prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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In triangle $\triangle ABC$ in the adjoining figure, $AD$ and $AE$ trisect $\angle BAC$. The lengths of $BD$, $DE$, and $EC$ are $2$, $3$, and $6$, respectively. Find the length of the shortest side of $\triangle ABC$. The original answer is in the form $x\sqrt{y}$, please give the value of $x + y$. | 12 |
If $N=\lfloor \frac{2}{5} \rfloor + \lfloor \frac{2^2}{5} \rfloor + \dots + \lfloor \frac{2^{2009}}{5} \rfloor$, find the remainder when $2^{2010}$ is divided by $N$. | 5024 |
Let $ABC$ be a triangle with $|AB| = 14$, $|BC| = 12$, $|AC| = 10$. Let $D$ be a point on $[AC]$ and $E$ be a point on $[BC]$ such that $|AD| = 4$ and $\text{Area}(ABC) = 2 \times \text{Area}(CDE)$. Find $\text{Area}(ABE)$ in the form of $m\sqrt{n}$, where $m$ and $n$ are integers. Please give the value of $m + n$. | 10 |
How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation: \[(1+2a)(2+2b)(2a+b) = 32ab\]? | 1 |
Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, what number could be obtained? | 119 |
A plane intersects a right circular cylinder of radius $1$ forming an ellipse. If the major axis of the ellipse is $50\%$ longer than the minor axis, what is the length of the major axis? | 3 |
For the integer numbers $i, j, k$ satisfying the condition $i^2 + j^2 + k^2 = 2011$, find the largest value of $i + j + k$. | 77 |
Three times Dick's age plus Tom's age equals twice Harry's age. Double the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age. Their respective ages are relatively prime to each other. What is the sum of the squares of their ages? | 42 |
A sector with an acute central angle \(\theta\) is cut from a circle of radius 6. What is the radius of the circle circumscribed about the sector in terms of \(\theta\)? The original answer is in the form \(k \sec \frac{m}{n} \theta\). Please find the value of k + m + n. | 6 |
Let $p$ be the greatest prime factor of 9991. What is the sum of the digits of $p$? | 4 |
How many ordered triples of integers \((a, b, c)\), with \(a \ge 2\), \(b \ge 1\), and \(c \ge 0\), satisfy both \(\log_a b = c^{2005}\) and \(a + b + c = 2005\)? | 2 |
A store prices an item in dollars and cents so that when 4\% sales tax is added, no rounding is necessary because the result is exactly $n$ dollars where $n$ is a positive integer. Find the smallest value of $n$. | 13 |
A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the side... | 4 |
Assuming \(a\neq3\), \(b\neq4\), and \(c\neq5\), find the integer value of the following expression: \[\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}\] Please provide your answer as an integer. | -1 |
A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50, 20, 20, 5,$ and $5$. Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is no... | -135 |
On an algebra quiz, $10\%$ of the students scored $70$ points, $35\%$ scored $80$ points, $30\%$ scored $90$ points, and the rest scored $100$ points. Find the difference between the mean and median score of the students' scores on this quiz. | 3 |
Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window? | 26 |
Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is $50\%$ more than the regular. After both consume $\frac{3}{4}$ of their drinks, Ann gives Ed a third of what she has left, and $2$ additional ounces. When they finish their lemonades they realize that they ... | 40 |
Suppose $A>B>0$ and A is $x$% greater than $B$. If the answer is in the form \frac{m(A-B)}{nB}, please find the value of m + n. | 101 |
For how many positive integers $n$ is $\frac{n}{30-n}$ also a positive integer? | 7 |
There are four more girls than boys in Ms. Raub's class of $28$ students. What is the ratio of the number of girls to the number of boys in her class? Provide your answer in the form of $a : b$, and then find the value of $a + b$. | 7 |
Among the following expressions, which one is a perfect square? \(\dfrac{14!15!}{2}\), \(\dfrac{15!16!}{2}\), \(\dfrac{16!17!}{2}\), \(\dfrac{17!18!}{2}\), \(\dfrac{18!19!}{2}\). Provide the expression that represents a perfect square in the form \(\dfrac{k!m!}{2}\), and give the value of k + m. | 35 |
