sungyub/dapo-math-17k-verl huggingface dataset:
user: Let \(N\) be a positive multiple of \(5\). One red ball and \(N\) green balls are arranged in a line in random order. Let \(P(N)\) be the probability that at least \(\tfrac{3}{5}\) of the green balls are on the same side of the red ball. Observe that \(P(5)=1\) and that \(P(N)\) approaches \(\tfrac{4}{5}\) as \(N\) grows large. Find the sum of the digits of the least value of \(N\) such that \(P(N) < \tfrac{321}{400}\).

Answer: 12
user: 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 the length of \(PQ\) in the form \(\frac{a}{b}\sqrt{c}\), where \(a\), \(b\), and \(c\) are integers? Please find the value of \(a + b + c\).

Answer: 28
user: For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. Find the number of positive integer divisors of the number $81n^4$.

Answer: 325
user: What is the area of the region enclosed by the graph of the equation \(x^2+y^2=|x|+|y|?\) Express your answer in the form \(a\pi + b\), where \(a\) and \(b\) are constants. Please provide the value of \(a + b\).

Answer: 3
user: In rectangle $ABCD,$ $AB=6$ and $BC=3$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$ are such that $BE=EF=FC$. Segments $\overline{AE}$ and $\overline{AF}$ intersect $\overline{BD}$ at $P$ and $Q$, respectively. The ratio $BP:PQ:QD$ can be written as $r:s:t$ where the greatest common factor of $r,s,$ and $t$ is $1.$ What is the value of $r+s+t$?

Answer: 20
user: Rectangle \(ABCD\) has \(AB=5\) and \(BC=4\). Point \(E\) lies on \(\overline{AB}\) so that \(EB=1\), point \(G\) lies on \(\overline{BC}\) so that \(CG=1\), and point \(F\) lies on \(\overline{CD}\) so that \(DF=2\). Segments \(\overline{AG}\) and \(\overline{AC}\) intersect \(\overline{EF}\) at \(Q\) and \(P\), respectively. The value of \(\frac{PQ}{EF}\) is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).

Answer: 101
user: Jerry starts at $0$ on the real number line. He tosses a fair coin $8$ times. When he gets heads, he moves $1$ unit in the positive direction; when he gets tails, he moves $1$ unit in the negative direction. The probability that he reaches $4$ at some time during this process is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. What is the value of $a + b$? (For example, he succeeds if his sequence of tosses is $HTHHHHHH.$)

Answer: 151
user: For some particular value of $N$, when $(a+b+c+d+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly $1001$ terms that include all four variables $a, b, c,$ and $d$, each to some positive power. What is the value of $N$?

Answer: 14
user: A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). The solution to the equation $2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find the value of $p+q.$

Answer: 109
user: A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A$?

Answer: 385
user: The least common multiple of $a$ and $b$ is $12$, and the least common multiple of $b$ and $c$ is $15$. What is the least possible value of the least common multiple of $a$ and $c$?

Answer: 20
user: Circles with centers \(P, Q\) and \(R\), having radii \(1, 2\) and \(3\), respectively, lie on the same side of line \(l\) and are tangent to \(l\) at \(P', Q'\) and \(R'\), respectively, with \(Q'\) between \(P'\) and \(R'\). The circle with center \(Q\) is externally tangent to each of the other two circles. Find the area of triangle \(PQR\). The original answer is in the form \(\sqrt{k} - \sqrt{m}\), please give the value of \(k + m\).

Answer: 8
user: What is the value of \(x\) such that the area of the region enclosed by the graph of the equation \(x^2+y^2=|x|+|y|\) is \(\pi + x\)?

Answer: 2
user: A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side?

Answer: 500
user: Let \(ABCD\) be a unit square. Let \(Q_1\) be the midpoint of \(\overline{CD}\). For \(i=1,2,\dots,\) let \(P_i\) be the intersection of \(\overline{AQ_i}\) and \(\overline{BD}\), and let \(Q_{i+1}\) be the foot of the perpendicular from \(P_i\) to \(\overline{CD}\). Find the value of 
\[\sum_{i=1}^{\infty} \text{Area of } \triangle DQ_i P_i \,.\]
The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 5
user: 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? Express your answer in the form of \(\frac{k}{m}\) as a common fraction, and give the value of \(k + m\).

Answer: 7
user: For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have?

Answer: 325
user: A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. Find the number of sets of three teams $\{A, B, C\}$ in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A$.

Answer: 385
user: How many ordered triples $(x,y,z)$ of positive integers satisfy ${lcm}(x,y) = 72, {lcm}(x,z) = 600$ and $lcm(y,z)=900$?

Answer: 15
user: What is the volume of the region in three-dimensional space defined by the inequalities \( |x|+|y|+|z|\le1 \) and \( |x|+|y|+|z-1|\le1 \)? Express your answer in the form \(\frac{k}{m}\) where k and m are integers. Please provide the value of \(k + m\).

