rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: Find all injective functions  $ f:\mathbb{N} \to \mathbb{N} $  such that   $$  f^{f\left(a\right)}\left(b\right)f^{f\left(b\right)}\left(a\right)=\left(f\left(a+b\right)\right)^2  $$   holds for all  $ a,b \in \mathbb{N} $ . Note that  $ f^{k}\left(n\right) $  means  $ \underbrace{f(f(\ldots f}_{k}(n) \ldots )) $ 
Answer: f(x) = x+1
Problem: A line through a vertex of a non-degenerate triangle cuts it in two similar triangles with  $\sqrt{3}$  as the ratio between correspondent sides.
Find the angles of the given triangle.
Answer: 30^
\circ, 60^
\circ, 90^
\circ
Problem: In a table consisting of  $n$  by  $n$  small squares some squares are coloured black and the other squares are coloured white. For each pair of columns and each pair of rows the four squares on the intersections of these rows and columns must not all be of the same colour.
What is the largest possible value of  $n$ ?
Answer:  n = 4
Problem: Let  $u, v$  be real numbers. The minimum value of  $\sqrt{u^2+v^2} +\sqrt{(u-1)^2+v^2}+\sqrt {u^2+ (v-1)^2}+ \sqrt{(u-1)^2+(v-1)^2}$  can be written as  $\sqrt{n}$ . Find the value of  $10n$ .
Answer: 80
Problem: What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours?
Answer: 3
Problem: Find the smallest  $n$  such that every subset of  $\{1, 2, 3, . . . , 2004 \}$  with  $n$  elements contains at least two elements that are relatively prime.
Answer: 1003
Problem: Let  $f(x) = x^2 + x + 1$ . Determine the ordered pair  $(p,q)$  of primes satisfying  $f(p) = f(q) + 242$ .
Answer: (61, 59)
Problem: Find the maximum of  $x+y$  given that  $x$  and  $y$  are positive real numbers that satisfy \[x^3+y^3+(x+y)^3+36xy=3456.\]
Answer: 12
Problem:  $K, L, M$  and  $N$  are points on sides  $AB, BC, CD$  and  $DA$ , respectively, of the unit square  $ABCD$  such that  $KM$  is parallel to  $BC$  and  $LN$  is parallel to  $AB$ . The perimeter of triangle  $KLB$  is equal to  $1$ . What is the area of triangle  $MND$ ?
Answer: \frac{1}{4}
Problem: The degree measures of the angles of a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.
Answer: 143
Problem: Find all triples  $(x, y, z)$  of positive integers such that 
\[x^2 + 4^y = 5^z. \]
Answer:  (x, y, z) = (3, 2, 2) , (1, 1, 1) , and (11, 1, 3)
Problem: A fly is being chased by three spiders on the edges of a regular octahedron. The fly has a speed of  $50$  meters per second, while each of the spiders has a speed of  $r$  meters per second. The spiders choose their starting positions, and choose the fly's starting position, with the requirement that the fly must begin at a vertex. Each bug knows the position of each other bug at all times, and the goal of the spiders is for at least one of them to catch the fly. What is the maximum  $c$  so that for any  $r<c,$   the fly can always avoid being caught?
Answer: 25
Problem: Suppose the side lengths of triangle  $ABC$  are the roots of polynomial  $x^3 - 27x^2 + 222x - 540$ . What is the product of its inradius and circumradius?
Answer: 10
Problem: How many ways are there to choose three digits  $A,B,C$  with  $1 \le A \le 9$  and  $0 \le B,C \le 9$  such that  $\overline{ABC}_b$  is even for all choices of base  $b$  with  $b \ge 10$ ?
Answer: 450
Problem: Let  $N$  be the number of ways of choosing a subset of  $5$  distinct numbers from the set  $$ {10a+b:1\leq a\leq 5, 1\leq b\leq 5} $$  where  $a,b$  are integers, such that no two of the selected numbers have the same units digits and no two have the same tens digit. What is the remainder when  $N$  is divided by  $73$ ?