In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$? $$ \begin{array}{cccccc}&A&B&B&C&B\\+&B&C&A&D&A\\\hline&D&B&D&D&D\end{array} $$ Provide the number of different possible values for $D$. | 7 |
Five positive consecutive integers starting with $a$ have an average of $b$. What is the average of 5 consecutive integers that start with $b$? Provide your answer in the form of an expression involving $a$ as $a + k$. Find the value of $k$. | 4 |
Five positive consecutive integers starting with $a$ have an average of $b$. What is the average of 5 consecutive integers that start with $b$? Provide your answer in the form of $a + k$, where $k$ is an integer. Please find the value of $k$. | 4 |
In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. Find how many different values are possible for $D$. \[\begin{array}[t]{r} ABBCB \\ + \ BCADA \\ \hline DBDDD \end{array}\] | 7 |
Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length \(1\). The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. The length of one of the two congruent sides of one of the isosceles triangles is ... | 6 |
David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How many miles is the airpor... | 210 |
Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. Find the number of ways he can reac... | 4 |
A regular hexagon has side length \(6\). Congruent arcs with radius \(3\) are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown. The original answer is in the form \(a\sqrt{b}-c\pi\). Please find the value of a + b... | 75 |
Two circles intersect at points \(A\) and \(B\). The minor arcs \(AB\) measure \(30^\circ\) on one circle and \(60^\circ\) on the other circle. Find the ratio of the area of the larger circle to the area of the smaller circle. The answer is in the format \(k + m\sqrt{3}\), please give the value of \(k + m\). | 3 |
A set S consists of triangles whose sides have integer lengths less than 5, and no two elements of S are congruent or similar. What is the largest number of elements that S can have? | 9 |
A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What is the probability that a reader guessing at random will match all three correctly? Pro... | 7 |
Equilateral \(\triangle ABC\) has side length \(1\), and squares \(ABDE\), \(BCHI\), \(CAFG\) lie outside the triangle. Find the area of hexagon \(DEFGHI\). The answer is in the form \(k+\sqrt{m}\), please give the value of \(k+m\). | 6 |
Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\triangle{ABC}$? Provide your answer in the form $a\sqrt{b}$, where $a$ and $b$ are integers. Please find the value of a + b. | 6 |
A fancy bed and breakfast inn has $5$ rooms, each with a distinctive color-coded decor. One day $5$ friends arrive to spend the night. There are no other guests that night. The friends can room in any combination they wish, but with no more than $2$ friends per room. In how many ways can the innkeeper assign the guests... | 2220 |
Real numbers \(a\) and \(b\) are chosen with \(1<a<b\) such that no triangle with positive area has side lengths \(1\), \(a\), and \(b\) or \(\frac{1}{b}\), \(\frac{1}{a}\), and \(1\). The smallest possible value of \(b\) is given in the form \(\frac{k+\sqrt{m}}{n}\). Please find the value of \(k+m+n\). | 10 |
In rectangle \(ABCD\), \(DC = 2 \cdot CB\) and points \(E\) and \(F\) lie on \(\overline{AB}\) so that \(\overline{ED}\) and \(\overline{FD}\) trisect \(\angle ADC\) as shown. Find the ratio of the area of \(\triangle DEF\) to the area of rectangle \(ABCD\). The original answer is in the format \(\frac{\sqrt{k}}{m}\), ... | 9 |
A five-digit palindrome is a positive integer with respective digits $abcba$, where $a$ is non-zero. Let $S$ be the sum of all five-digit palindromes. What is the sum of the digits of $S$? | 18 |
When $p = \sum\limits_{k=1}^{6} k { ln }{k}$, the number $e^p$ is an integer. What is the largest integer $x$ such that $2^x$ is a factor of $e^p$? | 16 |
