meetforever/GRPO_CL huggingface dataset:
user: For odd positive integers $n$, define $f(n)$ to be the smallest odd integer greater than $n$ that is not relatively prime to $n$. Compute the smallest $n$ such that $f(f(n))$ is not divisible by $3$.
Answer: 161
user: Reimu invented a new number base system that uses exactly five digits. The number $0$ in the decimal system is represented as $00000$. Whenever a number is incremented, Reimu finds the leftmost digit (of the five digits) that is equal to the "units" (rightmost) digit, increments this digit, and sets all the digits to its right to 0. 

For example, an analogous system that uses three digits would begin with $000$, $100$, $110$, $111$, $200$, $210$, $211$, $220$, $221$, $222$, $300$, $\ldots$.

Compute the decimal representation of the number that Reimu would write as $98765$.
Answer: 1727
user: Let $A, B, C, D$ be points, in order, on a straight line such that $AB = BC = CD$. Let $E$ be a point closer to $B$ than $D$ such that $BE = EC = CD$ and let $F$ be the midpoint of $DE$. Let $AF$ intersect $EC$ at $G$ and let $BF$ intersect $EC$ at $H$. If $[BHC] + [GHF] = 1$, then $AD^2 = \frac{a\sqrt{b}}{c}$ where $a, b,$ and $c$ are positive integers, $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a + b + c$.
Answer: 374
user: In the diagram below, $\angle B = 43^\circ$ and $\angle D = 102^\circ$. Find $\angle A + \angle B + \angle C + \angle D + \angle E + \angle F$.
Answer: 720
user: Noew is writing a $15$-problem mock AIME consisting of four subjects: algebra, geometry, combinatorics, and number theory. The AIME is considered somewhat evenly distributed if there is at least one problem of each subject and at least six combinatorics problems. Two AIMEs are considered similar if they have the same subject distribution (same number of each subject). How many non-similar somewhat evenly distributed mock AIMEs can Noew write?
Answer: 84
user: A trapezoid $ABCD$ lies on the $xy$-plane. The slopes of lines $BC$ and $AD$ are both $\frac{1}{3}$, and the slope of line $AB$ is $-\frac{2}{3}$. Given that $AB = CD$ and $BC < AD$, the absolute value of the slope of line $CD$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are two relatively prime positive integers. Find $100m + n$.
Answer: 1706
user: Compute \[\sum_{n_{60}=0}^2\sum_{n_{59}=0}^{n_{60}}\cdots\sum_{n_2=0}^{n_3}\sum_{n_1=0}^{n_2}\sum_{n_0=0}^{n_1}1.\]
Answer: 1953
user: Let $a$ be the sum of the numbers:

$$99 \times 0.9$$
$$999 \times 0.9$$
$$9999 \times 0.9$$
$$\vdots$$
$$999\cdots 9 \times 0.9$$

where the final number in the list is $0.9$ times a number written as a string of $101$ digits all equal to $9$. Find the sum of the digits in the number $a$.
Answer: 891
user: The $260$ volumes of the *Encyclopedia Galactica* are out of order in the library. Fortunately for the librarian, the books are numbered. Due to his religion, which holds both encyclopedias and the concept of parity in high esteem, the librarian can only rearrange the books two at a time, and then only by switching the position of an even-numbered volume with that of an odd-numbered volume. Find the minimum number of such transpositions sufficient to get the books back into ordinary sequential order, regardless of the starting positions of the books. (Find the minimum number of transpositions in the worst-case scenario.)
Answer: 130
user: All of the roots of $x^3 + ax^2 + bx + c$ are positive integers greater than 2, and the coefficients satisfy $a + b + c + 1 = -2009$. Find $a$.
Answer: -58
user: Find the sum of all natural numbers from $1$ to $500$ that are not divisible by $5$ or $7$.
Answer: 85783
user: Find the largest divisor of $2800$ that is a perfect square.
Answer: 400
user: At a meeting, each person is familiar with 22 people. If two persons $A$ and $B$ know each other, among the remaining people, they do not have a common friend. For each pair of individuals $A$ and $B$ who are not familiar with each other, there are six common acquaintances among the remaining people. How many people were at the meeting?
Answer: 100
user: Let $f(X, Y, Z) = X^5Y - XY^5 + Y^5Z - YZ^5 + Z^5X - ZX^5$.