Answer: 47
Problem: Consider a chessboard that is infinite in all directions. Alex the T-rex wishes to place a positive integer in each square in such a way that:


- No two numbers are equal.
- If a number  $m$  is placed on square  $C$ , then at least  $k$  of the squares orthogonally adjacent to  $C$  have a multiple of  $m$  written on them.

What is the greatest value of  $k$  for which this is possible?
Answer:  k = 2
Problem: Alice the ant starts at vertex  $A$  of regular hexagon  $ABCDEF$  and moves either right or left each move with equal probability. After  $35$  moves, what is the probability that she is on either vertex  $A$  or  $C$ ?
Answer: 0
Problem: Find the minimum value of  $c$  such that for any positive integer  $n\ge 4$  and any set  $A\subseteq \{1,2,\cdots,n\}$ , if  $|A| >cn$ , there exists a function  $f:A\to\{1,-1\}$  satisfying  $$ \left| \sum_{a\in A}a\cdot f(a)\right| \le 1. $$  
Answer:  \frac{2}{3}
Problem: A team of four students goes to LMT, and each student brings a lunch. However, on the bus, the students’ lunches get mixed up, and during lunch time, each student chooses a random lunch to eat (no two students may eat the same lunch). What is the probability that each student chooses his or her own lunch correctly?
Answer: \frac{1}{24}
Problem: Guy has 17 cards. Each of them has an integer written on it (the numbers are not necessarily positive, and not necessarily different from each other). Guy noticed that for each card, the square of the number written on it equals the sum of the numbers on the 16 other cards.

What are the numbers on Guy's cards? Find all of the options.
Answer: 0 \text{ or } 16
Problem: Starting with a list of three numbers, the “*Make-My-Day*” procedure creates a new list by replacing each number by the sum of the other two. For example, from  $\{1, 3, 8\}$  “*Make-My-Day*” gives  $\{11, 9, 4\}$  and a new “*MakeMy-Day*” leads to  $\{13, 15, 20\}$ . If we begin with  $\{20, 1, 8\}$ , what is the maximum difference between two numbers on the list after  $2018$  consecutive “*Make-My-Day*”s?
Answer: 19
Problem: Determine the primes  $p$  for which the numbers  $2\lfloor p/k\rfloor - 1, \ k = 1,2,\ldots, p,$  are all quadratic residues modulo  $p.$ *Vlad Matei*
Answer:  p = 2
Problem: Suppose that  $x$  is measured in radians. Find the maximum value of \[\frac{\sin2x+\sin4x+\sin6x}{\cos2x+\cos4x+\cos6x}\] for  $0\le x\le \frac{\pi}{16}$ 
Answer: 1
Problem: Find the smallest positive integer  $n$  such that if we color in red  $n$  arbitrary vertices of the cube , there will be a vertex of the cube which has the three vertices adjacent to it colored in red.
Answer: 5
Problem: Line  $l$  tangents unit circle  $S$  in point  $P$ . Point  $A$  and circle  $S$  are on the same side of  $l$ , and the distance from  $A$  to  $l$  is  $h$  ( $h > 2$ ). Two tangents of circle  $S$  are drawn from  $A$ , and intersect line  $l$  at points  $B$  and  $C$  respectively. 

Find the value of  $PB \cdot PC$ .
Answer: h^2 - 1
Problem: Find all possible values of integer  $n > 3$  such that there is a convex  $n$ -gon in which, each diagonal is the perpendicular bisector of at least one other diagonal.

Proposed by Mahdi Etesamifard
Answer:  n = 4
Problem: Determine all pairs  $(p,m)$  consisting of a prime number  $p$  and a positive integer  $m$ , 
for which  $p^3 + m(p + 2) = m^2 + p + 1$  holds.
Answer: (2, 5)
Problem: In a qualification football round there are six teams and each two play one versus another exactly once. No two matches are played at the same time. At every moment the difference between the number of already played matches for any two teams is  $0$  or  $1$ . A win is worth  $3$  points, a draw is worth  $1$  point and a loss is worth  $0$  points. Determine the smallest positive integer  $n$  for which it is possible that after the  $n$ -th match all teams have a different number of points and each team has a non-zero number of points.
