rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: Let  $p$  be a prime number,  $p \ge 5$ , and  $k$  be a digit in the  $p$ -adic representation of positive integers. Find the maximal length of a non constant arithmetic progression whose terms do not contain the digit  $k$  in their  $p$ -adic representation.
Answer: p - 1
Problem: The work team was working at a rate fast enough to process  $1250$  items in ten hours. But after working for six hours, the team was given an additional  $150$  items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours?
Answer: 30\%
Problem: Determine the number of  $2021$ -tuples of positive integers such that the number  $3$  is an element of the tuple and consecutive elements of the tuple differ by at most  $1$ .
Answer: 3^{2021} - 2^{2021}
Problem: Let  $a, b, c, d$  be distinct non-zero real numbers satisfying the following two conditions: $ac = bd$  and   $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}= 4$ .
Determine the largest possible value of the expression  $\frac{a}{c}+\frac{c}{a}+\frac{b}{d}+\frac{d}{b}$ .

Answer: 4
Problem: You are tossing an unbiased coin. The last  $ 28 $  consecutive ﬂips have all resulted in heads. Let  $ x $  be the expected number of additional tosses you must make before you get  $ 60 $  consecutive heads. Find the sum of all distinct prime factors in  $ x $ .
Answer: 5
Problem: A circle is randomly chosen in a circle of radius  $1$  in the sense that a point is randomly chosen for its center, then a radius is chosen at random so that the new circle is contained in the original circle. What is the probability that the new circle contains the center of the original circle?
Answer: \frac{1}{4}
Problem: Let  $\{x_1, x_2, x_3, ..., x_n\}$  be a set of  $n$  distinct positive integers, such that the sum of any  $3$  of them is a prime number. What is the maximum value of  $n$ ?
Answer: 4
Problem: Let  $p > 2$  be a fixed prime number. Find all functions  $f: \mathbb Z \to \mathbb Z_p$ , where the  $\mathbb Z_p$  denotes the set  $\{0, 1, \ldots , p-1\}$ , such that  $p$  divides  $f(f(n))- f(n+1) + 1$  and  $f(n+p) = f(n)$  for all integers  $n$ .
Answer: g(x) = x
Problem: A pen costs  $11$  € and a notebook costs  $13$  €. Find the number of ways in which a person can spend exactly  $1000$  € to buy pens and notebooks.
Answer: 7
Problem:  $N$  is an integer whose representation in base  $b$  is  $777$ .  Find the smallest integer  $b$  for which  $N$  is the fourth power of an integer.
Answer: 18
Problem: Let  $M= \{ 1, 2, \cdots, 19 \}$  and  $A = \{ a_{1}, a_{2}, \cdots, a_{k}\}\subseteq M$ . Find the least  $k$  so that for any  $b \in M$ , there exist  $a_{i}, a_{j}\in A$ , satisfying  $b=a_{i}$  or  $b=a_{i}\pm a_{i}$  ( $a_{i}$  and  $a_{j}$  do not have to be different) .
Answer: 5
Problem: Find all natural numbers  $n$  for which  $n + 195$  and  $n - 274$  are  perfect cubes.
Answer: 2002
Problem: If  $x > 10$ , what is the greatest possible value of the expression
\[
  {( \log x )}^{\log \log \log x} - {(\log \log x)}^{\log \log x} ? 
\]
All the logarithms are base 10.
Answer: 0
Problem: Suppose you are given that for some positive integer  $n$ ,  $1! + 2! + \ldots + n!$  is a perfect square. Find the sum of all possible values of  $n$ .
Answer: 4
Problem: The tetrahedron  $ S.ABC$  has the faces  $ SBC$  and  $ ABC$  perpendicular. The three angles at  $ S$  are all  $ 60^{\circ}$  and  $ SB \equal{} SC \equal{} 1$ . Find the volume of the tetrahedron.
