Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/i]

In the square $ABCD$ on the sides $AD$ and $DC$, the points $M$ and $N$ are selected so that $\angle BMA = \angle NMD = 60 { } ^ \circ $. Find the value of the angle $MBN$.

A circle $\omega$ and a point $T$ outside the circle is given. Let a tangent from $T$ to $\omega$ touch $\omega$ at $A$, and take points $B,C$ lying on $\omega$ such that $T,B,C$ are colinear. The bisector of $\angle ATC$ intersects $AB$ and $AC$ at $P$ and $Q$,respectively. Prove that $PA=\sqrt{PB\cdot QC}$

Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ and plynomials $P(x),Q(x),R(x)$ with positive real coefficients such that $Q(-1)=-1$ and for all positive reals $x,y$:$$f(\frac{x}{y}+R(y))=\frac{f(x)}{Q(y)}+P(y).$$ [i]Proposed by Alireza Danaie, Ali Mirazaie Anari[/i] [b]Rated 2[/b]

Prove that there exists $1904$-element subset of the set $\{1,2,\ldots,1992\}$, which doesn’t contain an arithmetic progression consisting of $41$ terms. [i](Ivan Tonov)[/i]

Let $\mathbb N$ be the set of positive integers. Find all functions $f: \mathbb N \to \mathbb N$ that satisfy the equation \[ f^{abc-a}(abc) + f^{abc-b}(abc) + f^{abc-c}(abc) = a + b + c \] for all $a,b,c \ge 2$. (Here $f^1(n) = f(n)$ and $f^k(n) = f(f^{k-1}(n))$ for every integer $k$ greater than $1$.)

Find all pairs $ (x,y)$ of real numbers which satisfy $ x^3\minus{}y^3\equal{}4(x\minus{}y)$ and $ x^3\plus{}y^3\equal{}2(x\plus{}y)$.

Solve $|x-1|-2|x+5|>3+x$.

Let $e > 0$ be a given real number. Find the least value of $f(e)$ (in terms of $e$ only) such that the inequality $a^{3}+ b^{3}+ c^{3}+ d^{3} \leq e^{2}(a^{2}+b^{2}+c^{2}+d^{2}) + f(e)(a^{4}+b^{4}+c^{4}+d^{4})$ holds for all real numbers $a, b, c, d$.

Find the number of partitions of the set $\{1,2,3,\cdots ,11,12\}$ into three nonempty subsets such that no subset has two elements which differ by $1$. [i]Proposed by Nathan Ramesh

On an island live $n$ ($n \ge 2$) $xyz$s. Any two $xyz$s are either friends or enemies. Every $xyz$ wears a necklace made of colored beads such that any two $xyz$s that are befriended have at least one bead of the same color and any two $xyz$s that are enemies do not have any common colors in their necklaces. It is also possible for some necklaces not to have any beads. What is the minimum number of colors of beads that is sufficient to manufacture such necklaces regardless on the relationship between the $xyz$s?

Let $n$ be a given positive integer. Find the solution set of the equation $\sum_{k=1}^{2n} \sqrt{x^2 -2kx + k^2} =|2nx - n - 2n^2|$  

Let $(m,n,N)$ be a triple of positive integers. Bruce and Duncan play a game on an m\times n array, where the entries are all initially zeroes. The game has the following rules. $\bullet$ The players alternate turns, with Bruce going first. $\bullet$ On Bruce's turn, he picks a row and either adds $1$ to all of the entries in the row or subtracts $1$ from all the entries in the row. $\bullet$ On Duncan's turn, he picks a column and either adds $1$ to all of the entries in the column or subtracts $1$ from all of the entries in the column. $\bullet$ Bruce wins if at some point there is an entry $x$ with $|x|\ge N$. Find all triples $(m, n,N)$ such that no matter how Duncan plays, Bruce has a winning strategy.

Find all functions $f : R \to R$ that satisfies the condition $$f(f(x - y)) = f(x)f(y) - f(x) + f(y) - xy$$ for all real numbers $x, y$.

In a cube-shaped box with an edge equal to $5$, there are two balls. The radius of one of the balls is $2$. Find the radius of the other ball if one of the balls touches the base and two side faces of the cube, and the other ball touches the first ball, base and two other side faces of the cube.