Answer: 9
Problem: Let  $T = (a,b,c)$  be a triangle with sides  $a,b$  and  $c$  and area  $\triangle$ .  Denote by  $T' = (a',b',c')$  the triangle whose sides are the altitudes of  $T$  (i.e.,  $a' = h_a, b' = h_b, c' = h_c$ ) and denote its area by  $\triangle '$ .  Similarly, let  $T'' = (a'',b'',c'')$  be the triangle formed from the altitudes of  $T'$ , and denote its area by  $\triangle ''$ . Given that  $\triangle ' = 30$  and  $\triangle '' = 20$ , find  $\triangle$ .
Answer: 45
Problem: You are given  $n$  not necessarily distinct real numbers  $a_1, a_2, \ldots, a_n$ . Let's consider all  $2^n-1$  ways to select some nonempty subset of these numbers, and for each such subset calculate the sum of the selected numbers. What largest possible number of them could have been equal to  $1$ ?

For example, if  $a = [-1, 2, 2]$ , then we got  $3$  once,  $4$  once,  $2$  twice,  $-1$  once,  $1$  twice, so the total number of ones here is  $2$ .
Answer: 2^{n-1}
Problem: For reals  $x_1, x_2, x_3, \dots, x_{333} \in [-1, \infty)$ , let  $S_k = \displaystyle \sum_{i = 1}^{333} x_i^k$  for each  $k$ . If  $S_2 = 777$ , compute the least possible value of  $S_3$ .
Answer: 999
Problem: Let  $ABCD$  be a cyclic quadrilateral, and suppose that  $BC = CD = 2$ . Let  $I$  be the incenter of triangle  $ABD$ . If  $AI = 2$  as well, find the minimum value of the length of diagonal  $BD$ .
Answer: 2\sqrt{3}
Problem: Find all  $x \in  R$  such that    $$  x - \left[ \frac{x}{2016} \right]= 2016 $$  , where  $[k]$  represents the largest smallest integer or equal to  $k$ .
Answer: 2017
Problem: Find all positive integers  $n$  with  $n \ge 2$  such that the polynomial \[ P(a_1, a_2, ..., a_n) = a_1^n+a_2^n + ... + a_n^n - n a_1 a_2 ... a_n \] in the  $n$  variables  $a_1$ ,  $a_2$ ,  $\dots$ ,  $a_n$  is irreducible over the real numbers, i.e. it cannot be factored as the product of two nonconstant polynomials with real coefficients.
Answer:  n \ge 4
Problem: Find all polynomials  $P(x)$  of degree  $1$  such that $\underset {a\le x\le b}{max} P(x) - \underset {a\le x\le b}{min}  P(x) =b-a$   ,   $\forall a,b\in R$  where  $a < b$ 
Answer: P(x) = x + c \text{ and } P(x) = -x + c
Problem: There are  $13$  positive integers greater than  $\sqrt{15}$  and less than  $\sqrt[3]{B}$ . What is the smallest integer value of  $B$ ?
Answer: 4097
Problem: In how many ways every unit square of a  $2018$  x  $2018$  board can be colored in red or white such that number of red unit squares in any two rows are distinct and number of red squares in any two columns are distinct.
Answer: 2 \cdot (2018!)^2
Problem: Evan has  $10$  cards numbered  $1$  through  $10$ . He chooses some of the cards and takes the product of the numbers on them. When the product is divided by  $3$ , the remainder is  $1$ . Find the maximum number of cards he could have chose.
Answer: 6
Problem: Two planes,  $P$  and  $Q$ , intersect along the line  $p$ . The point  $A$  is given in the plane  $P$ , and the point  $C$  in the plane  $Q$ ; neither of these points lies on the straight line  $p$ . Construct an isosceles trapezoid  $ABCD$  (with  $AB \parallel CD$ ) in which a circle can be inscribed, and with vertices  $B$  and  $D$  lying in planes  $P$  and  $Q$  respectively.