Answer: \frac{1}{8}
Problem: Let  $a$  and  $b$  be two fixed positive real numbers. Find all real numbers  $x$ , such that inequality holds   $$ \frac{1}{\sqrt{x}}+\frac{1}{\sqrt{a+b-x}} < \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}} $$  
Answer:  x \in (\min(a, b), \max(a, b))
Problem: The set  $M=\{1,2,\ldots,2007\}$  has the following property: If  $n$  is an element of  $M$ , then all terms in the arithmetic progression with its first term  $n$  and common difference  $n+1$ , are in  $M$ . Does there exist an integer  $m$  such that all integers greater than  $m$  are elements of  $M$ ?
Answer:  m = 2007
Problem: 	1. Baron Munchhausen was told that some polynomial  $P(x)=a_{n} x^{n}+\ldots+a_{1} x+a_{0}$  is such that  $P(x)+P(-x)$  has exactly 45 distinct real roots. Baron doesn't know the value of  $n$ . Nevertheless he claims that he can determine one of the coefficients  $a_{n}, \ldots, a_{1}, a_{0}$  (indicating its position and value). Isn't Baron mistaken?
	
	Boris Frenkin
Answer:  a_0 = 0
Problem: It is known that the equation $ |x - 1| + |x - 2| +... + |x - 2001| = a$  has exactly one solution. Find  $a$ .
Answer: 1001000
Problem: Find all prime numbers  $p$  and  $q$  such that  $2^2+p^2+q^2$  is also prime.

Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )
Answer: (3, 2) \text{ and } (2, 3)
Problem: Find all functions  $f:\mathbb{R}\rightarrow \mathbb{R}$  such that $\forall x\in \mathbb{R} \ \ f(x) = max(2xy-f(y))$  where  $y\in \mathbb{R}$ .
Answer: f(x) = x^2
Problem: Let  $n$  be a positive integer. E. Chen and E. Chen play a game on the  $n^2$  points of an  $n \times n$  lattice grid. They alternately mark points on the grid such that no player marks a point that is on or inside a non-degenerate triangle formed by three marked points. Each point can be marked only once. The game ends when no player can make a move, and the last player to make a move wins. Determine the number of values of  $n$  between  $1$  and  $2013$  (inclusive) for which the first player can guarantee a win, regardless of the moves that the second player makes.
Answer: 1007
Problem: A natural number  $n$  is called *perfect* if it is equal to the sum of all its natural divisors other than  $n$ . For example, the number  $6$  is perfect because  $6 = 1 + 2 + 3$ . Find all even perfect numbers that can be given as the sum of two cubes positive integers.

Answer: 28
Problem: Find the largest constant  $K>0$  such that for any  $0\le k\le K$  and non-negative reals  $a,b,c$  satisfying  $a^2+b^2+c^2+kabc=k+3$  we have  $a+b+c\le 3$ .
Answer:  K = 1
Problem: Find all real numbers  $x, y, z$  that satisfy the following system  $$ \sqrt{x^3 - y} = z - 1 $$    $$ \sqrt{y^3 - z} = x - 1 $$    $$ \sqrt{z^3 - x} = y - 1 $$  
Answer:  (1, 1, 1)
Problem: Find all positive integers  $n$  for which there exist non-negative integers  $a_1, a_2, \ldots, a_n$  such that
\[
\frac{1}{2^{a_1}} + \frac{1}{2^{a_2}} + \cdots + \frac{1}{2^{a_n}} = 
\frac{1}{3^{a_1}} + \frac{2}{3^{a_2}} + \cdots + \frac{n}{3^{a_n}} = 1.
\]
Answer: n \equiv 1, 2 \pmod{4}
Problem: Find all functions  $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$  such that if  $a,b,c$  are the length sides of a triangle, and  $r$  is the radius of its incircle, then  $f(a),f(b),f(c)$  also form a triangle where its radius of the incircle is  $f(r)$ .
Answer:  f(x) = kx
Problem: For every positive integeer  $n>1$ , let  $k(n)$  the largest positive integer  $k$  such that there exists a positive integer  $m$  such that  $n = m^k$ .