Find \( \frac{f(2009, 2010, 2011) + f(2010, 2011, 2009) - f(2011, 2010, 2009)}{f(2009, 2010, 2011)} \)
Answer: 3
user: A positive integer is called *cool* if it can be expressed in the form $a! \cdot b! + 315$ where $a$ and $b$ are positive integers. For example, $1! \cdot 1! + 315 = 316$ is a cool number. Find the sum of all cool numbers that are also prime numbers.
Answer: 317
user: A regular pentagon can have the line segments forming its boundary extended to lines, resulting in an arrangement of lines that intersect at ten points. How many ways are there to choose five points from these ten such that no three of the points are collinear?
Answer: 12
user: A nine-digit number has the form $\overline{6ABCDEFG3}$, where every three consecutive digits sum to $13$. Find $D$.
Answer: 4
user: We randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of the fourth largest number?
Answer: 606
user: Let $\triangle ABC$ be a triangle in a plane such that $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point in three-dimensional space such that $\angle{BDC}=\angle{CDA}=\angle{ADB}=90^\circ$. Let $d$ be the distance from $D$ to the plane containing $\triangle ABC$. The value $d^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
Answer: 511
user: Consider a rectangle $ABCD$ with $AB = 30$ and $BC = 60$. Construct a circle $T$ whose diameter is $AD$. Construct another circle $S$ whose diameter is $AB$. Let circles $S$ and $T$ intersect at a point $P$ such that $P \neq A$. Let the line $AP$ intersect $BC$ at point $E$. Define point $F$ on $AB$ such that the line $EF$ is tangent to the circle with diameter $AD$. Find the area of triangle $\triangle AEF$.
Answer: 75
user: There are $36$ participants at a BdMO event. Some of the participants shook hands with each other, but no two participants shook hands more than once. Each participant recorded the number of handshakes they made. It was found that no two participants with the same number of handshakes had shaken hands with each other. Find the maximum possible number of handshakes at the party, providing proof.

(When two participants shake hands with each other, this is counted as one handshake.)
Answer: 546
user: Which is the largest positive integer $n$ satisfying the inequality: $n^{2007} > (2007)^n$?
Answer: 2006
user: Let $\triangle ABC$ be an equilateral triangle and $P$ a point on $\overline{BC}$. If $PB=50$ and $PC=30$, compute $PA$.
Answer: 70
user: A dartboard consists of three concentric circles with radii 4, 6, and 8. Three darts are thrown at the board, sticking at random locations. Determine the probability that each dart lands in a different region of the dartboard. The probability can be expressed as \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. Calculate \( m + n \).
Answer: 617
user: Let $\triangle ABC$ be a triangle with side lengths $AB = 9$, $BC = 10$, $CA = 11$. Let $O$ be the circumcenter of $\triangle ABC$. Denote $D = AO \cap BC$, $E = BO \cap CA$, $F = CO \cap AB$. If $\frac{1}{AD} + \frac{1}{BE} + \frac{1}{CF}$ can be written in simplest form as $\frac{a \sqrt{b}}{c}$, find $a + b + c$.
Answer: 43
user: In a right-angled triangle where all side lengths are integers, one of the catheti has a length of $1994$. Determine the length of the hypotenuse.
Answer: 994010
user: Determine the last three digits of $374^{2022}$.
Answer: 376
user: Determine the largest prime which divides both $2^{24} - 1$ and $2^{16} - 1$.
Answer: 17
user: Consider a triangle $ABC$ and let $M$ be the midpoint of the side $BC$. Suppose $\angle MAC = \angle ABC$ and $\angle BAM = 105^\circ$. Find the measure of $\angle ABC$.
Answer: 30
user: Find the sum of the indexes of the singular points other than zero of the vector field:
\[ z\overline{z}^2 + z^4 + 2\overline{z}^4 \]
Answer: 3
user: What is the remainder when $1234^{567} + 89^{1011}$ is divided by $12$?
Answer: 9
user: Compute the number of ordered pairs $(a,b)$ of positive integers satisfying $a^b=2^{100}$.
Answer: 9
user: Consider an arithmetic progression made up of $100$ terms. If the sum of all the terms of the progression is $150$ and the sum of the even terms is $50$, find the sum of the squares of the $100$ terms of the progression.
Answer: 83550
user: Each of the numbers $x_1, x_2, \ldots, x_{101}$ is $\pm 1$. What is the smallest positive value of $\sum_{1\leq i < j \leq 101} x_i x_j$?
Answer: 10
user: Let $ABCD$ be a square with side length $10$. Point $P$ is located on side $BC$. By folding the paper along line $AP$, point $B$ maps to point $Q$. The line $PQ$ intersects side $CD$ at point $R$. Calculate the perimeter of triangle $PCR$.
Answer: 20
user: Three palaces, each rotating on a duck leg, make a full round in $30$, $50$, and $70$ days, respectively. Today, at noon, all three palaces face northwards. In how many days will they all face southwards?
Answer: 525
user: Let $f(n)$ be the sum of the digits of $n$. Find $\sum_{n=1}^{99}f(n)$.
Answer: 900
user: Find the value of $|xy|$ given that $x$ and $y$ are integers and $6x^2y^2 + 5x^2 - 18y^2 = 17253$.
Answer: 54
user: In the diagram below, three squares are inscribed in right triangles. Their areas are $A$, $M$, and $N$, as indicated in the diagram. If $M = 5$ and $N = 12$, then $A$ can be expressed as $a + b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of any prime. Compute $a + b + c$.
Answer: 36
user: How many positive integers $n$ are there such that $n$ is an exact divisor of at least one of the numbers $10^{40}$ and $20^{30}$?
Answer: 2301
user: Let $N_{15}$ be the answer to problem 15.