Answer: ABCD
Problem: The admission fee for an exhibition is  $ \$ 25 $ per adult and $  \ $12$  per child.  Last Tuesday, the exhibition collected  $ \$ 1950 $ in admission fees from at least one adult and at least one child.  Of all the possible ratios of adults to children at the exhibition last Tuesday, which one is closest to $  1$?
Answer: \frac{27}{25}
Problem: Find the total length of the set of real numbers satisfying \[\frac{x^2 - 80x + 1500}{x^2 - 55x + 700} < 0.\]
Answer: 25
Problem: There are  $100$  students who want to sign up for the class Introduction to Acting. There are three class sections for Introduction to Acting, each of which will fit exactly  $20$  students. The  $100$  students, including Alex and Zhu, are put in a lottery, and 60 of them are randomly selected to fill up the classes. What is the probability that Alex and Zhu end up getting into the same section for the class?
Answer: \frac{19}{165}
Problem: We have  $n$  positive integers greater than  $1$  and less than  $10000$  such that neither of them is prime but any two of them are relative prime. Find the maximum value of  $n $ .
Answer: 25
Problem: Let  $x_n=2^{2^{n}}+1$  and let  $m$  be the least common multiple of  $x_2, x_3, \ldots, x_{1971}.$  Find the last digit of  $m.$ 
Answer: 9
Problem: Let  $\lfloor x \rfloor$  denote the largest integer less than or equal to  $x$ , and let  $\{x\}$  denote the fractional part of  $x$ . For example,  $\lfloor \pi \rfloor=3$ , and  $\{\pi\}=0.14159\dots$ , while  $\lfloor 100 \rfloor=100$  and  $\{100\}=0$ . If  $n$  is the largest solution to the equation  $\frac{\lfloor n \rfloor}{n}=\frac{2015}{2016}$ , compute  $\{n\}$ . 
Answer: \frac{2014}{2015}
Problem: A group of n  friends wrote a math contest consisting of eight short-answer problem  $S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8$ , and four full-solution problems  $F_1, F_2, F_3, F_4$ . Each person in the group correctly solved exactly 11 of the  12 problems. We create an 8 x 4 table. Inside the square located in the  $i$ th row and  $j$ th column, we write down the number of people who correctly solved both problem  $S_i$  and  $F_j$ . If the 32 entries in the table sum to 256, what is the value of n?
Answer: 32
Problem: A number is called *purple* if it can be expressed in the form  $\frac{1}{2^a 5^b}$  for positive integers  $a > b$ . The sum of all purple numbers can be expressed as  $\frac{a}{b}$  for relatively prime positive integers  $a, b$ . Compute  $100a + b$ .
Answer: 109
Problem: Find all positive integers  $n$  such that  $6^n+1$  it has all the same digits when it is writen in decimal representation.
Answer: n = 1 \text{ and } n = 5
Problem: Find all pairs  $ (m, n) $  of natural numbers such that  $ n ^ 4 \ | \ 2m ^ 5 - 1 $  and  $ m ^ 4 \ | \ 2n ^ 5 + 1 $ . 
Answer:  (m, n) = (1, 1)
Problem: For some positive integer  $n$ , the sum of all odd positive integers between  $n^2-n$  and  $n^2+n$  is a number between  $9000$  and  $10000$ , inclusive. Compute  $n$ .
Answer: 21
Problem: Determine the largest integer  $n$  such that  $2^n$  divides the decimal representation given by some permutation of the digits  $2$ ,  $0$ ,  $1$ , and  $5$ . (For example,  $2^1$  divides  $2150$ . It may start with  $0$ .)
Answer: 4
Problem: Let  $ABC$  be a triangle with  $AB=5$ ,  $BC=6$ ,  $CA=7$ . Let  $D$  be a point on ray  $AB$  beyond  $B$  such that  $BD=7$ ,  $E$  be a point on ray  $BC$  beyond  $C$  such that  $CE=5$ , and  $F$  be a point on ray  $CA$  beyond  $A$  such that  $AF=6$ . Compute the area of the circumcircle of  $DEF$ .
