetry?
[asy] unitsize(5mm); defaultpen(linewidth(1pt)); draw(shift(2,0)*unitsquare); draw(shift(2,1)*unitsquare); draw(shift(2,2)*unitsquare); draw(shift(1,2)*unitsquare); draw(shift(0,2)*unitsquare); draw(shift(2,4)*unitsquare); draw(shift(2,5)*unitsquare); draw(shift(2,6)*unitsquare); draw(shift(1,5)*unitsquare); draw(shift(0,5)*unitsquare); draw(shift(4,8)*unitsquare); draw(shift(3,8)*unitsquare); draw(shift(2,8)*unitsquare); draw(shift(1,8)*unitsquare); draw(shift(0,8)*unitsquare); draw(shift(6,8)*unitsquare); draw(shift(7,8)*unitsquare); draw(shift(8,8)*unitsquare); draw(shift(9,8)*unitsquare); draw(shift(9,9)*unitsquare); draw(shift(6,5)*unitsquare); draw(shift(7,5)*unitsquare); draw(shift(8,5)*unitsquare); draw(shift(7,6)*unitsquare); draw(shift(7,4)*unitsquare); draw(shift(6,1)*unitsquare); draw(shift(7,1)*unitsquare); draw(shift(8,1)*unitsquare); draw(shift(6,0)*unitsquare); draw(shift(7,2)*unitsquare); draw(shift(11,8)*unitsquare); draw(shift(12,8)*unitsquare); draw(shift(13,8)*unitsquare); draw(shift(14,8)*unitsquare); draw(shift(13,9)*unitsquare); draw(shift(11,5)*unitsquare); draw(shift(12,5)*unitsquare); draw(shift(13,5)*unitsquare); draw(shift(11,6)*unitsquare); draw(shift(13,4)*unitsquare); draw(shift(11,1)*unitsquare); draw(shift(12,1)*unitsquare); draw(shift(13,1)*unitsquare); draw(shift(13,2)*unitsquare); draw(shift(14,2)*unitsquare); draw(shift(16,8)*unitsquare); draw(shift(17,8)*unitsquare); draw(shift(18,8)*unitsquare); draw(shift(17,9)*unitsquare); draw(shift(18,9)*unitsquare); draw(shift(16,5)*unitsquare); draw(shift(17,6)*unitsquare); draw(shift(18,5)*unitsquare); draw(shift(16,6)*unitsquare); draw(shift(18,6)*unitsquare); draw(shift(16,0)*unitsquare); draw(shift(17,0)*unitsquare); draw(shift(17,1)*unitsquare); draw(shift(18,1)*unitsquare); draw(shift(18,2)*unitsquare);[/asy] | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$ is equilateral with $AB=1$. Points $E$ and $G$ are on $\overline{AC}$ and points $D$ and $F$ are on $\overline{AB}$ such that both $\overline{DE}$ and $\overline{FG}$ are parallel to $\overline{BC}$. Furthermore, triangle $ADE$ and trapezoids $DFGE$ and $FBCG$ all have the same perimeter. What is $DE+FG$? | {
"answer": "\\frac{21}{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $D$ is on $AC$ and $F$ is on $BC$. Also, $AB \perp AC$, $AF \perp BC$, and $BD=DC=FC=1$. Find $AC$. | {
"answer": "\\sqrt[3]{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $b$ men take $c$ days to lay $f$ bricks, then the number of days it will take $c$ men working at the same rate to lay $b$ bricks, is | {
"answer": "\\frac{b^2}{f}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Gilda has a bag of marbles. She gives $20\%$ of them to her friend Pedro. Then Gilda gives $10\%$ of what is left to another friend, Ebony. Finally, Gilda gives $25\%$ of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself? | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three congruent circles with centers $P$, $Q$, and $R$ are tangent to the sides of rectangle $ABCD$ as shown. The circle centered at $Q$ has diameter $4$ and passes through points $P$ and $R$. The area of the rectangle is | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Walter has exactly one penny, one nickel, one dime and one quarter in his pocket. What percent of one dollar is in his pocket? | {