For a positive integer $x$ expressed in base ten, let $x'$ be the result of swapping its first and last digits. For example, if $x = 2024$, then $x' = 4022$. Let $C$ be the number of $N_{15}$-digit positive integers $x$ with a nonzero leading digit that satisfy the property that both $x$ and $x'$ are divisible by $11$. Note: $x'$ is allowed to have a leading digit of zero. Compute the sum of the digits of $C$ when $C$ is expressed in base ten.
Answer: 9
user: Find the smallest positive integer $n$ such that both the sum $n + 10$ and the product $n \times 10$ are perfect squares.
Answer: 90
user: Each of the natural numbers from $1$ to $n$ is colored either red or blue, with each color being used at least once. It turns out that:
- Every red number is a sum of two distinct blue numbers; and
- Every blue number is a difference between two red numbers.

Determine the smallest possible value of $n$ for which such a coloring exists.
Answer: 9
user: Wendy takes Honors Biology at school, a small class with only fourteen students (including Wendy) who sit around a circular table. Wendy's friends Lucy, Starling, and Erin are also in that class. Last Monday, none of the fourteen students were absent from class. Before the teacher arrived, Lucy and Starling stretched out a blue piece of yarn between them. Then Wendy and Erin stretched out a red piece of yarn between them at about the same height so that the yarn would intersect if possible. If all possible positions of the students around the table are equally likely, let $\frac{m}{n}$ be the probability that the yarns intersect, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
Answer: 4
user: A class of $10$ students took a math test. Each problem was solved by exactly $7$ of the students. If the first nine students each solved $4$ problems, how many problems did the tenth student solve?
Answer: 6
user: Let $N$ be the greatest positive integer that can be expressed using all seven Roman numerals $I$, $V$, $X$, $L$, $C$, $D$, and $M$ exactly once each, and let $n$ be the least positive integer that can be expressed using these numerals exactly once each. Find $N - n$. Note that the arrangement $CM$ is never used in a number along with the numeral $D$.
Answer: 222
user: Fred chooses a positive two-digit number with distinct nonzero digits. Laura takes Fred’s number and swaps its digits. She notices that the sum of her number and Fred’s number is a perfect square, and the positive difference between them is a perfect cube. Find the greater of the two numbers.
Answer: 74
user: Find the number of ordered pairs of integers $(m, n)$ that satisfy $20m - 10n = mn$.
Answer: 24
user: You are given an unlimited supply of red, blue, and yellow cards to form a hand. Each card has a point value, and your score is the sum of the point values of those cards. The point values are as follows: the value of each red card is $1$, the value of each blue card is equal to twice the number of red cards, and the value of each yellow card is equal to three times the number of blue cards. What is the maximum score you can get with fifteen cards?