Answer: 7
user: A binary operation $\diamondsuit$ has the properties that $a\diamondsuit (b\diamondsuit c) = (a\diamondsuit b)\cdot c$ and that $a\diamondsuit a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). The solution to the equation $2016 \diamondsuit (6\diamondsuit x)=100$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is the value of $p+q$?

Answer: 109
user: In regular hexagon \(ABCDEF\), points \(W\), \(X\), \(Y\), and \(Z\) are chosen on sides \(\overline{BC}\), \(\overline{CD}\), \(\overline{EF}\), and \(\overline{FA}\) respectively, so lines \(AB\), \(ZW\), \(YX\), and \(ED\) are parallel and equally spaced. The original answer for the ratio of the area of hexagon \(WCXYFZ\) to the area of hexagon \(ABCDEF\) is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).

Answer: 38
user: Three numbers in the interval \([0,1]\) are chosen independently and at random. Find the probability that the chosen numbers are the side lengths of a triangle with positive area. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 3
user: Two congruent circles centered at points $A$ and $B$ each pass through the other circle's center. The line containing both $A$ and $B$ is extended to intersect the circles at points $C$ and $D$. The circles intersect at two points, one of which is $E$. What is the degree measure of $\angle CED$? If the answer is $x^\circ$, what is the value of $x$?

Answer: 120
user: A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. Find the length of the fourth side.

Answer: 500
user: 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
user: There is a smallest positive real number $a$ such that there exists a positive real number $b$ such that all the roots of the polynomial $x^3-ax^2+bx-a$ are real. In fact, for this value of $a$ the value of $b$ is unique. What is this value of $b$?

Answer: 9
user: There are exactly $77,000$ ordered quadruplets $(a, b, c, d)$ such that $\gcd(a, b, c, d) = 77$ and $\operatorname{lcm}(a, b, c, d) = n$. What is the smallest possible value for $n$? Please provide your answer as an integer without any commas.

Answer: 27720
user: How many ordered triples $(x,y,z)$ of positive integers satisfy ${lcm}(x,y) = 72, {lcm}(x,z) = 600 { and ~lcm}(y,z)=900$? Provide your answer as an integer.

Answer: 15
user: The sequence \((a_n)\) is defined recursively by \(a_0=1\), \(a_1=\sqrt[19]{2}\), and \(a_n=a_{n-1}a_{n-2}^2\) for \(n\geq 2\). Find the smallest positive integer \(k\) such that the product \(a_1a_2\cdots a_k\) is an integer.

Answer: 17
user: What is the value of the expression \((2(2(2(2(2(2+1)+1)+1)+1)+1)+1)\)?

Answer: 127
user: Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11$. Then she switched the digits of the result, obtaining a number between $71$ and $75$, inclusive. What was Mary's original number?

Answer: 12
user: Pablo buys popsicles for his friends. The store sells single popsicles for $1 each, 3-popsicle boxes for $2, and 5-popsicle boxes for $3. What is the greatest number of popsicles that Pablo can buy with $8?

Answer: 13
user: Pablo buys popsicles for his friends. The store sells single popsicles for $1 each, 3-popsicle boxes for $2 each, and 5-popsicle boxes for $3. What is the greatest number of popsicles that Pablo can buy with $8?

Answer: 13
user: Tamara has three rows of two $6$-feet by $2$-feet flower beds in her garden. The beds are separated and also surrounded by $1$-foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?

Answer: 78
user: Real numbers \( x \), \( y \), and \( z \) satisfy the inequalities \( 0<x<1 \), \( -1<y<0 \), and \( 1<z<2 \). Among the following expressions, find the one that is necessarily positive: \( y + x^2 \), \( y + xz \), \( y + y^2 \), \( y + 2y^2 \), \( y + z \). The original answer is in the form \( a \cdot y + b \cdot z \), where \( a \) and \( b \) are non-zero integers. Please find the value of \( a + b \).

Answer: 2
user: Samia set off on her bicycle to visit her friend, traveling at an average speed of $17 ~kilometers~ per~ hour$. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at $5 ~kilometers~ per ~hour$. In all it took her $44~ minutes$ to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk? If the answer is $x$ kilometers, what is the value of $10x$?

Answer: 28
user: The data set $[6, 19, 33, 33, 39, 41, 41, 43, 51, 57]$ has median $Q_2 = 40$, first quartile $Q_1 = 33$, and third quartile $Q_3 = 43$. An outlier in a data set is a value that is more than $1.5$ times the interquartile range below the first quartile ($Q_1$) or more than $1.5$ times the interquartile range above the third quartile ($Q_3$), where the interquartile range is defined as $Q_3 - Q_1$. How many outliers does this data set have? Provide your answer as an integer.

