sungyub/dapo-math-17k-verl huggingface dataset:
user: Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were $71$, $76$, $80$, $82$, and $91$. What was the last score Mrs. Walter entered?

Answer: 80
user: $AL$, $BM$, and $CN$ are the medians of $\triangle ABC$. $K$ is the intersection of the medians. If $C, K, L, M$ are concyclic and $AB = \sqrt{3}$, find the length of the median $CN$. The original answer is in the form \(\frac{k\sqrt{3}}{n}\), where k and n are integers. Please find the value of k + n.

Answer: 5
user: The traffic on a certain east-west highway moves at a constant speed of 60 miles per hour in both directions. An eastbound driver passes 20 west-bound vehicles in a five-minute interval. Assume vehicles in the westbound lane are equally spaced. What is the number of westbound vehicles present in a 100-mile section of highway?

Answer: 200
user: Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token, and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?

Answer: 103
user: What is the maximum value of the difference between the largest real root and the smallest real root of the equation system \[\begin{array}{rcl} ax^2 + bx+ c &=& 0 \\ bx^2 + cx+ a &=& 0 \\ cx^2 + ax+ b &=& 0 \end{array}\], where at least one of the reals $a,b,c$ is non-zero?

Answer: 0
user: For a rhombus with a side length of 5, the length of one of its diagonals is not larger than $6$, and the length of the other diagonal is not smaller than $6$. Determine the maximum value of the sum of the two diagonals.

Answer: 14
user: In a right triangle with sides \(a\) and \(b\), and hypotenuse \(c\), the altitude drawn on the hypotenuse is \(x\). Find the relationship between \(a\), \(b\), and \(x\) that satisfies \(\frac{k}{x^2} = \frac{m}{a^2} + \frac{n}{b^2}\). Please provide the sum of the coefficients \(k + m + n\).

Answer: 3
user: 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 $x + y$.

Answer: 12
user: 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 $N$.

Answer: 5024
user: 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 $m + n$.

Answer: 10
user: How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation:
\[(1+2a)(2+2b)(2a+b) = 32ab\]?

Answer: 1
user: Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, what number could be obtained?

Answer: 119
user: 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?

Answer: 3
user: For the integer numbers $i, j, k$ satisfying the condition $i^2 + j^2 + k^2 = 2011$, find the largest value of $i + j + k$.

Answer: 77
user: 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?

Answer: 42
user: 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.

Answer: 6
user: Let $p$ be the greatest prime factor of 9991. What is the sum of the digits of $p$?

Answer: 4
user: 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 \(a + b + c = 2005\)?

Answer: 2
user: 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 $n$.

Answer: 13
user: 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 sides and brought together to meet at the center of the top of the box, point $A$ in the figure on the right. The box has base length $w$ and height $h$. The area of the sheet of wrapping paper can be expressed in the form $k(w+h)^m$. Please find the value of $k + m$.

Answer: 4
user: 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.

Answer: -1
user: 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 noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. Calculate the integer value of \((t-s) \times 10\).

Answer: -135
user: 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.

Answer: 3
user: 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?

Answer: 26
user: 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 both drank the same amount. How many ounces of lemonade did they drink together?

Answer: 40
user: 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.

Answer: 101
user: For how many positive integers $n$ is $\frac{n}{30-n}$ also a positive integer?

Answer: 7
user: 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 $a + b$.

Answer: 7
user: 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.

Answer: 35
user: 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 $D$.

Answer: 7
user: 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 $k$.

Answer: 4
user: 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 $k$.

Answer: 4
user: In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. Find how many different values are possible for $D$.

\[\begin{array}[t]{r} ABBCB \\ + \ BCADA \\ \hline DBDDD \end{array}\]

Answer: 7
user: 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 given in the form \(\frac{\sqrt{k}}{m}\). Please provide the value of k + m.

Answer: 6
user: 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 airport from his home?

Answer: 210
user: 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 reach Jill's house by biking a total of five blocks.

Answer: 4
user: 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 + c.

Answer: 75
user: 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 \(k + m\).

Answer: 3
user: 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?

Answer: 9
user: 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? Provide your answer in the form of a simplified fraction \frac{k}{m}. Please give the value of k + m.

Answer: 7
user: 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 \(k+m\).

Answer: 6
user: 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.

Answer: 6
user: 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 to the rooms?

Answer: 2220
user: 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 \(k+m+n\).

Answer: 10
user: 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 \(\frac{\sqrt{k}}{m}\), please give the value of k + m.

Answer: 9
user: 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 $S$?

Answer: 18
user: 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 $e^p$?

Answer: 16
user: 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 $z$?

Answer: 90
user: 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 $k$?

Answer: 991
user: 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 \(k + m\).

Answer: 29
user: 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 $n$?

Answer: 1005
user: 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 value of \(k + m + n\).

Answer: 11
user: 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
user: 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 $m+n$.

Answer: 271
user: 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.

Answer: 41
user: 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 give the value of \(k + m\).

Answer: 4
user: 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 $N$?

Answer: 78
user: 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 give the value of k + m + n.

Answer: 10
user: 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 in the form \(\frac{k}{m}\), where the fraction is fully simplified. Please find the value of k + m.

Answer: 59
user: For how many integers $x$ is the number $x^4-51x^2+50$ negative? Provide your answer as an integer.

Answer: 12
user: 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 $BE+DE+CD$.

Answer: 14
user: For how many positive integers $x$ is $\log_{10}(x-40) + \log_{10}(60-x) < 2$?

Answer: 18
user: 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?

Answer: 1
user: 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 \(k + m\).

Answer: 5
user: 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 $\overline{AE}$.

Answer: 20
user: 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.

Answer: 8
user: 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 
$$5^n<2^m<2^{m+2}<5^{n+1}.$$

Answer: 279
user: 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? Provide your answer as a fraction in simplest form, \(\frac{m}{n}\), and give the value of \(m + n\).

Answer: 209
user: 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 line to create a new shape with area \(B\). The ratio of the new shape's area to the original area is given in the form \(\frac{k}{m}\). Please calculate the value of \(k + m\).

Answer: 5
user: 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 \(\frac{a+b\sqrt{c}}{d}\). Please find the value of a + b + c + d.

Answer: 11
user: The number $2017$ is prime. Let $S = \sum \limits_{k=0}^{62} \binom{2014}{k}$. What is the remainder when $S$ is divided by $2017$?

Answer: 1024
user: 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 numbers is the date of my birthday later this month.
Ashley: And the sum of your two uniform numbers is today's date.
What number does Caitlin wear?

Answer: 11
user: 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?

Answer: 996506
user: 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 $m+n$.

Answer: 391
user: 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 $10x$?

Answer: 35
user: 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 
\[5^n<2^m<2^{m+2}<5^{n+1}.\]

Answer: 279
user: 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? Express your answer in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Calculate the value of \(k+m\).

Answer: 209
user: 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 \(x\)?

Answer: 1080
user: 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.

Answer: 6
user: 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?

Answer: 930
user: 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 find the value of HH + MM.

Answer: 33
user: 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. Who finished in $8$th place? Please provide the position of the first letter of their name in the alphabet.

Answer: 8
user: 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 $N$?

Answer: 37
user: 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 odd? If the answer is $x$, what is the value of $100x$?

Answer: 21