Answer: \frac{251}{3} \pi
Problem: 
	 Consider a  $2 \times n$  grid of points and a path consisting of  $2n-1$  straight line segments connecting all these  $2n$  points, starting from the bottom left corner and ending at the upper right corner. Such a path is called  $\textit{efficient}$  if each point is only passed through once and no two line segments intersect. How many efficient paths are there when  $n = 2016$ ?
Answer: \binom{4030}{2015}
Problem: **H**orizontal parallel segments  $AB=10$  and  $CD=15$  are the bases of trapezoid  $ABCD$ . Circle  $\gamma$  of radius  $6$  has center within the trapezoid and is tangent to sides  $AB$ ,  $BC$ , and  $DA$ . If side  $CD$  cuts out an arc of  $\gamma$  measuring  $120^{\circ}$ , find the area of  $ABCD$ .
Answer: \frac{225}{2}
Problem: A quadrilateral  $ABCD$  is inscribed in a circle. Suppose that  $|DA| =|BC|= 2$  and $ |AB|  = 4$ . Let  $E $  be the intersection  point of lines  $BC$  and  $DA$ . Suppose that  $\angle AEB = 60^o$  and that  $|CD| <|AB|$ . Calculate the radius of the circle.
Answer: 2
Problem: A subset  $M$  of  $\{1, 2, . . . , 2006\}$  has the property that for any three elements  $x, y, z$  of  $M$  with  $x < y < z$ ,  $x+ y$  does not divide  $z$ . Determine the largest possible size of  $M$ .
Answer: 1004
Problem: How many ordered triples of nonzero integers  $(a, b, c)$  satisfy  $2abc = a + b + c + 4$ ?
Answer: 6
Problem: Real numbers  $x,y,z$  satisfy   $$ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}+x+y+z=0 $$   and none of them lies in the open interval  $(-1,1)$ . Find the maximum value of  $x+y+z$ .
Answer: 0
Problem: Aurick throws  $2$  fair  $6$ -sided dice labeled with the integers from  $1$  through  $6$ . What is the probability that the sum of the rolls is a multiple of  $3$ ?
Answer: \frac{1}{3}
Problem:  $ABCD$  is a square of side 1.  $P$  and  $Q$  are points on  $AB$  and  $BC$  such that  $\widehat{PDQ} = 45^{\circ}$ . Find the perimeter of  $\Delta PBQ$ .
Answer: 2
Problem: How many integers  $n$  are there such that  $0  \le n \le 720$  and  $n^2 \equiv 1$  (mod  $720$ )?
Answer: 16
Problem: Find all monotonic positive functions  $f(x)$  defined on the positive reals such that  $f(xy) f\left( \frac{f(y)}{x}\right) = 1$  for all  $x, y$ .	
Answer: f(x) = \frac{1}{x}
Problem: The integers  $a, b,$  and  $c$  form a strictly increasing geometric sequence. Suppose that  $abc = 216$ . What is the maximum possible value of  $a + b + c$ ?
Answer: 43
Problem: The quadrilateral  $ABCD$  with perpendicular diagonals is inscribed in the circle with center  $O$ , the points  $M,N$  are the midpoints of  $[BC]$  and  $[CD]$  respectively. Find the ratio of areas of the figures  $OMCN$  and  $ABCD$ 
Answer: \frac{1}{4}
Problem: The sequence  $\{a_{n}\}$  is defined as follows. \[a_{1}=\frac{\pi}{4},\ a_{n}=\int_{0}^{\frac{1}{2}}(\cos \pi x+a_{n-1})\cos \pi x\ dx\ \ (n=2,3,\cdots)\] Find  $\lim_{n\to\infty}a_{n}$ .