Find   $$ lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n} $$  
Answer: 1
Problem: Triangle  $ABC$  is isosceles with  $AB=AC$ . The bisectors of angles  $ABC$  and  $ACB$  meet at  $I$ . If the measure of angle  $CIA$  is  $130^\circ$ , compute the measure of angle  $CAB$ .
Answer: 80^	extcirc
Problem: For each positive integer  $k$ , let  $d(k)$  be the number of positive divisors of  $k$  and  $\sigma(k)$  be the sum of positive divisors of  $k$ . Let  $\mathbb N$  be the set of all positive integers. Find all functions  $f: \mathbb{N} \to \mathbb N$  such that \begin{align*}
			f(d(n+1)) &= d(f(n)+1)\quad \text{and} 
			f(\sigma(n+1)) &= \sigma(f(n)+1)
		\end{align*}
	for all positive integers  $n$ .
Answer:  f(n) = n
Problem: Let  $\alpha ,\ \beta$  be the distinct positive roots of the equation of  $2x=\tan x$ .

Evaluate the following definite integral.
\[\int_{0}^{1}\sin \alpha x\sin \beta x\ dx \]
Answer: 0
Problem: In an acute triangle  $ABC$ , the segment  $CD$  is an altitude and  $H$  is the orthocentre. Given that the circumcentre of the triangle lies on the line containing the bisector of the angle  $DHB$ , determine all possible values of  $\angle CAB$ .
Answer: 60^{\circ}
Problem: Let  $p=2^{16}+1$  be a prime. A sequence of  $2^{16}$  positive integers  $\{a_n\}$  is *monotonically bounded* if  $1\leq a_i\leq i$  for all  $1\leq i\leq 2^{16}$ . We say that a term  $a_k$  in the sequence with  $2\leq k\leq 2^{16}-1$  is a *mountain* if  $a_k$  is greater than both  $a_{k-1}$  and  $a_{k+1}$ . Evan writes out all possible monotonically bounded sequences. Let  $N$  be the total number of mountain terms over all such sequences he writes. Find the remainder when  $N$  is divided by  $p$ .
Answer: 49153
Problem: Determine all triples (x, y, z) of non-negative real numbers that satisfy the following system of equations $\begin{cases} x^2 - y = (z - 1)^2
y^2 - z = (x - 1)^2 
z^2 - x = (y -1)^2 \end{cases}$ .
Answer: (x, y, z) = (1, 1, 1)
Problem: Let  $x_i\  (i = 1, 2, \cdots 22)$  be reals such that  $x_i \in [2^{i-1},2^i]$ . Find the maximum possible value of  $$ (x_1+x_2+\cdots +x_{22})(\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_{22}}) $$  
Answer: 2^{24}
Problem: Determine all real  $ x$  satisfying the equation \[ \sqrt[5]{x^3 \plus{} 2x} \equal{} \sqrt[3]{x^5\minus{}2x}.\] Odd roots for negative radicands shall be included in the discussion.
Answer: 0
Problem: We define the ridiculous numbers recursively as follows:
[list=a]
[*]1 is a ridiculous number.
[*]If  $a$  is a ridiculous number, then  $\sqrt{a}$  and  $1+\sqrt{a}$  are also ridiculous numbers. 
[/list]
A closed interval  $I$  is ``boring'' if


- $I$  contains no ridiculous numbers, and
- There exists an interval  $[b,c]$  containing  $I$  for which  $b$  and  $c$  are both ridiculous numbers.

The smallest non-negative  $l$  such that there does not exist a boring interval with length  $l$  can be represented in the form  $\dfrac{a + b\sqrt{c}}{d}$  where  $a, b, c, d$  are integers,  $\gcd(a, b, d) = 1$ , and no integer square greater than 1 divides  $c$ . What is  $a + b + c + d$ ?
