Ana plays a game on a $100\times 100$ chessboard. Initially, there is a white pawn on each square of the bottom row and a black pawn on each square of the top row, and no other pawns anywhere else.\\ Each white pawn moves toward the top row and each black pawn moves toward the bottom row in one of the following ways: [list] [*] it moves to the square directly in front of it if there is no other pawn on it; [*] it [b]captures[/b] a pawn on one of the diagonally adjacent squares in the row immediately in front of it if there is a pawn of the opposite color on it. [/list] (We say a pawn $P$ [b]captures[/b] a pawn $Q$ of the opposite color if we remove $Q$ from the board and move $P$ to the square that $Q$ was previously on.)\\ \\ Ana can move any pawn (not necessarily alternating between black and white) according to those rules. What is the smallest number of pawns that can remain on the board after no more moves can be made? [i]Proposed by José Alejandro Reyes González, Mexico[/i]

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

Suppose that the circumradius $R$ and the inradius $r$ of a triangle $ABC$ satisfy $R = 2r$. Prove that the triangle is equilateral.

Each grid point of a cartesian plane is colored with one of three colors, whereby all three colors are used. Show that one can always find a right-angled triangle, whose three vertices have pairwise different colors.

The Fibonacci numbers $F_n$ are defined as $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for all $n> 2$. Let $A$ be the minimum area of a (possibly degenerate) convex polygon with $2020$ sides, whose side lengths are the first $2020$ Fibonacci numbers $F_1$, $F_2$, $...$ , $F_{2020}$ (in any order). A degenerate convex polygon is a polygon where all angles are $\le 180^o$. If $A$ can be expressed in the form $$\frac{\sqrt{(F_a-b)^2-c}}{d}$$ , where $a, b, c$ and $d$ are positive integers, compute the minimal possible value of $a + b + c + d$.

Find all functions $f\colon R \to R$ such that \[f\left(x^{2}+yf(x)\right) = f(x)^{2}+xf(y)\] for all reals $x,y$.

Find all real numbers $x$ which satisfy $\frac{n}{3n+1}\leq x\leq \frac{4n+1}{2n-1}$, for all $n\in\mathbb{N}^*$. [i]Gheorghe Boroica[/i]

Let $a,b,c$ be positive real numbers such that $a + b + c = 1$. Show that $$\left(\frac{1}{a} + \frac{1}{bc}\right)\left(\frac{1}{b} + \frac{1}{ca}\right)\left(\frac{1}{c} + \frac{1}{ab}\right) \geq 1728$$

Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?

Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

The sequence $(a_{n})$ is defined by $a_1=1$ and $a_n=n(a_1+a_2+\cdots+a_{n-1})$ , $\forall n>1$. [b](a)[/b] Prove that for every even $n$, $a_{n}$ is divisible by $n!$. [b](b)[/b] Find all odd numbers $n$ for the which $a_{n}$ is divisible by $n!$.

Three lines are drawn in a plane. Which of the following could NOT be the total number of points of intersections? (A) $0$ (B) $1$ (C) $2$ (D) $3$ (E) They all could.

Sides $\overline{AB}$ and $\overline{AC}$ of equilateral triangle $ABC$ are tangent to a circle at points $B$ and $C$ respectively. What fraction of the area of $\triangle ABC$ lies outside the circle? $ \textbf{(A) }\dfrac{4\sqrt{3}\pi}{27}-\frac{1}{3}\qquad \textbf{(B) } \frac{\sqrt{3}}{2}-\frac{\pi}{8}\qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) }\sqrt{3}-\frac{2\sqrt{3}\pi}{9}\qquad \textbf{(E) } \frac{4}{3}-\dfrac{4\sqrt{3}\pi}{27}$

Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.

Let $x$ be an integer of $n$ digits, all equal to $ 1$. Show that if $x$ is prime, then $n$ is also prime.

For positive numbers $ a $, $ b $ and $ c $ satisfying the equality $ a + b + c = 1 $, prove the inequality $$ \frac {a ^ 7 + b ^ 7} {a ^ 5 + b ^ 5} + \frac {b ^ 7 + c ^ 7} {b ^ 5 + c ^ 5} + \frac {c ^ 7 + a ^ 7} {c ^ 5 + a ^ 5} \geq \frac {1} {3}. $$

$P(x,y)$ is polynomial with real coefficients and $P(x+2y,x+y)=P(x,y)$. Prove that exists polynomial $Q(t)$ such that $P(x,y)=Q((x^2-2y^2)^2)$ [i]A. Golovanov[/i]

What is the last two digits of base-$3$ representation of $2005^{2003^{2004}+3}$? $ \textbf{(A)}\ 21 \qquad\textbf{(B)}\ 01 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 02 \qquad\textbf{(E)}\ 22 $

Let $Z_1,Z_2,\cdots ,Z_n$ be complex numbers satisfying $|Z_1|+|Z_2|+\cdots +|Z_n|=1$. Show that there exist some among the $n$ complex numbers such that the modulus of the sum of these complex numbers is not less than $1/6$.

Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $$\sqrt{\frac{a+b}{b+1}}+\sqrt{\frac{b+c}{c+1}}+\sqrt{\frac{c+a}{a+1}} \ge 3$$

What is the remainder when $7^{8^9}$ is divided by $1000?$

In triangle $ABC$, $\angle C = 30^{\circ}$, $O$ and $I$ are the circumcenter and incenter respectively, Points $D \in AC$ and $E \in BC$, such that $AD = BE = AB$. Prove that $OI = DE$ and $OI \bot DE$.

A semicircle with diameter $AB = 2r$ is divided into two sectors by an arbitrary radius. To each of the sectors a circle is inscribed. These two circles touch A$B$ at $S$ and $T$. Show that $ST \ge 2r(\sqrt{2}-1)$.

The integers from $1$ to $n$ are written, one on each of $n$ cards. The first player removes one card. Then the second player removes two cards with consecutive integers. After that the first player removes three cards with consecutive integers. Finally, the second player removes four cards with consecutive integers. What is th smallest value of $n$ for which the second player can ensure that he competes both his moves?

Let $F=\{1,2,...,100\}$ and let $G$ be any $10$-element subset of $F$. Prove that there exist two disjoint nonempty subsets $S$ and $T$ of $G$ with the same sum of elements.