Answer: \frac{\pi}{4(\pi - 1)}
Problem: Let  $A(x)=\lfloor\frac{x^2-20x+16}{4}\rfloor$ ,  $B(x)=\sin\left(e^{\cos\sqrt{x^2+2x+2}}\right)$ ,  $C(x)=x^3-6x^2+5x+15$ ,  $H(x)=x^4+2x^3+3x^2+4x+5$ ,  $M(x)=\frac{x}{2}-2\lfloor\frac{x}{2}\rfloor+\frac{x}{2^2}+\frac{x}{2^3}+\frac{x}{2^4}+\ldots$ ,  $N(x)=\textrm{the number of integers that divide }\left\lfloor x\right\rfloor$ ,  $O(x)=|x|\log |x|\log\log |x|$ ,  $T(x)=\sum_{n=1}^{\infty}\frac{n^x}{\left(n!\right)^3}$ , and  $Z(x)=\frac{x^{21}}{2016+20x^{16}+16x^{20}}$  for any real number  $x$  such that the functions are defined. Determine   $$ C(C(A(M(A(T(H(B(O(N(A(N(Z(A(2016)))))))))))))). $$  *2016 CCA Math Bonanza Lightning #5.3*
Answer: 3
Problem: Let  $p$  be a monic cubic polynomial such that  $p(0)=1$  and such that all the zeroes of  $p^\prime (x)$  are also zeroes of  $p(x)$ .  Find  $p$ .  Note: monic means that the leading coefficient is  $1$ .
Answer: p(x) = (x + 1)^3
Problem: Define a sequence  $(a_n)$  by  $a_0 =0$  and  $a_n = 1 +\sin(a_{n-1}-1)$  for  $n\geq 1$ . Evaluate  $$ \lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^{n} a_k. $$  
Answer: 1
Problem: Given the Fibonacci sequence with  $f_0=f_1=1$ and for  $n\geq 1, f_{n+1}=f_n+f_{n-1}$ , find all real solutions to the equation:   $$ x^{2024}=f_{2023}x+f_{2022}. $$  
Answer: \frac{1 \pm \sqrt{5}}{2}
Problem: Find the smallest positive number  $\lambda $  , such that for any complex numbers  ${z_1},{z_2},{z_3}\in\{z\in C\big| |z|<1\}$  ,if   $z_1+z_2+z_3=0$ , then   $$ \left|z_1z_2 +z_2z_3+z_3z_1\right|^2+\left|z_1z_2z_3\right|^2 <\lambda . $$  
Answer: \lambda = 1
Problem: Let  $n\geq 3$  be an odd number. We will play a game using a  $n$  by  $n$  grid. The game is comprised of  $n^2$  turns, in every turn, we will perform the following operation sequentially. $\bullet$  We will choose a square with an unwritten integer, and write down an integer among 1 through  $n^2$ . We can write down any integer only at once through the game. $\bullet$  For each row, colum including the square, if the sum of integers is a multiple of  $n$ , then we will get 1 point (both of each sum is a multiple of  $n$ , we will get 2 points).

Determine the maximum possible value of the points as the total sum that we can obtain by the end of the game.
Answer: n(n+1)
Problem: The five numbers  $a, b, c, d,$  and  $e$  satisfy the inequalities  $$ a+b < c+d < e+a < b+c < d+e. $$  Among the five numbers, which one is the smallest, and which one is the largest?
Answer:  a
Problem: Find all pairs  $(n, p)$  of positive integers such that  $p$  is prime and
\[ 1 + 2 + \cdots + n = 3 \cdot (1^2 + 2^2 + \cdot + p^2). \]
Answer: (n, p) = (5, 2)
Problem: An acute triangle  $ABC$  has side lenghths  $a$ ,  $b$ ,  $c$  such that  $a$ ,  $b$ ,  $c$  forms an arithmetic sequence. Given that the area of triangle  $ABC$  is an integer, what is the smallest value of its perimeter?
Answer: 18
Problem: Find all increasing sequences  $a_1,a_2,a_3,...$  of natural numbers such that for each  $i,j\in \mathbb N$ , number of the divisors of  $i+j$  and  $a_i+a_j$  is equal. (an increasing sequence is a sequence that if  $i\le j$ , then  $a_i\le a_j$ .)