Answer: 168
user: Sam spends his days walking around the following $2 \times 2$ grid of squares:
\[
\begin{array}{|c|c|}
\hline
1 & 2 \\
\hline
4 & 3 \\
\hline
\end{array}
\]
Two squares are considered adjacent if they share a side. Sam starts at the square labeled $1$ and every second, he walks to an adjacent square. How many paths can Sam take such that the sum of the numbers on every square he visits in his path is equal to $20$, excluding the square he started on?
Answer: 167
user: Given a real number $a$, the floor of $a$, written $\lfloor a \rfloor$, is the greatest integer less than or equal to $a$. For how many real numbers $x$ such that $1 \le x \le 20$ is \[ x^2 + \lfloor 2x \rfloor = \lfloor x^2 \rfloor + 2x \? \]
Answer: 362
user: Vera has several identical matches, from which she forms a triangle. She wants any two sides of this triangle to differ in length by at least $10$ matches. However, it is impossible to create such a triangle with the available matches (without leaving any extra matches). What is the maximum number of matches Vera can have?
Answer: 62
user: Let $L$ be the number formed by 2022 digits equal to $1$, that is, $L=111\ldots111$ (2022 times). Compute the sum of the digits of the number $9L^2 + 2L$.
Answer: 4044
user: While entertaining his younger sister Alexis, Michael drew two different cards from an ordinary deck of playing cards. Let $a$ be the probability that the cards are of different ranks. Compute $\lfloor 1000a \rfloor$.
Answer: 941
user: Each of the integers $1, 2, \ldots, 729$ is written in its base-$3$ representation without leading zeroes. The numbers are then joined together in that order to form a continuous string of digits: $12101112202122\ldots$ How many times in this string does the substring $012$ appear?
Answer: 148
user: Consider the set $A = \{0, 1, 2, \dots , 9 \}$ and let $(B_1,B_2, \dots , B_k)$ be a collection of nonempty subsets of $A$ such that $B_i \cap B_j$ has at most two elements for $i \neq j$. What is the maximal value of $k$?
Answer: 175
user: Compute the number of natural numbers $1 \leq n \leq 10^6$ such that the least prime divisor of $n$ is $17$.
Answer: 11284
user: Find the number of ordered quadruples of positive integers \((a,b,c,d)\) such that \(a,b,c,\) and \(d\) are all (not necessarily distinct) factors of \(30\) and \(abcd > 900\).
Answer: 1940
user: How many pairs of positive integers \((a, b)\) satisfy the equation \(\log_a 16 = b\)?
Answer: 3
user: The product of $10$ integers is $1024$. What is the greatest possible sum of these $10$ integers?
Answer: 1033
user: One can define the greatest common divisor of two positive rational numbers as follows: for $a$, $b$, $c$, and $d$ positive integers with $\gcd(a,b)=\gcd(c,d)=1$, write:  
\[ \gcd\left(\frac{a}{b},\frac{c}{d}\right) = \frac{\gcd(ad,bc)}{bd}. \]  

For all positive integers $K$, let $f(K)$ denote the number of ordered pairs of positive rational numbers $(m,n)$ with $m<1$ and $n<1$ such that:  
\[ \gcd(m,n)=\frac{1}{K}. \]  