Answer: 9
Problem: Let  $M$  be on segment $ BC$  of  $\vartriangle ABC$  so that  $AM = 3$ ,  $BM = 4$ , and  $CM = 5$ . Find the largest possible area of  $\vartriangle ABC$ .
Answer: \frac{27}{2}
Problem: Let  $ABCD$  be a square, such that the length of its sides are integers. This square is divided in  $89$  smaller squares,  $88$  squares that have sides with length  $1$ , and  $1$  square that has sides with length  $n$ , where  $n$  is an integer larger than  $1$ . Find all possible lengths for the sides of  $ABCD$ .
Answer: 13 \text{ and } 23
Problem: The numbers  $12,14,37,65$  are one of the solutions of the equation  $xy-xz+yt=182$ . What number corresponds to which letter?
Answer:  x = 12, y = 37, z = 65, t = 14
Problem: Find all prime numbers  $p$ , for which there exist  $x, y \in \mathbb{Q}^+$  and  $n \in \mathbb{N}$ , satisfying $x+y+\frac{p}{x}+\frac{p}{y}=3n$ .
Answer:  p
Problem: A rectangle  $ABEF$  is drawn on the leg  $AB$  of a right triangle  $ABC$ , whose apex  $F$  is on the leg  $AC$ . Let  $X$  be the intersection of the diagonal of the rectangle  $AE$  and the hypotenuse  $BC$  of the triangle. In what ratio does point  $X$  divide the hypotenuse  $BC$  if it is known that  $| AC | = 3 | AB |$  and  $| AF | = 2 | AB |$ ?
Answer: 2:3
Problem: The lengths of the three sides of a right triangle form a geometric sequence. The sine of the smallest of the angles in the triangle is  $\tfrac{m+\sqrt{n}}{k}$  where  $m$ ,  $n$ , and  $k$  are integers, and  $k$  is not divisible by the square of any prime. Find  $m + n + k$ .
Answer: 6
Problem: Find all prime numbers  $p$  such that the number  $$ 3^p+4^p+5^p+9^p-98 $$  has at most  $6$  positive divisors.
Answer: 2, 3
Problem: Determine all numbers  $x, y$  and  $z$  satisfying the system of equations  $$ \begin{cases} x^2 + yz = 1  y^2 - xz = 0  z^2 + xy = 1\end{cases} $$  
Answer:  x = y = z = \pm \frac{\sqrt{2}}{2}
Problem: Triangle  $ABC$  has  $AB = 4$ ,  $AC = 5$ , and  $BC = 6$ .  An angle bisector is drawn from angle  $A$ , and meets  $BC$  at  $M$ .  What is the nearest integer to  $100 \frac{AM}{CM}$ ?
Answer: 80
Problem: Find all integers  $n$  greater than or equal to  $4$  that satisfy the following conditions:


- Take an arbitrary convex  $n$ -gon  $P$  on the coordinate plane whose vertices are lattice points (points whose coordinates are both integers). There are a total of  $n$  triangles that share two sides with  $P$ . Let  $S_1, S_2, \ldots, S_n$  be their areas, and let  $S$  be the area of  $P$ . Then, the greatest common divisor of  $2S_1, 2S_2, \ldots, 2S_n$  divides the integer  $2S$ .

Answer:  n = 4 }  or  n = 5
Problem: There are  $n \ge 2$  houses on the northern side of a street. Going from the west to the east, the houses are numbered from 1 to  $n$ . The number of each house is shown on a plate. One day the inhabitants of the street make fun of the postman by shuffling their number plates in the following way: for each pair of neighbouring houses, the currnet number plates are swapped exactly once during the day.
How many different sequences of number plates are possible at the end of the day?
Answer: 2^{n-1}
Problem: Let  $p,q$  and  $s{}$  be prime numbers such that  $2^sq =p^y-1$  where  $y > 1.$  Find all possible values of  $p.$ 
Answer: 3 \text{ and } 5
Problem: Let  $ABCD$  be a convex quadrilateral with  $\angle ABD = \angle BCD$ ,  $AD = 1000$ ,  $BD = 2000$ ,  $BC = 2001$ , and  $DC = 1999$ . Point  $E$  is chosen on segment  $DB$  such that  $\angle ABD = \angle ECD$ . Find  $AE$ .