Answer:  a_n = n
Problem: Let  $n \geq 2$  be an integer and let \[M=\bigg\{\frac{a_1 + a_2 + ... + a_k}{k}: 1 \le k \le n\text{ and }1 \le a_1 < \ldots < a_k \le n\bigg\}\] be the set of the arithmetic means of the elements of all non-empty subsets of  $\{1, 2, ..., n\}$ . Find \[\min\{|a - b| : a, b \in M\text{ with } a \neq b\}.\]
Answer: \begin{cases} \frac{1}{2} & n=2 \\ \frac{1}{(n-1)(n-2)} & n \geq 3 \end{cases}
Problem: In triangle  $ABC$  we have  $|AB| \ne |AC|$ . The bisectors of  $\angle ABC$  and  $\angle  ACB$  meet  $AC$  and  $AB$  at  $E$  and  $F$ , respectively, and intersect at I. If  $|EI| = |FI|$  find the measure of  $\angle  BAC$ .
Answer: 60^\circ
Problem: Throw  $ n$  balls in to  $ 2n$  boxes.　Suppose each ball comes into each box with equal probability of entering in any boxes.
Let  $ p_n$  be the probability such that any box has ball less than or equal to one. Find the limit  $ \lim_{n\to\infty} \frac{\ln p_n}{n}$ 
Answer: 0
Problem: What is the smallest positive integer  $k$  such that  $k(3^3 + 4^3 + 5^3) = a^n$  for some positive integers  $a$  and  $n$ , with  $n > 1$ ?
Answer: 1
Problem: Determine all intergers  $n\geq 2$  such that  $a+\sqrt{2}$  and  $a^n+\sqrt{2}$  are both rational for some real number  $a$  depending on  $n$ 
Answer:  n = 2
Problem: Find all functions  $f(x)$  determined on interval  $[0,1]$ , satisfying following conditions   $$ \{f(x)\}\sin^{2}x+\{x\}\cos f(x)\cos x=f(x) $$     $$ f(f(x))=f(x) $$   Here  $\{y\}$  means a fractional part of number  $y$ 
Answer: f(x) = x \text{ for all } x \in [0,1]
Problem: It is known that  $a^{2005} + b^{2005}$  can be expressed as the polynomial of  $a + b$  and  $ab$ . Find the coefficients' sum of this polynomial.
Answer: 1
Problem: Determine the greatest positive integer \(n\) for which there exists a sequence of distinct positive integers \(s_1\), \(s_2\), \(\ldots\), \(s_n\) satisfying \[s_1^{s_2}=s_2^{s_3}=\cdots=s_{n-1}^{s_n}.\]
Answer: n = 5
Problem: Olya wrote fractions of the form  $1 / n$  on cards, where  $n$  is all possible divisors the numbers  $6^{100}$  (including the unit and the number itself). These cards she laid out in some order. After that, she wrote down the number on the first card, then the sum of the numbers on the first and second cards, then the sum of the numbers on the first three cards, etc., finally, the sum of the numbers on all the cards. Every amount Olya recorded on the board in the form of irreducible fraction. What is the least different denominators could be on the numbers on the board?
Answer: 2
Problem: Jane is 25 years old.  Dick is older than Jane.  In  $n$  years, where  $n$  is a positive integer, Dick's age and Jane's age will both be two-digit number and will have the property that Jane's age is obtained by interchanging the digits of Dick's age.  Let  $d$  be Dick's present age.  How many ordered pairs of positive integers  $(d,n)$  are possible?
Answer: 25
Problem: Establish necessary and sufficient conditions on the constant  $k$  for the existence of a continuous real valued function  $f(x)$  satisfying   $$ f(f(x))=kx^9 $$   for all real  $x$ .
Answer: k \geq 0
Problem: Find all positive integers  $n$  such that there exist non-constant polynomials with integer coefficients  $f_1(x),...,f_n(x)$  (not necessarily distinct) and  $g(x)$  such that   $$ 1 + \prod_{k=1}^{n}\left(f^2_k(x)-1\right)=(x^2+2013)^2g^2(x) $$  
Answer: 2
Problem: Let  $a,b, c$  denote the real numbers such that  $1 \le  a, b, c\le  2$ . 
Consider  $T = (a - b)^{2018} + (b - c)^{2018} + (c - a)^{2018}$ .
Determine the largest possible value of  $T$ .
Answer: 2