What is $f(2017)-f(2016)$?
Answer: 1152
user: Fedja used matches to form a parallelogram, ensuring that the sides are of equal length and the vertices do not lie on a single line. It is observed that exactly 7 or 9 matches fit into the diagonals. Determine the number of matches that make up the perimeter of the parallelogram.
Answer: 22
user: Given $1994$ points on a plane, it is known that any $100$ of them can be selected such that $98$ of these can be enclosed by a circle with a diameter of $1$. Determine the minimum number of circles with a radius of $1$ required to cover all $1994$ points.
Answer: 21
user: Consider an orange and black coloring of a $20 \times 14$ square grid. Let $n$ be the number of colorings such that every row and column has an even number of orange squares. Evaluate $\log_2 n$.
Answer: 247
user: How many natural numbers exist such that their square results in a thirty-digit number with the following property: When divided from left to right into three groups of ten digits, the numbers formed by the middle group and the right group are both four times the number formed by the left group?
Answer: 18377
user: The expression $\circ 1 \circ 2 \circ 3 \circ \dots \circ 2012$ is written on a blackboard. Catherine places a $+$ sign or a $-$ sign into each blank. She then evaluates the expression and finds the remainder when it is divided by 2012. How many possible values are there for this remainder?
Answer: 1006
user: Let $a$ and $b$ be the two possible values of $\tan\theta$ given that \[\sin\theta + \cos\theta = \frac{193}{137}.\] If $a+b=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $m+n$.
Answer: 28009
user: Convex pentagon $ABCDE$ satisfies $AB \parallel DE$, $BE \parallel CD$, $BC \parallel AE$, $AB = 30$, $BC = 18$, $CD = 17$, and $DE = 20$. Find its area.
Answer: 612
user: For how many integers $n$ between $1$ and $2021$ does the infinite nested expression $\sqrt{n + \sqrt{n + \sqrt{n + \sqrt{\ldots}}}}$ give a rational number?
Answer: 44
user: Consider all functions $f: \mathbb{N} \to \mathbb{N}$ satisfying $f(t^2 f(s)) = s(f(t))^2$ for all $s$ and $t$ in $\mathbb{N}$. Determine the least possible value of $f(1998)$.
Answer: 120
user: Let $a$ and $b$ be positive integers such that $(2a + b)(2b + a) = 4752$. Find the value of $ab$.
Answer: 520
user: Let $P$ be the intersection point of the diagonals of a convex quadrilateral $ABCD$. It is known that the areas of triangles $\triangle ABC$, $\triangle BCD$, and $\triangle DAP$ are $8 \text{ cm}^2$, $9 \text{ cm}^2$, and $10 \text{ cm}^2$, respectively. Find the area of the quadrilateral $ABCD$.
Answer: 24
user: Let $r_1$, $r_2$, $\ldots$, $r_{20}$ be the roots of the polynomial $x^{20}-7x^3+1$.  If 
\[\dfrac{1}{r_1^2+1}+\dfrac{1}{r_2^2+1}+\cdots+\dfrac{1}{r_{20}^2+1}\] 
can be written in the form $\tfrac{m}{n}$ where $m$ and $n$ are positive coprime integers, find $m+n$.
Answer: 240
user: The diagram below shows rectangle $ABDE$ where $C$ is the midpoint of side $\overline{BD}$, and $F$ is the midpoint of side $\overline{AE}$. If $AB=10$ and $BD=24$, find the area of the shaded region.

\[\text{[asy]}
\text{size(300);}
\text{defaultpen(linewidth(0.8));}
\text{pair A = (0,10), B=origin, C=(12,0), D=(24,0), E=(24,10), F=(12,10), G=extension(C,E,D,F);}
\text{filldraw(A--C--G--F--cycle, gray(0.7));}
\text{draw(A--B--D--E--F^^E--G--D);}
\text{label("$A$", A, NW);}
\text{label("$B$", B, SW);}
\text{label("$C$", C, S);}
\text{label("$D$", D, SE);}
\text{label("$E$", E, NE);}
\text{label("$F$", F, N);}
\text{[/asy]}
\]
Answer: 90
user: In trapezoid $ABCD$, where $BC \parallel AD$, $AB = 13$, $BC = 15$, $CD = 14$, and $DA = 30$. Find the area of $ABCD$.
Answer: 252
user: What is the smallest integer $n$ for which $\frac{10!}{n}$ is a perfect square?
Answer: 7
user: For every positive integer $n$, define $S_n$ to be the sum
\[
  S_n = \sum_{k = 1}^{2010} \left( \cos \frac{k! \pi}{2010} \right)^n .
\]
As $n$ approaches infinity, what value does $S_n$ approach?
Answer: 1944
user: Ten test papers are to be prepared for the National Olympiad. Each paper has 4 problems, and no two papers have more than 1 problem in common. What is the minimum number of distinct problems needed?
Answer: 13
user: There are $2014$ students from high schools nationwide sitting around a round table in an arbitrary manner. The organizers want to rearrange the students so that those from the same school sit next to each other. This is done by performing swaps between two adjacent groups of students. 