Answer: 1000
Problem: Find all  $ f:N\rightarrow N$ , such that  $\forall m,n\in N $   $ 2f(mn) \geq f(m^2+n^2)-f(m)^2-f(n)^2 \geq 2f(m)f(n) $ 
Answer: f(n) = n^2 \quad \forall n \in \mathbb{N}
Problem: Find all polynomials  $f(x)$  with integer coefficients such that  $f(n)$  and  $f(2^{n})$  are co-prime for all natural numbers  $n$ .
Answer:  f(x) \equiv 1  \text{ or } f(x) \equiv -1
Problem: Three equal circles of radius  $r$  each pass through the centres of the other two.
What is the area of intersection that is common to all the three circles?
Answer: r^2 \left( \frac{\pi - \sqrt{3}}{2} \right)
Problem: Let  $A_1B_1C_1D_1A_2B_2C_2D_2$  be a unit cube, with  $A_1B_1C_1D_1$  and  $A_2B_2C_2D_2$  opposite square faces, and let  $M$  be the center of face  $A_2 B_2 C_2 D_2$ . Rectangular pyramid  $MA_1B_1C_1D_1$  is cut out of the cube. If the surface area of the remaining solid can be expressed in the form  $a + \sqrt{b}$ , where  $a$  and  $b$  are positive integers and  $b$  is not divisible by the square of any prime, find  $a+b$ .
Answer: 11
Problem: A line parallel to the side  $AC$  of a triangle  $ABC$  with  $\angle C = 90$  intersects side  $AB$  at  $M$  and side  $BC$  at  $N$ , so that  $CN/BN = AC/BC = 2/1$ . The segments  $CM$  and  $AN$  meet at  $O$ . Let  $K$  be a point on the segment  $ON$  such that  $MO+OK = KN$ . The bisector of  $\angle ABC$  meets the line through  $K$  perpendicular to  $AN$  at point  $T$ .
Determine  $\angle MTB$ .
 
Answer:  \angle MTB = 90^	au
Problem: Let  $n$  points with integer coordinates be given in the  $xy$ -plane. What is the minimum value of  $n$  which will ensure that three of the points are the vertices of a triangel with integer (possibly, 0) area?
Answer: 5
Problem: Determine all pairs of polynomials  $(P, Q)$  with real coefficients satisfying  $$ P(x + Q(y)) = Q(x + P(y)) $$  for all real numbers  $x$  and  $y$ .
Answer: P(x) = Q(x) \text{ or } P(x) = x + a, Q(x) = x + b \text{ for some constants } a \text{ and } b
Problem: In quadrilateral  $ABCD$ , diagonals  $AC$  and  $BD$  intersect at  $O$ . If the area of triangle  $DOC$  is  $4$  and the area of triangle  $AOB$  is  $36$ , compute the minimum possible value of the area of  $ABCD$ .
Answer: 80
Problem: Let  $a=1+10^{-4}$ . Consider some  $2023\times 2023$  matrix with each entry a real in  $[1,a]$ . Let  $x_i$  be the sum of the elements of the  $i$ -th row and  $y_i$  be the sum of the elements of the  $i$ -th column for each integer  $i\in [1,n]$ . Find the maximum possible value of  $\dfrac{y_1y_2\cdots y_{2023}}{x_1x_2\cdots x_{2023}}$  (the answer may be expressed in terms of  $a$ ).
Answer: 1
Problem: For a positive integer  $n$  determine all  $n\times n$  real matrices  $A$  which have only real eigenvalues and such that there exists an integer  $k\geq n$  with  $A + A^k = A^T$ .
