rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: Let  $\triangle P_1P_2P_3$  be an equilateral triangle. For each  $n\ge 4$ , *Mingmingsan* can set  $P_n$  as the circumcenter or orthocenter of  $\triangle P_{n-3}P_{n-2}P_{n-1}$ . Find all positive integer  $n$  such that *Mingmingsan* has a strategy to make  $P_n$  equals to the circumcenter of  $\triangle P_1P_2P_3$ .
Answer:  n \equiv 0 \pmod{4}
Problem: William is biking from his home to his school and back, using the same route. When he travels to school, there is an initial  $20^\circ$  incline for  $0.5$  kilometers, a flat area for  $2$  kilometers, and a  $20^\circ$  decline for  $1$  kilometer. If William travels at  $8$  kilometers per hour during uphill  $20^\circ$  sections,  $16$  kilometers per hours during flat sections, and  $20$  kilometers per hour during downhill  $20^\circ$  sections, find the closest integer to the number of minutes it take William to get to school and back.
Answer: 29
Problem: An integer  $ m > 1$  is given. The inﬁnite sequence  $ (x_n)_{n\ge 0}$  is deﬁned by  $ x_i\equal{}2^i$  for  $ i<m$  and  $ x_i\equal{}x_{i\minus{}1}\plus{}x_{i\minus{}2}\plus{}\cdots \plus{}x_{i\minus{}m}$  for  $ i\ge m$ .
Find the greatest natural number  $ k$  such that there exist  $ k$  successive terms of this sequence which are divisible by  $ m$ .
Answer: m
Problem: Find all prime numbers of the form  $\tfrac{1}{11} \cdot \underbrace{11\ldots 1}_{2n \textrm{ ones}}$ , where  $n$  is a natural number.
Answer: 101
Problem: Find the smallest positive integer  $n$  such that a cube with sides of length  $n$  can be divided up into exactly  $2007$  smaller cubes, each of whose sides is of integer length.
Answer:  n = 13
Problem: Find all triplets  $ (x,y,z) $  of real numbers such that
\[ 2x^3 + 1 = 3zx \]\[ 2y^3 + 1 = 3xy \]\[ 2z^3 + 1 =  3yz \]
Answer:  (1, 1, 1)
Problem: In the acute-angled triangle  $ABC$ , the altitudes  $BP$  and  $CQ$  were drawn, and the point  $T$  is the intersection point  of the altitudes of  $\Delta PAQ$ . It turned out that  $\angle CTB = 90 {} ^ \circ$ . Find the measure of  $\angle BAC$ .
Answer: 45^	ext{\circ}
Problem: Find all real numbers  $a$  for which there exists a function   $f: R \to R$  asuch that  $x + f(y) =a(y + f(x))$  for all real numbers  $x,y\in R$ .

I.Voronovich
Answer: a = \pm 1
Problem: Find all real solutions of the system  $$ \begin{cases}  x_1 +x_2 +...+x_{2000} = 2000  x_1^4 +x_2^4 +...+x_{2000}^4= x_1^3 +x_2^3 +...+x_{2000}^3\end{cases} $$  
Answer:  x_1 = x_2 = \cdots = x_{2000} = 1
Problem: Ana has  $22$  coins. She can take from her friends either  $6$  coins or  $18$  coins, or she can give  $12$  coins to her friends. She can do these operations many times she wants. Find the least number of coins Ana can have.
Answer: 4
Problem: There is a point source of light in an empty universe. What is the minimum number of solid balls (of any size) one must place in space so that any light ray emanating from the light source intersects at least one ball?
Answer: 4
Problem: Find all bounded sequences  $(a_n)_{n=1}^\infty$  of natural numbers such that for all  $n \ge 3$ , \[ a_n = \frac{a_{n-1} + a_{n-2}}{\gcd(a_{n-1}, a_{n-2})}. \]
Answer: a_i = 2 \text{ for all } i
Problem:  Compute the minimum value of  $cos(a-b) + cos(b-c) + cos(c-a)$  as  $a,b,c$   ranges over the real numbers.