The circumference of the circle with center $O$ is divided into $12$ equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$? If the answer is $z$ degrees, what is the value of $z$? | 90 |
The product $(8)(888\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$. What is the value of $k$? | 991 |
Three fair six-sided dice are rolled. Find the probability that the values shown on two of the dice sum to the value shown on the remaining die. Express your answer as a simplified fraction in the form \(\frac{k}{m}\). Please provide the value of \(k + m\). | 29 |
What is the greatest power of $2$ that is a factor of $10^{1002} - 4^{501}$? If the answer is $2^n$, what is the value of $n$? | 1005 |
A \(4\times 4\times h\) rectangular box contains a sphere of radius \(2\) and eight smaller spheres of radius \(1\). The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. The original answer is in the form of \(k + m\sqrt{n}\). Please find the v... | 11 |
A square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square? | 17 |
The domain of the function $f(x)=\log_{\frac12}(\log_4(\log_{\frac14}(\log_{16}(\log_{\frac1{16}}x))))$ is an interval of length $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$. | 271 |
Four cubes with edge lengths $1$, $2$, $3$, and $4$ are stacked as shown. Find the length of the portion of $\overline{XY}$ contained in the cube with edge length $3$. The original answer is in the format \(\frac{k\sqrt{n}}{m}\). Please calculate the value of k + m + n. | 41 |
Two concentric circles have radii \(1\) and \(2\). Two points on the outer circle are chosen independently and uniformly at random. Find the probability that the chord joining the two points intersects the inner circle. Express your answer in the form \(\frac{k}{m}\) where \(\frac{k}{m}\) is a simplified fraction, and ... | 4 |
There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and \[5x^2+kx+12=0\] has at least one integer solution for $x$. What is the value of $N$? | 78 |
A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. Find the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone. The original answer is in \(\frac{k+\sqrt{m}}{n}\) format, please gi... | 10 |
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? The answer should be i... | 59 |
For how many integers $x$ is the number $x^4-51x^2+50$ negative? Provide your answer as an integer. | 12 |
In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively. Find the minimum possible value of $BE+DE+CD$. | 14 |
For how many positive integers $x$ is $\log_{10}(x-40) + \log_{10}(60-x) < 2$? | 18 |
For every real number $x$, let $\lfloor x\rfloor$ denote the greatest integer not exceeding $x$, and let $$f(x)=\lfloor x\rfloor(2014^{x-\lfloor x\rfloor}-1).$$ The set of all numbers $x$ such that $1\leq x<2014$ and $f(x)\leq 1$ is a union of disjoint intervals. What is the sum of the lengths of those intervals? | 1 |
In the figure, \(ABCD\) is a square of side length \(1\). The rectangles \(JKHG\) and \(EBCF\) are congruent. The original answer is in the form \(k - \sqrt{m}\). Find the value of \(k + m\). | 5 |
In rectangle $ABCD$, $\overline{AB}=20$ and $\overline{BC}=10$. Let $E$ be a point on $\overline{CD}$ such that $\angle CBE=15^\circ$. Find the length of $\overline{AE}$. | 20 |
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? The original answer is in the form \(\frac{\sqrt{k}-m}{n}\), please give the value of k + m + n. | 8 |
The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. Find how many pairs of integers $(m,n)$ there are such that $1\leq m\leq 2012$ and $$5^n<2^m<2^{m+2}<5^{n+1}.$$ | 279 |
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 fr... | 209 |
A rectangular piece of paper whose length is \(\sqrt{3}\) times the width has area \(A\). The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted... | 5 |
A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone? Provide your answer as a simplified expression in the form of \(\... | 11 |