Answer: A = 0
Problem: Solve the equation, where  $x$  and  $y$  are positive integers:   $$  x^3-y^3=999 $$  
Answer:  (x, y) = (12, 9) and (x, y) = (10, 1)
Problem: Determine the number of all positive integers which cannot be written in the form  $80k + 3m$ , where  $k,m \in N = \{0,1,2,...,\}$ 
Answer: 79
Problem: Find all the real roots of the system of equations:  $$  \begin{cases} x^3+y^3=19  x^2+y^2+5x+5y+xy=12 \end{cases}   $$  
Answer: (3, -2) 	ext{ or } (-2, 3)
Problem: Given a positive integer  $n,$  what is the largest  $k$  such that the numbers  $1,2,\dots,n$  can be put into  $k$  boxes so that the sum of the numbers in each box is the same?

[When  $n=8,$  the example  $\{1,2,3,6\},\{4,8\},\{5,7\}$  shows that the largest  $k$  is *at least* 3.]
Answer: \max(k) =
\begin{cases}
\frac{n+1}{2}, & \text{if } n \equiv 1 \pmod{2} \\
\frac{n}{2}, & \text{if } n \equiv 0 \pmod{2}
\end{cases}
Problem: Let  $n>1$  be a natural number. Find the real values of the parameter  $a$ , for which the equation  $\sqrt[n]{1+x}+\sqrt[n]{1-x}=a$  has a single real root.
Answer: a = 2
Problem: There are 2 pizzerias in a town, with 2010 pizzas each. Two scientists  $A$  and  $B$  are taking turns ( $A$  is first), where on each turn one can eat as many pizzas as he likes from one of the pizzerias or exactly one pizza from each of the two. The one that has eaten the last pizza is the winner. Which one of them is the winner, provided that they both use the best possible strategy?
Answer: B
Problem: For what integers  $ n\ge 3$  is it possible to accommodate, in some order, the numbers  $ 1,2,\cdots, n$  in a circular form such that every number divides the sum of the next two numbers, in a clockwise direction?
Answer: 3
Problem: Ankit, Box, and Clark are taking the tiebreakers for the geometry round, consisting of three problems. Problem  $k$  takes each  $k$  minutes to solve. If for any given problem there is a  $\frac13$  chance for each contestant to solve that problem first, what is the probability that Ankit solves a problem first?
Answer: \frac{19}{27}
Problem: Let  $n$  be a natural number greater than 2.  $l$  is a line on a plane. There are  $n$  distinct points  $P_1$ ,  $P_2$ , …,  $P_n$  on  $l$ . Let the product of distances between  $P_i$  and the other  $n-1$  points be  $d_i$  ( $i = 1, 2,$  …,  $n$ ). There exists a point  $Q$ , which does not lie on  $l$ , on the plane. Let the distance from  $Q$  to  $P_i$  be  $C_i$  ( $i = 1, 2,$  …,  $n$ ). Find  $S_n = \sum_{i = 1}^{n} (-1)^{n-i} \frac{c_i^2}{d_i}$ .
Answer:  S_n = 0
Problem: A  $3n$ -table is a table with three rows and  $n$  columns containing all the numbers  $1, 2, …, 3n$ . Such a table is called *tidy* if the  $n$  numbers in the first row appear in ascending order from left to right, and the three numbers in each column appear in ascending order from top to bottom. How many tidy  $3n$ -tables exist?
Answer: \frac{(3n)!}{6^n \cdot n!}
Problem: Let  $ n$  be a given positive integer.  Find the smallest positive integer  $ u_n$  such that for any positive integer  $ d$ , in any  $ u_n$  consecutive odd positive integers, the number of them that can be divided by  $ d$  is not smaller than the number of odd integers among  $ 1, 3, 5, \ldots, 2n \minus{} 1$  that can be divided by  $ d$ .
Answer: u_n = 2n-1
Problem: Find the number of positive integers  $n$  such that the highest power of  $7$  dividing  $n!$  is  $8$ .