Answer: 1
user: Joy has $30$ thin rods, one each of every integer length from $1 cm$ through $30 cm$. She places the rods with lengths $3 cm$, $7 cm$, and $15 cm$ on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?

Answer: 17
user: If the degree measures of the angles of a triangle are in the ratio $3:3:4$, and if the answer is $x^\circ$, what is the value of $x$?

Answer: 72
user: Samia set off on her bicycle to visit her friend, traveling at an average speed of $17$ kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at $5$ kilometers per hour. In all it took her $44$ minutes to reach her friend's house. In kilometers, rounded to the nearest tenth, how far did Samia walk? After finding the distance, multiply it by 10 and provide the resulting integer value.

Answer: 28
user: The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. The original answer is in the form \(\frac{k\sqrt{m} - n}{p}\), where \(k\), \(m\), \(n\), and \(p\) are integers. What is the value of \(k + m + n + p\)?

Answer: 9
user: Points $A(11, 9)$ and $B(2, -3)$ are vertices of $\triangle ABC$ with $AB=AC$. The altitude from $A$ meets the opposite side at $D(-1, 3)$. What are the coordinates of point $C$? Provide your answer as an ordered pair \((m, n)\). Please give the value of m + n.

Answer: 5
user: The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216 \pi$. What is the length of $\overline{AB}$?

Answer: 20
user: Malcolm wants to visit Isabella after school today and knows the street where she lives but doesn't know her house number. She tells him, "My house number has two digits, and exactly three of the following four statements about it are true."

It is prime.

It is even.

It is divisible by 7.

One of its digits is 9.

This information allows Malcolm to determine Isabella's house number. What is the units digit of her house number?

Answer: 8
user: Minnie rides on a flat road at $20$ kilometers per hour (kph), downhill at $30$ kph, and uphill at $5$ kph. Penny rides on a flat road at $30$ kph, downhill at $40$ kph, and uphill at $10$ kph. Minnie goes from town $A$ to town $B$, a distance of $10$ km all uphill, then from town $B$ to town $C$, a distance of $15$ km all downhill, and then back to town $A$, a distance of $20$ km on the flat. Penny goes the other way around using the same route. How many more minutes does it take Minnie to complete the $45$-km ride than it takes Penny?

Answer: 65
user: A radio program has a quiz consisting of $3$ multiple-choice questions, each with $3$ choices. A contestant wins if he or she gets $2$ or more of the questions right. The contestant answers randomly to each question. What is the probability of winning? Express your answer in the form \frac{k}{m}, where k and m are integers, and provide the value of k + m.

Answer: 34
user: Joy has $30$ thin rods, one each of every integer length from $1$ cm through $30$ cm. She places the rods with lengths $3$ cm, $7$ cm, and $15$ cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?

Answer: 17
user: Chloe chooses a real number uniformly at random from the interval \([ 0,2017 ]\). Independently, Laurent chooses a real number uniformly at random from the interval \([ 0 , 4034 ]\). Find the probability that Laurent's number is greater than Chloe's number. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 7
user: At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ say that they dislike it, and the rest say that they like it. What is the value of $x$ if $x\%$ of students who say they dislike dancing actually like it?

Answer: 25
user: Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?

Answer: 143
user: Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, $3$, $23578$, and $987620$ are monotonous, but $88$, $7434$, and $23557$ are not. Find the total number of monotonous positive integers.

Answer: 1524
user: There are $10$ horses, named Horse $1$, Horse $2, \ldots,$ Horse $10$. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time $0$ all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time $S > 0$, in minutes, at which all $10$ horses will again simultaneously be at the starting point is $S = 2520$. Let $T>0$ be the least time, in minutes, such that at least $5$ of the horses are again at the starting point. What is the sum of the digits of $T$?

Answer: 3
user: What is the sum of the roots of $z^{12}=64$ that have a positive real part? Express your answer in the form $a\sqrt{b}+c\sqrt{d}$, where $a$, $b$, $c$, and $d$ are integers. Please find the value of a + b + c + d.

Answer: 11
user: Define a sequence recursively by $F_{0}=0,~F_{1}=1,$ and $F_{n}=$ the remainder when $F_{n-1}+F_{n-2}$ is divided by $3,$ for all $n\geq 2.$ Thus the sequence starts $0,1,1,2,0,2,\cdots$ Find the value of $F_{2017}+F_{2018}+F_{2019}+F_{2020}+F_{2021}+F_{2022}+F_{2023}+F_{2024}.$

Answer: 9
user: Peter, Emma, and Kyler played chess with each other. Peter won $4$ games and lost $2$ games. Emma won $3$ games and lost $3$ games. If Kyler lost $3$ games, how many games did he win? Fill in the number of games Kyler won.