Answer: -\frac{3}{2}
Problem: Find the smallest possible side of a square in which five circles of radius  $1$  can be placed, so that no two of them have a common interior point.
Answer: 2 + 2\sqrt{2}
Problem: Find the real number  $t$ , such that the following system of equations has a unique real solution  $(x, y, z, v)$ :

\[ \left\{\begin{array}{cc}x+y+z+v=0 (xy + yz +zv)+t(xz+xv+yv)=0\end{array}\right. \]
Answer:  t \in \left( \frac{3 - \sqrt{5}}{2}, \frac{3 + \sqrt{5}}{2} \right)
Problem: Let  $\mathbb{R}^*$  be the set of all real numbers, except  $1$ . Find all functions  $f:\mathbb{R}^* \rightarrow \mathbb{R}$  that satisfy the functional equation   $$ x+f(x)+2f\left(\frac{x+2009}{x-1}\right)=2010 $$  .
Answer: f(x) = \frac{1}{3}\left(x + 2010 - 2\frac{x+2009}{x-1}\right)
Problem: Construct the  $ \triangle ABC$ , given  $ h_a$ ,  $ h_b$  (the altitudes from  $ A$  and  $ B$ ) and  $ m_a$ , the median from the vertex  $ A$ .
Answer:  \triangle ABC
Problem: Let  $f(x)=cx(x-1)$ , where  $c$  is a positive real number.  We use  $f^n(x)$  to denote the polynomial obtained by composing  $f$  with itself  $n$  times.  For every positive integer  $n$ , all the roots of  $f^n(x)$  are real.  What is the smallest possible value of  $c$ ?
Answer: 4
Problem: Find the smallest number  $n$  such that there exist polynomials  $f_1, f_2, \ldots , f_n$  with rational coefficients satisfying \[x^2+7 = f_1\left(x\right)^2 + f_2\left(x\right)^2 + \ldots + f_n\left(x\right)^2.\]
Answer: 5
Problem: Let  $n$  be a positive integer. Determine, in terms of  $n$ , the greatest integer which divides
every number of the form  $p + 1$ , where  $p \equiv 2$  mod  $3$  is a prime number which does not divide  $n$ .

Answer: 6
Problem: Seller reduced price of one shirt for  $20\%$ ,and they raised it by  $10\%$ . Does he needs to reduce or raise the price and how many, so that price of shirt will be reduced by  $10\%$  from the original price
Answer: 2 \text{ dollars}
Problem: What is the smallest positive integer  $n$  such that  $n=x^3+y^3$  for two different positive integer tuples  $(x,y)$ ?
Answer: 1729
Problem: Suppose that two curves  $ C_1 : y\equal{}x^3\minus{}x, C_2 : y\equal{}(x\minus{}a)^3\minus{}(x\minus{}a)$  have two intersection points.

Find the maximum area of the figure bounded by  $ C_1,\ C_2$ .
Answer: \frac{3}{4}
Problem: For how many numbers  $n$  does  $2017$  divided by  $n$  have a remainder of either  $1$  or  $2$ ?
Answer: 43
Problem: How many ordered triples  $(a, b, c)$  of odd positive integers satisfy  $a + b + c = 25?$ 
Answer: 78
Problem: How many pairs of integers  $a$  and  $b$  are there such that  $a$  and  $b$  are between  $1$  and  $42$  and  $a^9 = b^7 \mod 43$ ?
Answer: 42
Problem: Six distinguishable players are participating in a tennis tournament. Each player plays one match of tennis against every other player. The outcome of each tennis match is a win for one player and a loss for the other players; there are no ties. Suppose that whenever  $A$  and  $B$  are players in the tournament for which  $A$  won (strictly) more matches than  $B$  over the course of the tournament, it is also the case that  $A$  won the match against  $B$  during the tournament. In how many ways could the tournament have gone?