"answer": "41\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Set $u_0 = \frac{1}{4}$, and for $k \ge 0$ let $u_{k+1}$ be determined by the recurrence
\[u_{k+1} = 2u_k - 2u_k^2.\]This sequence tends to a limit; call it $L$. What is the least value of $k$ such that
\[|u_k-L| \le \frac{1}{2^{1000}}?\] | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each day Maria must work $8$ hours. This does not include the $45$ minutes she takes for lunch. If she begins working at $\text{7:25 A.M.}$ and takes her lunch break at noon, then her working day will end at | {
"answer": "\\text{4:10 P.M.}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A number $m$ is randomly selected from the set $\{11,13,15,17,19\}$, and a number $n$ is randomly selected from $\{1999,2000,2001,\ldots,2018\}$. What is the probability that $m^n$ has a units digit of $1$? | {
"answer": "\\frac{7}{20}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rectangle $ABCD$ has $AB = 6$ and $BC = 3$. Point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$. What is the degree measure of $\angle AMD$? | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One of the sides of a triangle is divided into segments of $6$ and $8$ units by the point of tangency of the inscribed circle. If the radius of the circle is $4$, then the length of the shortest side is | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Estimate the year in which the population of Nisos will be approximately 6,000. | {
"answer": "2075",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A decorative window is made up of a rectangle with semicircles at either end. The ratio of $AD$ to $AB$ is $3:2$. And $AB$ is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircles? | {
"answer": "6:\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the sum $1 + 2 + 3 + \cdots + K$ is a perfect square $N^2$ and if $N$ is less than $100$, then the possible values for $K$ are: | {
"answer": "1, 8, and 49",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $y=f(x)=\frac{x+2}{x-1}$, then it is incorrect to say: | {
"answer": "$f(1)=0$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sides $AB$, $BC$, and $CD$ of (simple*) quadrilateral $ABCD$ have lengths $4$, $5$, and $20$, respectively.
If vertex angles $B$ and $C$ are obtuse and $\sin C = - \cos B = \frac{3}{5}$, then side $AD$ has length
A polygon is called “simple” if it is not self intersecting. | {
"answer": "25",
"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 flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations? | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\{a_k\}_{k=1}^{2011}$ be the sequence of real numbers defined by $a_1=0.201,$ $a_2=(0.2011)^{a_1},$ $a_3=(0.20101)^{a_2},$ $a_4=(0.201011)^{a_3}$, and in general,
\[a_k=\begin{cases}(0.\underbrace{20101\cdots 0101}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if }k\text{ is odd,}\\(0.\underbrace{20101\cdots 01011}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if }k\text{ is even.}\end{cases}\]Rearranging the numbers in the sequence $\{a_k\}_{k=1}^{2011}$ in decreasing order produces a new sequence $\{b_k\}_{k=1}^{2011}$. What is the sum of all integers $k$, $1\le k \le 2011$, such that $a_k=b_k?$ | {
"answer": "1341",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The addition below is incorrect. The display can be made correct by changing one digit $d$, wherever it occurs, to another digit $e$. Find the sum of $d$ and $e$.