The number $2017$ is prime. Let $S = \sum \limits_{k=0}^{62} \binom{2014}{k}$. What is the remainder when $S$ is divided by $2017$? | 1024 |
Three members of the Euclid Middle School girls' softball team had the following conversation. Ashley: I just realized that our uniform numbers are all $2$-digit primes. Bethany : And the sum of your two uniform numbers is the date of my birthday earlier this month. Caitlin: That's funny. The sum of your two uniform nu... | 11 |
A sequence of natural numbers is constructed by listing the first $4$, then skipping one, listing the next $5$, skipping $2$, listing $6$, skipping $3$, and on the $n$th iteration, listing $n+3$ and skipping $n$. The sequence begins $1,2,3,4,6,7,8,9,10,13$. What is the $500,000$th number in the sequence? | 996506 |
Let $ABCDE$ be a pentagon inscribed in a circle such that $AB = CD = 3$, $BC = DE = 10$, and $AE= 14$. The sum of the lengths of all diagonals of $ABCDE$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$. | 391 |
One day the Beverage Barn sold $252$ cans of soda to $100$ customers, and every customer bought at least one can of soda. If the maximum possible median number of cans of soda bought per customer on that day is $x$, what is the value of $10x$? | 35 |
The number \(5^{867}\) is between \(2^{2013}\) and \(2^{2014}\). Find how many pairs of integers \((m,n)\) there are such that \(1\leq m\leq 2012\) and \[5^n<2^m<2^{m+2}<5^{n+1}.\] | 279 |
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 fr... | 209 |
Find the sum of all the positive solutions of \[2\cos2x \left(\cos2x - \cos{\left( \frac{2014\pi^2}{x} \right) } \right) = \cos4x - 1\] If the answer is \(x\pi\), what is the value of \(x\)? | 1080 |
What is the value of \((2^0-1+5^2-0)^{-1}\times5?\) Provide your answer in the form \(\frac{k}{m}\), where k and m are integers. Then, calculate the sum k + m. | 6 |
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. At what time, in minutes after midnight (use a 24-hour clock), does she finish the third task? | 930 |
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. If she finishes the third task at $x$ PM, express the time $x$ in the format HH:MM PM, where HH and MM are two-digit numbers (possibly with leading zeros), and ... | 33 |
David, Hikmet, Jack, Marta, Rand, and Todd were in a $12$-person race with $6$ other people. Rand finished $6$ places ahead of Hikmet. Marta finished $1$ place behind Jack. David finished $2$ places behind Hikmet. Jack finished $2$ places behind Todd. Todd finished $1$ place behind Rand. Marta finished in $6$th place. ... | 8 |
The Tigers beat the Sharks $2$ out of the $3$ times they played. They then played $N$ more times, and the Sharks ended up winning at least $95\%$ of all the games played. What is the minimum possible value for $N$? | 37 |
Back in $1930$, Tillie had to memorize her multiplication facts from $0 \times 0$ to $12 \times 12$. The multiplication table she was given had rows and columns labeled with the factors, and the products formed the body of the table. To the nearest hundredth, what fraction of the numbers in the body of the table are od... | 21 |
Consider the operation "minus the reciprocal of," defined by $a\diamond b=a-\frac{1}{b}$. Find the value of $((1\diamond2)\diamond3)-(1\diamond(2\diamond3))$. The original answer is in the form -\frac{m}{n}, where m and n are integers. Please provide the value of m + n. | 37 |
What is the smallest whole number larger than the perimeter of any triangle with a side of length $5$ and a side of length $19$? Fill in the blank with the correct number. | 48 |
A box contains $2$ red marbles, $2$ green marbles, and $2$ yellow marbles. Carol takes $2$ marbles from the box at random; then Claudia takes $2$ of the remaining marbles at random; and then Cheryl takes the last $2$ marbles. What is the probability that Cheryl gets $2$ marbles of the same color? The original answer is... | 6 |
Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn, the probability that a player knocks the bottle off the ledge is \(\tfrac{1}{2}\), independently of what has happened befo... | 5 |