Answer: 720
Problem: What is the least positive integer by which  $2^5 \cdot  3^6 \cdot  4^3 \cdot  5^3 \cdot 6^7$  should be multiplied so that, the product is a perfect square?
Answer: 15
Problem: Let  $M$  be the intersection point of the medians  $AD$  and  $BE$  of a right triangle  $ABC$  ( $\angle C=90^\circ$ ). It is known that the circumcircles of triangles  $AEM$  and  $CDM$  are tangent. Find the angle  $\angle BMC.$ 
Answer: 90^
Problem: Define  $ \{ a_n \}_{n\equal{}1}$  as follows:  $ a_1 \equal{} 1989^{1989}; \ a_n, n > 1,$  is the sum of the digits of  $ a_{n\minus{}1}$ . What is the value of  $ a_5$ ?
Answer: 9
Problem: Let be given  $a < b < c$  and  $f(x) =\frac{c(x - a)(x - b)}{(c - a)(c - b)}+\frac{a(x - b)(x - c)}{(a - b)(a -c)}+\frac{b(x -c)(x - a)}{(b - c)(b - a)}$ . 
Determine  $f(2014)$ .
Answer: 2014
Problem: rainbow is the name of a bird. this bird has  $n$  colors and it's colors in two consecutive days are not equal. there doesn't exist  $4$  days in this bird's life like  $i,j,k,l$  such that  $i<j<k<l$  and the bird has the same color in days  $i$  and  $k$  and the same color in days  $j$  and  $l$  different from the colors it has in days  $i$  and  $k$ . what is the maximum number of days rainbow can live in terms of  $n$ ?
Answer: 2n - 1
Problem: A cube with sides of length 3cm is painted red
and then cut into 3 x 3 x 3 = 27 cubes with sides of length 1cm.
If a denotes the number of small cubes (of 1cm x 1cm x 1cm) that
are not painted at all, b the number painted on one sides, c the
number painted on two sides, and d the number painted on three
sides, determine the value a-b-c+d.
Answer: -9
Problem: Let  $ABC$  be a triangle with  $\angle BAC=40^\circ $ ,  $O$  be the center of its circumscribed circle and  $G$  is its centroid. Point  $D$  of line  $BC$  is such that  $CD=AC$  and  $C$  is between  $B$  and  $D$ . If  $AD\parallel OG$ , find  $\angle ACB$ .
Answer: 70^
Problem: Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units.  Given that the volume of this set is  $(m + n \pi)/p$ , where  $m$ ,  $n$ , and  $p$  are positive integers, and  $n$  and  $p$  are relatively prime, find  $m + n + p$ .
Answer: 505
Problem: A particle starts at  $(0,0,0)$  in three-dimensional space. Each second, it randomly selects one of the eight lattice points a distance of  $\sqrt{3}$  from its current location and moves to that point. What is the probability that, after two seconds, the particle is a distance of  $2\sqrt{2}$  from its original location?
Answer: \frac{3}{8}
Problem: Define a list of number with the following properties:
- The first number of the list is a one-digit natural number.
- Each number (since the second) is obtained by adding  $9$  to the number before in the list.
- The number  $2012$  is in that list.
Find the first number of the list.
Answer: 5
Problem: In a class, the total numbers of boys and girls are in the ratio  $4 : 3$ . On one day it was found that  $8$  boys and  $14$  girls were absent from the class, and that the number of boys was the square of the number of girls. What is the total number of students in the class?
Answer: 42
Problem: Let  $f$  be a non-constant polynomial such that \[ f(x-1) + f(x) + f(x+1) = \frac {f(x)^2}{2013x} \] for all nonzero real numbers  $x$ . Find the sum of all possible values of  $f(1)$ .
Answer: 6039
Problem: A coin is flipped  $20$  times. Let  $p$  be the probability that each of the following sequences of flips occur exactly twice:

        
- one head, two tails, one head
- one head, one tails, two heads.