Answer: 1
user: An integer $N$ is selected at random in the range $1\leq N \leq 2020$. Find the probability that the remainder when $N^{16}$ is divided by $5$ is $1$. Express your answer as a fraction \frac{k}{m} in its simplest form, and then give the value of k + m.

Answer: 9
user: Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. Find the number of ways for the five of them to sit in a row of $5$ chairs under these conditions.

Answer: 28
user: Chloe and Zoe are both students in Ms. Demeanor's math class. Last night, they each solved half of the problems in their homework assignment alone and then solved the other half together. Chloe had correct answers to only $80\%$ of the problems she solved alone, but overall $88\%$ of her answers were correct. Zoe had correct answers to $90\%$ of the problems she solved alone. What was Zoe's overall percentage of correct answers? If the answer is x%, what is the value of x?

Answer: 93
user: Chloe chooses a real number uniformly at random from the interval \([0, 2017]\). Independently, Laurent chooses a real number uniformly at random from the interval \([0, 4034]\). Find the probability that Laurent's number is greater than Chloe's number. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 7
user: Rectangle \(ABCD\) has \(AB=3\) and \(BC=4\). Point \(E\) is the foot of the perpendicular from \(B\) to diagonal \(\overline{AC}\). What is the area of \(\triangle AED\)? The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

Answer: 79
user: Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB'=3AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC'=3BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA'=3CA$. Find the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$. If the ratio is $x:1$, what is the value of $x$?

Answer: 37
user: In the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in the middle, a path only allows moves from one letter to an adjacent (above, below, left, or right, but not diagonal) letter. One example of such a path is traced in the picture.

fill((0.5, 4.5)--(1.5,4.5)--(1.5,2.5)--(0.5,2.5)--cycle,lightgray); fill((1.5,3.5)--(2.5,3.5)--(2.5,1.5)--(1.5,1.5)--cycle,lightgray); label("$8$", (1, 0)); label("$C$", (2, 0)); label("$8$", (3, 0)); label("$8$", (0, 1)); label("$C$", (1, 1)); label("$M$", (2, 1)); label("$C$", (3, 1)); label("$8$", (4, 1)); label("$C$", (0, 2)); label("$M$", (1, 2)); label("$A$", (2, 2)); label("$M$", (3, 2)); label("$C$", (4, 2)); label("$8$", (0, 3)); label("$C$", (1, 3)); label("$M$", (2, 3)); label("$C$", (3, 3)); label("$8$", (4, 3)); label("$8$", (1, 4)); label("$C$", (2, 4)); label("$8$", (3, 4));[/asy]

Find the number of different paths.

Answer: 24
user: There are $10$ horses, named Horse $1$, Horse $2$, . . . , Horse $10$. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time $0$ all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time $S > 0$, in minutes, at which all $10$ horses will again simultaneously be at the starting point is $S=2520$. Let $T > 0$ be the least time, in minutes, such that at least $5$ of the horses are again at the starting point. What is the sum of the digits of $T?$

Answer: 3
user: In the figure below, semicircles with centers at \(A\) and \(B\) and with radii \(2\) and \(1\), respectively, are drawn in the interior of, and sharing bases with, a semicircle with diameter \(JK\). The two smaller semicircles are externally tangent to each other and internally tangent to the largest semicircle. A circle centered at \(P\) is drawn externally tangent to the two smaller semicircles and internally tangent to the largest semicircle. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 13
user: The number \(21! = 51,090,942,171,709,440,000\) has over \(60,000\) positive integer divisors. One of them is chosen at random. Find the probability that it is odd. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

Answer: 20
user: In the figure below, choose point \(D\) on \(\overline{BC}\) so that \(\triangle ACD\) and \(\triangle ABD\) have equal perimeters. What is the area of \(\triangle ABD\)?

draw((0,0)--(4,0)--(0,3)--(0,0)); label("$A$", (0,0), SW); label("$B$", (4,0), ESE); label("$C$", (0, 3), N); label("$3$", (0, 1.5), W); label("$4$", (2, 0), S); label("$5$", (2, 1.5), NE);[/asy]

The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

Answer: 17
user: Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, $3$, $23578$, and $987620$ are monotonous, but $88$, $7434$, and $23557$ are not. How many monotonous positive integers are there? Provide your answer as an integer.

Answer: 1524
user: The diameter \(AB\) of a circle of radius \(2\) is extended to a point \(D\) outside the circle so that \(BD=3\). Point \(E\) is chosen so that \(ED=5\) and line \(ED\) is perpendicular to line \(AD\). Segment \(AE\) intersects the circle at a point \(C\) between \(A\) and \(E\). Find the area of \(\triangle ABC\). The original answer is in \(\frac{m}{n}\) format, please give the value of \(m + n\).

Answer: 177
user: Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions? Provide your answer as an integer.

Answer: 28