$\begin{tabular}{ccccccc} & 7 & 4 & 2 & 5 & 8 & 6 \\ + & 8 & 2 & 9 & 4 & 3 & 0 \\ \hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \end{tabular}$ | {
"answer": "8",
"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"
} |
Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. What is the expected number of balls that occupy their original positions after these two successive transpositions? | {
"answer": "2.2",
"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"
} |
Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received $36$ votes, then how many votes were cast all together? | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The acronym AMC is shown in the rectangular grid below with grid lines spaced $1$ unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC$? | {
"answer": "13 + 4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\omega=-\tfrac{1}{2}+\tfrac{1}{2}i\sqrt3.$ Let $S$ denote all points in the complex plane of the form $a+b\omega+c\omega^2,$ where $0\leq a \leq 1,0\leq b\leq 1,$ and $0\leq c\leq 1.$ What is the area of $S$? | {
"answer": "\\frac{3}{2}\\sqrt3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For positive integers $n$, denote $D(n)$ by the number of pairs of different adjacent digits in the binary (base two) representation of $n$. For example, $D(3) = D(11_{2}) = 0$, $D(21) = D(10101_{2}) = 4$, and $D(97) = D(1100001_{2}) = 2$. For how many positive integers less than or equal to $97$ does $D(n) = 2$? | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be a triangle where $M$ is the midpoint of $\overline{AC}$, and $\overline{CN}$ is the angle bisector of $\angle{ACB}$ with $N$ on $\overline{AB}$. Let $X$ be the intersection of the median $\overline{BM}$ and the bisector $\overline{CN}$. In addition $\triangle BXN$ is equilateral with $AC=2$. What is $BX^2$? | {
"answer": "\\frac{10-6\\sqrt{2}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The roots of the equation $ax^2 + bx + c = 0$ will be reciprocal if: | {
"answer": "c = a",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A $2$ by $2$ square is divided into four $1$ by $1$ squares. Each of the small squares is to be painted either green or red. In how many different ways can the painting be accomplished so that no green square shares its top or right side with any red square? There may be as few as zero or as many as four small green squares. | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Nebraska, the home of the AMC, changed its license plate scheme. Each old license plate consisted of a letter followed by four digits. Each new license plate consists of three letters followed by three digits. By how many times has the number of possible license plates increased? | {
"answer": "\\frac{26^2}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$ has $AC=3$, $BC=4$, and $AB=5$. Point $D$ is on $\overline{AB}$, and $\overline{CD}$ bisects the right angle. The inscribed circles of $\triangle ADC$ and $\triangle BCD$ have radii $r_a$ and $r_b$, respectively. What is $r_a/r_b$? | {
"answer": "\\frac{3}{28} \\left(10 - \\sqrt{2}\\right)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S=\{(x,y) : x\in \{0,1,2,3,4\}, y\in \{0,1,2,3,4,5\},\text{ and } (x,y)\ne (0,0)\}$.
Let $T$ be the set of all right triangles whose vertices are in $S$. For every right triangle $t=\triangle{ABC}$ with vertices $A$, $B$, and $C$ in counter-clockwise order and right angle at $A$, let $f(t)=\tan(\angle{CBA})$. What is \[\prod_{t\in T} f(t)?\] | {
"answer": "\\frac{625}{144}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe? | {
"answer": "\\frac {16!}{2^8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define $n_a!$ for $n$ and $a$ positive to be
$n_a ! = n (n-a)(n-2a)(n-3a)...(n-ka)$
where $k$ is the greatest integer for which $n>ka$. Then the quotient $72_8!/18_2!$ is equal to | {
"answer": "4^9",
"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 $10,444$ and $3,245$, and LeRoy obtains the sum $S = 13,689$. 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"
} |
For every $m$ and $k$ integers with $k$ odd, denote by $\left[ \frac{m}{k} \right]$ the integer closest to $\frac{m}{k}$. For every odd integer $k$, let $P(k)$ be the probability that
\[\left[ \frac{n}{k} \right] + \left[ \frac{100 - n}{k} \right] = \left[ \frac{100}{k} \right]\]for an integer $n$ randomly chosen from the interval $1 \leq n \leq 99$. What is the minimum possible value of $P(k)$ over the odd integers $k$ in the interval $1 \leq k \leq 99$? | {
"answer": "\\frac{34}{67}",
"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"
} |
The graph shows the price of five gallons of gasoline during the first ten months of the year. By what percent is the highest price more than the lowest price? | {
"answer": "70",
"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"
} |
Two concentric circles have radii $1$ and $2$. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle? | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle of radius $2$ is centered at $O$. Square $OABC$ has side length $1$. Sides $AB$ and $CB$ are extended past $B$ to meet the circle at $D$ and $E$, respectively. What is the area of the shaded region in the figure, which is bounded by $BD$, $BE$, and the minor arc connecting $D$ and $E$? | {
"answer": "\\frac{\\pi}{3}+1-\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively.
