Let $f(x)=\sum_{i=0}^{n}a_ix^i$ and $g(x)=\sum_{i=0}^{n}b_ix^i$, where $a_n$,$b_n$ can be zero. Called $f(x)\ge g(x)$ if exist $r$ such that $\forall i>r,a_i=b_i,a_r>b_r$ or $f(x)=g(x)$. Prove that: if the leading coefficients of $f$ and $g$ are positive, then $f(f(x))+g(g(x))\ge f(g(x))+g(f(x))$

Let $\omega$ be the incircle of triangle $ABC$ and $\omega$ touches $BC$ at $D$. $AD$ meets $\omega$ again at $L$. Let $K$ be $A$-excenter, and $M,N$ be the midpoint of $BC,KM$, respectively. Prove that $B,C,N,L$ are concyclic.

Let the sequence $\{a_n\}$ satisfy $a_{n+1}=a_n^3+103,n=1,2,...$. Prove that at most one integer $n$ such that $a_n$ is a perfect square.

Let $ABC$ be a triangle. The points $K, L,$ and $M$ lie on the segments $BC, CA,$ and $AB,$ respectively, such that the lines $AK, BL,$ and $CM$ intersect in a common point. Prove that it is possible to choose two of the triangles $ALM, BMK,$ and $CKL$ whose inradii sum up to at least the inradius of the triangle $ABC$. Proposed by Estonia

We have $2^m$ sheets of paper, with the number $1$ written on each of them. We perform the following operation. In every step we choose two distinct sheets; if the numbers on the two sheets are $a$ and $b$, then we erase these numbers and write the number $a + b$ on both sheets. Prove that after $m2^{m -1}$ steps, the sum of the numbers on all the sheets is at least $4^m$ . Proposed by Abbas Mehrabian, Iran

Given a real number $t\neq -1$. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[(t+1)f(1+xy)-f(x+y)=f(x+1)f(y+1)\] for all $x,y\in\mathbb{R}$.

For a sequence $x_1,x_2,\ldots,x_n$ of real numbers, we define its $\textit{price}$ as \[\max_{1\le i\le n}|x_1+\cdots +x_i|.\] Given $n$ real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possible ways and finds the minimum possible price $D$. Greedy George, on the other hand, chooses $x_1$ such that $|x_1 |$ is as small as possible; among the remaining numbers, he chooses $x_2$ such that $|x_1 + x_2 |$ is as small as possible, and so on. Thus, in the $i$-th step he chooses $x_i$ among the remaining numbers so as to minimise the value of $|x_1 + x_2 + \cdots x_i |$. In each step, if several numbers provide the same value, George chooses one at random. Finally he gets a sequence with price $G$. Find the least possible constant $c$ such that for every positive integer $n$, for every collection of $n$ real numbers, and for every possible sequence that George might obtain, the resulting values satisfy the inequality $G\le cD$. Proposed by Georgia

Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively. Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes. Proposed by Tamas Fleiner and Peter Pal Pach, Hungary

Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. Let the line passing through $I$ and perpendicular to $CI$ intersect the segment $BC$ and the arc $BC$ (not containing $A$) of $\Omega$ at points $U$ and $V$ , respectively. Let the line passing through $U$ and parallel to $AI$ intersect $AV$ at $X$, and let the line passing through $V$ and parallel to $AI$ intersect $AB$ at $Y$ . Let $W$ and $Z$ be the midpoints of $AX$ and $BC$, respectively. Prove that if the points $I, X,$ and $Y$ are collinear, then the points $I, W ,$ and $Z$ are also collinear. Proposed by David B. Rush, USA

Let $ABC$ be a triangle with incircle $\omega$, incenter $I$ and circumcircle $\Gamma$. Let $D$ be the tangency point of $\omega$ with $BC$, let $M$ be the midpoint of $ID$, and let $A'$ be the diametral opposite of $A$ with respect to $\Gamma$. If we denote $X=A'M\cap \Gamma$, then prove that the circumcircle of triangle $AXD$ is tangent to $BC$.

Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$. Proposed by Netherlands

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. Proposed by Belgium