How many noncongruent integer-sided triangles with positive area and perimeter less than $15$ are neither equilateral, isosceles, nor right triangles? Provide your answer as an integer. | 5 |
Among the positive integers less than $100$, each of whose digits is a prime number, one is selected at random. What is the probability that the selected number is prime? Express your answer in the form of a simplified fraction \frac{k}{m}, and find the value of k + m. | 7 |
On a sheet of paper, Isabella draws a circle of radius $2$, a circle of radius $3$, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly $k \ge 0$ lines. How many different values of $k$ are possible? | 5 |
The line \(12x+5y=60\) forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle? The original answer is in \(\frac{m}{n}\) format, please give the value of m + n. | 294 |
For how many integers $x$ is the point $(x, -x)$ inside or on the circle of radius $10$ centered at $(5, 5)$? Provide your answer as an integer. | 11 |
The parabolas $y=ax^2 - 2$ and $y=4 - bx^2$ intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area $12$. If the answer is $x$, what is the value of $10x$ for $a+b$? | 15 |
Let $a$, $b$, and $c$ be three distinct one-digit numbers. Find the maximum value of the sum of the roots of the equation $(x-a)(x-b)+(x-b)(x-c)=0$. If the answer is $y$, what is the value of $10y$? | 165 |
The line \(12x+5y=60\) forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle? The original answer is in the format \(\frac{k}{m}\), please give the value of \(k + m\). | 294 |
Quadrilateral $ABCD$ is inscribed in a circle with $\angle BAC=70^{\circ}, \angle ADB=40^{\circ}, AD=4,$ and $BC=6$. What is the length of $AC$? | 6 |
How many subsets of two elements can be removed from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$ so that the mean (average) of the remaining numbers is $6$? Provide the number of such subsets. | 5 |
Let $a$, $b$, and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $(x-a)(x-b)+(x-b)(x-c)=0$? If the answer is $y$, what is the value of $10y$? | 165 |
Consider the set of all fractions $\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers. Find the number of these fractions such that if both numerator and denominator are increased by $1$, the value of the fraction is increased by $10\%$. | 1 |
The town of Hamlet has $3$ people for each horse, $4$ sheep for each cow, and $3$ ducks for each person. What is a number that could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet? | 47 |
What is the minimum number of digits to the right of the decimal point needed to express the fraction \(\frac{123456789}{2^{26}\cdot 5^4}\) as a decimal? | 26 |
At Rachelle's school, an A counts $4$ points, a B $3$ points, a C $2$ points, and a D $1$ point. Her GPA in the four classes she is taking is computed as the total sum of points divided by 4. She is certain that she will get A's in both Mathematics and Science and at least a C in each of English and History. She thinks... | 35 |
If $y+4 = (x-2)^2, x+4 = (y-2)^2$, and $x \neq y$, find the value of $x^2+y^2$. | 15 |
Al, Bill, and Cal will each randomly be assigned a whole number from \(1\) to \(10\), inclusive, with no two of them getting the same number. Find the probability that Al's number will be a whole number multiple of Bill's and Bill's number will be a whole number multiple of Cal's. The original answer is in \(\frac{k}{m... | 81 |
Tetrahedron \(ABCD\) has \(AB=5\), \(AC=3\), \(BC=4\), \(BD=4\), \(AD=3\), and \(CD=\tfrac{12}{5}\sqrt{2}\). Find the volume of the tetrahedron. The original answer is in the form \(\frac{k}{m}\), where k and m are integers. Please calculate the value of k + m. | 29 |
A regular hexagon with sides of length $6$ has an isosceles triangle attached to each side. Each of these triangles has two sides of length $8$. The isosceles triangles are folded to make a pyramid with the hexagon as the base of the pyramid. What is the volume of the pyramid? If the answer is in the form $a\sqrt{b}$, ... | 36 |
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