A third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. If $AB=20$, then the perimeter of $\triangle APR$ is | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sign at the fish market says, "50% off, today only: half-pound packages for just $3 per package." What is the regular price for a full pound of fish, in dollars? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to them in the circle. Then each person computes and announces the average of the numbers of their two neighbors. The figure shows the average announced by each person (not the original number the person picked.)
The number picked by the person who announced the average $6$ was | {
"answer": "1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ be the set of all points with coordinates $(x,y,z)$, where $x$, $y$, and $z$ are each chosen from the set $\{0,1,2\}$. How many equilateral triangles have all their vertices in $S$? | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A fair $6$-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number? | {
"answer": "\\frac{1}{20}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Call a positive real number special if it has a decimal representation that consists entirely of digits $0$ and $7$. For example, $\frac{700}{99}= 7.\overline{07}= 7.070707\cdots$ and $77.007$ are special numbers. What is the smallest $n$ such that $1$ can be written as a sum of $n$ special numbers? | {
"answer": "8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points $A$ and $C$ lie on a circle centered at $O$, each of $\overline{BA}$ and $\overline{BC}$ are tangent to the circle, and $\triangle ABC$ is equilateral. The circle intersects $\overline{BO}$ at $D$. What is $\frac{BD}{BO}$? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a $4 \times 4 \times 3$ rectangular parallelepiped, vertices $A$, $B$, and $C$ are adjacent to vertex $D$. The perpendicular distance from $D$ to the plane containing $A$, $B$, and $C$ is closest to | {
"answer": "2.1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The angles in a particular triangle are in arithmetic progression, and the side lengths are $4, 5, x$. The sum of the possible values of x equals $a+\sqrt{b}+\sqrt{c}$ where $a, b$, and $c$ are positive integers. What is $a+b+c$? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end of the tournament? | {
"answer": "5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$2.46 \times 8.163 \times (5.17 + 4.829)$ is closest to | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$. The polygons meet at a point $A$ in such a way that the sum of the three interior angles at $A$ is $360^{\circ}$. Thus the three polygons form a new polygon with $A$ as an interior point. What is the largest possible perimeter that this polygon can have? | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number $121_b$, written in the integral base $b$, is the square of an integer, for | {
"answer": "$b > 2$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled? | {
"answer": "1462",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \le 50$ is $f_{50}(n) = 12?$ | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$|3-\pi|=$ | {
"answer": "\\pi-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the value of $\sqrt{(3-2\sqrt{3})^2}+\sqrt{(3+2\sqrt{3})^2}$? | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The shaded region formed by the two intersecting perpendicular rectangles, in square units, is | {
"answer": "38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The two circles pictured have the same center $C$. Chord $\overline{AD}$ is tangent to the inner circle at $B$, $AC$ is $10$, and chord $\overline{AD}$ has length $16$. What is the area between the two circles? | {
"answer": "64 \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A top hat contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn? | {
"answer": "\\frac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For what value of $k$ does the equation $\frac{x-1}{x-2} = \frac{x-k}{x-6}$ have no solution for $x$? | {
"answer": "5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three identical square sheets of paper each with side length $6$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a-b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is $a+b+c?$ | {
"answer": "147",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum value of $\frac{(2^t-3t)t}{4^t}$ for real values of $t?$ | {
"answer": "\\frac{1}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The value of $\frac {1}{2 - \frac {1}{2 - \frac {1}{2 - \frac12}}}$ is | {
"answer": "\\frac{3}{4}",
"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"
} |
A paper triangle with sides of lengths $3,4,$ and $5$ inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease? | {
"answer": "$\\frac{15}{8}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
All lines with equation $ax+by=c$ such that $a,b,c$ form an arithmetic progression pass through a common point. What are the coordinates of that point? | {
"answer": "(-1,2)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two cyclists, $k$ miles apart, and starting at the same time, would be together in $r$ hours if they traveled in the same direction, but would pass each other in $t$ hours if they traveled in opposite directions. The ratio of the speed of the faster cyclist to that of the slower is: | {
"answer": "\\frac {r + t}{r - t}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The hypotenuse $c$ and one arm $a$ of a right triangle are consecutive integers. The square of the second arm is: | {
"answer": "c+a",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles already traveled. For each subsequent mile the speed is constant. If the second mile is traversed in $2$ hours, then the time, in hours, needed to traverse the $n$th mile is: | {
"answer": "2(n-1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Raashan, Sylvia, and Ted play the following game. Each starts with $1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $1$? (For example, Raashan and Ted may each decide to give $1$ to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $0$, Sylvia will have $2$, and Ted will have $1$, and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their $1$ to, and the holdings will be the same at the end of the second round.) | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x$ and $y$ are non-zero numbers such that $x=1+\frac{1}{y}$ and $y=1+\frac{1}{x}$, then $y$ equals | {
"answer": "x",
"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"
} |
A line initially 1 inch long grows according to the following law, where the first term is the initial length.