    Given that  $p$  can be expressed as  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers, compute  $\gcd (m,n)$ .
Answer: 1
Problem: Determine all the possible non-negative integer values that are able to satisfy the expression: $\frac{(m^2+mn+n^2)}{(mn-1)}$  
if  $m$  and  $n$  are non-negative integers such that  $mn \neq 1$ .
Answer: 0, 4, 7
Problem: Let  $f (x)$  be a function mapping real numbers to real numbers. Given that  $f (f (x)) =\frac{1}{3x}$ , and  $f (2) =\frac19$ , find  $ f\left(\frac{1}{6}\right)$ .
Answer: 3
Problem: Determine all the positive integers with more than one digit, all distinct, such that the sum of its digits is equal to the product of its digits.

Answer:  \{123, 132, 213, 231, 312, 321\}
Problem: Circle  $\omega_1$  and  $\omega_2$  have centers  $(0,6)$  and  $(20,0)$ , respectively.  Both circles have radius  $30$ , and intersect at two points  $X$  and  $Y$ .  The line through  $X$  and  $Y$  can be written in the form  $y = mx+b$ .  Compute  $100m+b$ .
Answer: 303
Problem: Compare the magnitude of the following three numbers.
      $$         \sqrt[3]{\frac{25}{3}} ,\, \sqrt[3]{\frac{1148}{135}} ,\, \frac{\sqrt[3]{25}}{3} + \sqrt[3]{\frac{6}{5}}
     $$  
Answer:  a < c < b
Problem: Tyler rolls two  $ 4025 $  sided fair dice with sides numbered  $ 1, \dots , 4025 $ . Given that the number on the ﬁrst die is greater than or equal to the number on the second die, what is the probability that the number on the ﬁrst die is less than or equal to  $ 2012 $ ?
Answer: \frac{1006}{4025}
Problem: Let  $a$ ,  $b$ , and  $c$  be real numbers such that  $0\le a,b,c\le 5$  and  $2a + b + c = 10$ . Over all possible values of  $a$ ,  $b$ , and  $c$ , determine the maximum possible value of  $a + 2b + 3c$ .
Answer: 25
Problem: Find all numbers  $ n $  for which there exist three (not necessarily distinct) roots of unity of order  $ n $  whose sum is  $ 
1. $ 
Answer: Any even positive integer
Problem: Let  $n \in \mathbb{N}_{\geq 2}.$  For any real numbers  $a_1,a_2,...,a_n$  denote  $S_0=1$  and for  $1 \leq k \leq n$  denote  $$ S_k=\sum_{1 \leq i_1 < i_2 < ... <i_k \leq n}a_{i_1}a_{i_2}...a_{i_k} $$  Find the number of  $n-$ tuples  $(a_1,a_2,...a_n)$  such that   $$ (S_n-S_{n-2}+S_{n-4}-...)^2+(S_{n-1}-S_{n-3}+S_{n-5}-...)^2=2^nS_n. $$  
Answer: 2^{n-1}
Problem: Let  $0=x_0<x_1<\cdots<x_n=1$  .Find the largest real number $ C$  such  that for any positive integer  $ n $  ,  we have   $$ \sum_{k=1}^n  x^2_k (x_k - x_{k-1})>C $$  
Answer: \frac{1}{3}
Problem: Let  $T$  denote the  $15$ -element set  $\{10a+b:a,b\in\mathbb{Z},1\le a<b\le 6\}$ . Let  $S$  be a subset of  $T$  in which all six digits  $1,2,\ldots ,6$  appear and in which no three elements together use all these six digits. Determine the largest possible size of  $S$ .