Answer: 7
Problem: Michelle is drawing segments in the plane. She begins from the origin facing up the  $y$ -axis and draws a segment of length  $1$ . Now, she rotates her direction by  $120^\circ$ , with equal probability clockwise or counterclockwise, and draws another segment of length  $1$  beginning from the end of the previous segment. She then continues this until she hits an already drawn segment. What is the expected number of segments she has drawn when this happens?
Answer: 5
Problem: Let  $a_1,a_2,a_3,\dots,a_6$  be an arithmetic sequence with common difference  $3$ . Suppose that  $a_1$ ,  $a_3$ , and  $a_6$  also form a geometric sequence. Compute  $a_1$ .
Answer: 12
Problem: A prime number  $p$  is a **moderate** number if for every  $2$  positive integers  $k > 1$  and  $m$ , there exists k positive integers  $n_1, n_2, ..., n_k $  such that \[ n_1^2+n_2^2+ ... +n_k^2=p^{k+m} \]
If  $q$  is the smallest **moderate** number, then determine the smallest prime  $r$  which is not moderate and  $q < r$ .
Answer:  7
Problem: Let  $A$  and  $B$  be two distinct points on the circle  $\Gamma$ , not diametrically opposite. The point  $P$ , distinct from  $A$  and  $B$ , varies on  $\Gamma$ . Find the locus of the orthocentre of triangle  $ABP$ .
Answer: \Gamma
Problem: Two convex polygons have a total of 33 sides and 243 diagonals. Find the number of diagonals in the polygon with the greater number of sides.
Answer: 189
Problem: In a school there are  $1200$  students. Each student is part of exactly  $k$  clubs. For any  $23$  students, they are part of a common club. Finally, there is no club to which all students belong. Find the smallest possible value of  $k$ .
Answer: 23
Problem: Find all f:N  $\longrightarrow$  N that:
[list]**a)**  $f(m)=1 \Longleftrightarrow m=1 $ **b)**  $d=gcd(m,n) f(m\cdot n)= \frac{f(m)\cdot f(n)}{f(d)} $ **c)**  $ f^{2000}(m)=f(m) $ [/list]
Answer: f(n) = n
Problem: Moor has  $2016$  white rabbit candies. He and his  $n$  friends split the candies equally amongst themselves, and they find that they each have an integer number of candies. Given that  $n$  is a positive integer (Moor has at least  $1$  friend), how many possible values of  $n$  exist?
Answer: 35
Problem: Let there be positive integers  $a, c$ . Positive integer  $b$  is a divisor of  $ac-1$ . For a positive rational number  $r$  which is less than  $1$ , define the set  $A(r)$  as follows.   $$ A(r) = \{m(r-ac)+nab| m, n \in \mathbb{Z} \} $$  Find all rational numbers  $r$  which makes the minimum positive rational number in  $A(r)$  greater than or equal to  $\frac{ab}{a+b}$ .
Answer: \frac{a}{a + b}
Problem: Let  $k$  and  $a$  are positive constants. Denote by  $V_1$  the volume of the solid generated by a rotation of the figure enclosed
by the curve  $C: y=\frac{x}{x+k}\ (x\geq 0)$ , the line  $x=a$  and the  $x$ -axis around the  $x$ -axis, and denote by  $V_2$  that of
the solid by a rotation of the figure enclosed by the curve  $C$ , the line  $y=\frac{a}{a+k}$  and the  $y$ -axis around the  $y$ -axis. 
Find the ratio  $\frac{V_2}{V_1}.$ 
Answer: k
Problem: For a positive integer  $n$  and any real number  $c$ , define  $x_k$  recursively by :
\[ x_0=0,x_1=1 \text{ and for }k\ge 0, \;x_{k+2}=\frac{cx_{k+1}-(n-k)x_k}{k+1} \]
Fix  $n$  and then take  $c$  to be the largest value for which  $x_{n+1}=0$ . Find  $x_k$  in terms of  $n$  and  $k,\; 1\le k\le n$ .
Answer:  x_k = \binom{n-1}{k-1}