\[1+\frac{1}{4}\sqrt{2}+\frac{1}{4}+\frac{1}{16}\sqrt{2}+\frac{1}{16}+\frac{1}{64}\sqrt{2}+\frac{1}{64}+\cdots\]
If the growth process continues forever, the limit of the length of the line is: | {
"answer": "\\frac{1}{3}(4+\\sqrt{2})",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When a bucket is two-thirds full of water, the bucket and water weigh $a$ kilograms. When the bucket is one-half full of water the total weight is $b$ kilograms. In terms of $a$ and $b$, what is the total weight in kilograms when the bucket is full of water? | {
"answer": "3a - 2b",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The 16 squares on a piece of paper are numbered as shown in the diagram. While lying on a table, the paper is folded in half four times in the following sequence:
(1) fold the top half over the bottom half
(2) fold the bottom half over the top half
(3) fold the right half over the left half
(4) fold the left half over the right half.
Which numbered square is on top after step 4? | {
"answer": "9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The product of the two $99$-digit numbers
$303,030,303,...,030,303$ and $505,050,505,...,050,505$
has thousands digit $A$ and units digit $B$. What is the sum of $A$ and $B$? | {
"answer": "8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$ and point $P$ in the same plane are given. Point $P$ is equidistant from $A$ and $B$, angle $APB$ is twice angle $ACB$, and $\overline{AC}$ intersects $\overline{BP}$ at point $D$. If $PB = 3$ and $PD= 2$, then $AD\cdot CD =$ | {
"answer": "5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.
For example, starting with an input of $N=7,$ the machine will output $3 \cdot 7 +1 = 22.$ Then if the output is repeatedly inserted into the machine five more times, the final output is $26.$ $7 \to 22 \to 11 \to 34 \to 17 \to 52 \to 26$ When the same $6$-step process is applied to a different starting value of $N,$ the final output is $1.$ What is the sum of all such integers $N?$ $N \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to 1$ | {
"answer": "83",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each principal of Lincoln High School serves exactly one $3$-year term. What is the maximum number of principals this school could have during an $8$-year period? | {
"answer": "4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular grazing area is to be fenced off on three sides using part of a $100$ meter rock wall as the fourth side. Fence posts are to be placed every $12$ meters along the fence including the two posts where the fence meets the rock wall. What is the fewest number of posts required to fence an area $36$ m by $60$ m? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $f(x) = 5x^2 - 2x - 1$, then $f(x + h) - f(x)$ equals: | {
"answer": "h(10x+5h-2)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ratio of $w$ to $x$ is $4:3$, the ratio of $y$ to $z$ is $3:2$, and the ratio of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y?$ | {
"answer": "16:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $r$ be the result of doubling both the base and exponent of $a^b$, and $b$ does not equal to $0$.
If $r$ equals the product of $a^b$ by $x^b$, then $x$ equals: | {
"answer": "4a",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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