Answer: 9
Problem: The USAMO is a  $6$  question test. For each question, you submit a positive integer number  $p$  of pages on which your solution is written. On the  $i$ th page of this question, you write the fraction  $i/p$  to denote that this is the  $i$ th page out of  $p$  for this question. When you turned in your submissions for the  $2017$  USAMO, the bored proctor computed the sum of the fractions for all of the pages which you turned in. Surprisingly, this number turned out to be  $2017$ . How many pages did you turn in?
Answer: 4028
Problem: For real numbers  $a$  and  $b$ , define  $$ f(a,b) = \sqrt{a^2+b^2+26a+86b+2018}. $$  Find the smallest possible value of the expression   $$ f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b). $$  
Answer: 4 \sqrt{2018}
Problem: Find the smallest odd integer  $ k$  such that: for every  $ 3\minus{}$ degree polynomials  $ f$  with integer coefficients, if there exist  $ k$  integer  $ n$  such that  $ |f(n)|$  is a prime number, then  $ f$  is irreducible in  $ \mathbb{Z}[n]$ .
Answer: 5
Problem: Compute the number of ways to rearrange nine white cubes and eighteen black cubes into a  $3\times 3\times 3$  cube such that each  $1\times1\times3$  row or column contains exactly one white cube. Note that rotations are considered  $\textit{distinct}$ .
Answer: 60480
Problem: Let  $AD,BF$  and  ${CE}$  be the altitudes of  $\vartriangle ABC$ . A line passing through  ${D}$  and parallel to  ${AB}$ intersects the line  ${EF}$ at the point  ${G}$ . If  ${H}$  is the orthocenter of  $\vartriangle ABC$ , find the angle  ${\angle{CGH}}$ .
Answer: \angle CGH = 90^\circ
Problem: Let  $a$ ,  $b$ ,  $c$ ,  $d$ ,  $e$  be positive reals satisfying \begin{align*} a + b &= c  a + b + c &= d  a + b + c + d &= e.\end{align*} If  $c=5$ , compute  $a+b+c+d+e$ .
Answer: 40
Problem: Screws are sold in packs of  $10$  and  $12$ . Harry and Sam independently go to the hardware store, and by coincidence each of them buys exactly  $k$  screws. However, the number of packs of screws Harry buys is different than the number of packs Sam buys. What is the smallest possible value of  $k$ ?
Answer: 60
Problem:  $ABCD$  is circumscribed in a circle  $k$ , such that  $[ACB]=s$ ,  $[ACD]=t$ ,  $s<t$ . Determine the smallest value of  $\frac{4s^2+t^2}{5st}$  and when this minimum is achieved.
Answer: \frac{4}{5}
Problem: For real numbers  $B,M,$  and  $T,$  we have  $B^2+M^2+T^2 =2022$  and  $B+M+T =72.$  Compute the sum of the minimum and maximum possible values of  $T.$ 
Answer: 48
Problem: Bryan Ai has the following 8 numbers written from left to right on a sheet of paper:  $$ \textbf{1 4 1 2 0 7 0 8} $$  Now in each of the 7 gaps between adjacent numbers, Bryan Ai wants to place one of ` $+$ ', ` $-$ ', or ` $\times$ ' inside that gap.
Now, Bryan Ai wonders, if he picks a random placement out of the  $3^7$  possible placements, what's the expected value of the expression (order of operations apply)?
Answer: 0
Problem: Let  $a$  and  $b$  be positive integers. When  $a^{2}+b^{2}$  is divided by  $a+b,$  the quotient is  $q$  and the remainder is  $r.$  Find all pairs  $(a,b)$  such that  $q^{2}+r=1977$ .
Answer: (a, b) = (50, 37) or (a, b) = (37, 50)
Problem: It is known that, for all positive integers  $k,$ 
\[1^{2}+2^{2}+3^{2}+\cdots+k^{2}=\frac{k(k+1)(2k+1)}{6}. \]Find the smallest positive integer  $k$  such that  $1^{2}+2^{2}+3^{2}+\cdots+k^{2}$  is a multiple of  $200.$ 
Answer: 112