The task is to find the smallest number $k$ such that the desired arrangement can be achieved with no more than $k$ swaps.

Illustration of a swap: 
$$...\underbrace{ABCD}_\text{1}\underbrace{EFG}_\text{2}... \to ...\underbrace{EFG}_\text{2}\underbrace{ABCD}_\text{1}...$$
Answer: 2013
user: How many solutions does the equation:
$$\left\lfloor \frac{x}{20} \right\rfloor = \left\lfloor \frac{x}{17} \right\rfloor$$
have over the set of positive integers?

$\left\lfloor a \right\rfloor$ denotes the largest integer that is less than or equal to $a$.
Answer: 56
user: How many cubic polynomials of the form $x^3 - ax^2 + (a+d)x - (a+2d)$ have roots that are all non-negative integers, where $a$ and $d$ are integers?
Answer: 3
user: Call a lattice point *visible* if the line segment connecting the point and the origin does not pass through another lattice point. Given a positive integer $k$, denote by $S_k$ the set of all visible lattice points $(x, y)$ such that $x^2 + y^2 = k^2$. Let $D$ denote the set of all positive divisors of $2021 \cdot 2025$. Compute the sum \[ \sum_{d \in D} |S_d| \]

Here, a lattice point is a point $(x, y)$ on the plane where both $x$ and $y$ are integers, and $|A|$ denotes the number of elements of the set $A$.
Answer: 20
user: How many seven-digit numbers are multiples of $388$ and end in $388$?
Answer: 93
user: Let $S$ be a subset of \( \{1, 2, \ldots, 500\} \) such that no two distinct elements of $S$ have a product that is a perfect square. Find, with proof, the maximum possible number of elements in $S$.
Answer: 306
user: Let $A$ be a positive integer which is a multiple of 3 but not a multiple of 9. If adding the product of each digit of $A$ to $A$ results in a multiple of 9, find the possible minimum value of $A$.
Answer: 138
user: Cat and Claire are having a conversation about Cat's favorite number.

Cat says, "My favorite number is a two-digit positive integer that is the product of three distinct prime numbers!"

Claire says, "I don't know your favorite number yet, but I do know that among four of the numbers that might be your favorite number, you could start with any one of them, add a second, subtract a third, and get the fourth!"

Cat says, "That's cool! My favorite number is not among those four numbers, though."

Claire says, "Now I know your favorite number!"

What is Cat's favorite number?
Answer: 70
user: The area of triangle $ABC$ is $2007$. One of its sides has length $18$, and the tangent of the angle opposite that side is $\frac{2007}{24832}$. When the altitude is dropped to the side of length $18$, it cuts that side into two segments. Find the sum of the squares of those two segments.
Answer: 194
user: Determine all positive integers $n$ for which $2^{n-1}n + 1$ is a perfect square.
Answer: 5
user: Students are in a classroom with $n$ rows. In each row, there are $m$ tables. It is given that $m, n \geq 3$. At each table, there is exactly one student. We call the neighbors of a student those sitting one place to the right, left, in front, and behind him. Each student shook hands with his neighbors. In the end, there were $252$ handshakes. How many students were in the classroom?
Answer: 153
user: Let $AX$ be a diameter of a circle $\Omega$ with radius $10$, and suppose that $C$ lies on $\Omega$ so that $AC = 16$. Let $D$ be the other point on $\Omega$ so that $CX = CD$. From here, define $D'$ to be the reflection of $D$ across the midpoint of $AC$, and $X'$ to be the reflection of $X$ across the midpoint of $CD$. If the area of triangle $CD'X'$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime, find $p + q$.
Answer: 1369
user: Find the integer part of \( \sum_{n=1}^{40000} \frac{1}{\sqrt{n}} \).
Answer: 398