A student puts $2010$ red balls and $1957$ blue balls into a box. Weiqing draws randomly from the box one ball at a time without replacement. She wins if, at anytime, the total number of blue balls drawn is more than the total number of red balls drawn. Assuming Weiqing keeps drawing balls until she either wins or runs out, ompute the probability that she eventually wins.

Triangle $ABC$ has side lengths $AB=12$, $BC=25$, and $CA=17$. Rectangle $PQRS$ has vertex $P$ on $\overline{AB}$, vertex $Q$ on $\overline{AC}$, and vertices $R$ and $S$ on $\overline{BC}$. In terms of the side length $PQ=w$, the area of $PQRS$ can be expressed as the quadratic polynomial \[\text{Area}(PQRS)=\alpha w-\beta\cdot w^2\] Then the coefficient $\beta=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Suppose that points $A$ and $B$ lie on circle $\Omega$, and suppose that points $C$ and $D$ are the trisection points of major arc $AB$, with $C$ closer to $B$ than $A$. Let $E$ be the intersection of line $AB$ with the line tangent to $\Omega$ at $C$. Suppose that $DC = 8$ and $DB = 11$. If $DE = a\sqrt{b}$ for integers $a$ and $b$ with $b$ squarefree, find $a + b$.

Let \( n \geq 4 \). Proof that \[ (2^x - 1)(5^x - 1) = y^n \] have no positive integer solution \((x, y)\).

Let $u_n$ denote the ramp function $$ u_n (x) =\begin{cases} -n \;\; \text{for} \;\; x \leq -n, \\ \; x \;\;\; \text{for} \;\; -n \leq x \leq n,\\ \;n \;\; \; \text{for} \;\; n \leq x, \end{cases}$$ and let $f$ be a real function of a real variable. Show that $f$ is continuous if and only if $u_n \circ f$ is continuous for all $n.$

A $8\times 8$ board is divided in dominoes (rectangles with dimensions $1 \times 2$ or $2 \times 1$). [list=a] [*] Prove that the total length of the border between horizontal and vertical dominoes is at most $52$. [*] Determine the maximum possible total length of the border between horizontal and vertical dominoes. [/list] [i]Proposed by B. Frenkin, A. Zaslavsky, E. Arzhantseva[/i]

A polynomial $P$ has integer coefficients and $P(3)=4$ and $P(4)=3$. For how many $x$ we might have $P(x)=x$?

In the film there is $n$ roles. For each $i$ ($1 \le i \le n$), the role of number $i$ can play $a_i$ a person, and one person can play only one role. Every day a casting is held, in which participate people for $n$ roles, and from each role only one person. Let $p$ be a prime number such that $p \ge a_1, \ldots, a_n, n$. Prove that you can have $p^k$ castings such that if we take any $k$ people who are tried in different roles, they together participated in some casting ($k$ is a natural number not exceeding $n$ ).

In ZS Chess, an Ivanight attacks like a knight, except that if the attacked square is out of range, it goes through the edge and comes out from the other side of the board, and attacks that square instead. The ZS chessboard is an $8 \times 8$ board, where cells are coloured with $n$ distinct colours, where $n$ is a natural number, such that a Ivanight placed on any square attacks $ 8 $ squares that consist of all $n$ colours, and the colours appear equally many times in those $ 8 $ squares. For which values of $n$ does such a ZS chess board exist?

In triangle $\triangle ABC$ with $BC = 1$, the internal angle bisector of $\angle A$ intersects $BC$ at $D$. $M$ is taken to be the midpoint of $BC$. Point $E$ is chosen on the boundary of $\triangle ABC$ such that $ME$ bisects its perimeter. The circumcircle $\omega$ of $\triangle DEC$ is taken, and the second intersection of $AD$ and $\omega$ is $K$, as well as the second intersection of $ME$ and $\omega$ being $L$. If $B$ lies on line $KL$ and $ED$ is parallel to $AB$, then the perimeter of $\triangle ABC$ can be written as a real number $S$. Compute $\lfloor 1000S\rfloor$. Proposed by Albert Wang (awang11)