Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$. [i]Proposed by Juhan Aru, Estonia[/i]

Let $f :\mathbb{N} \rightarrow\mathbb{N}$ be an injective function such that $f(1) = 2,~ f(2) = 4$ and \[f(f(m) + f(n)) = f(f(m)) + f(n)\] for all $m, n \in \mathbb{N}$. Prove that $f(n) = n + 2$ for all $n \ge 2$.

Given an equilateral triangle $ABC$ of side $a$ in a plane, let $M$ be a point on the circumcircle of the triangle. Prove that the sum $s = MA^4 +MB^4 +MC^4$ is independent of the position of the point $M$ on the circle, and determine that constant value as a function of $a$.

Let $f : \mathbb{R} \to \mathbb{R}$ be a function whose second derivative is continuous. Suppose that $f$ and $f''$ are bounded. Show that $f'$ is also bounded.

Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold: (i) $f(n!)=f(n)!$ for every positive integer $n$, (ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.

We say a finite set $S$ of points in the plane is [i]very[/i] if for every point $X$ in $S$, there exists an inversion with center $X$ mapping every point in $S$ other than $X$ to another point in $S$ (possibly the same point). (a) Fix an integer $n$. Prove that if $n \ge 2$, then any line segment $\overline{AB}$ contains a unique very set $S$ of size $n$ such that $A, B \in S$. (b) Find the largest possible size of a very set not contained in any line. (Here, an [i]inversion[/i] with center $O$ and radius $r$ sends every point $P$ other than $O$ to the point $P'$ along ray $OP$ such that $OP\cdot OP' = r^2$.) [i]Proposed by Sammy Luo[/i]

Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following. $f(xf(x)+f(x)f(y)+y-1)=f(xf(x)+xy)+y-1$

What is the maximum value of the function $$\frac{1}{\left|x+1\right|+\left|x+2\right|+\left|x-3\right|}?$$ $\text{(A) }\frac{1}{3}\qquad\text{(B) }\frac{1}{4}\qquad\text{(C) }\frac{1}{5}\qquad\text{(D) }\frac{1}{6}\qquad\text{(E) }\frac{1}{7}$

Let $n$ be a positive integer. $S_1, S_2, \ldots, S_n$ are pairwise non-intersecting sets, and $S_k $ has exactly $k$ elements $(k = 1, 2, \ldots, n)$. Define $S = S_1\cup S_2\cup\cdots \cup S_n$. The function $f: S \to S $ maps all elements in $S_k$ to a fixed element of $S_k$, $k = 1, 2, \ldots, n$. Find the number of functions $g: S \to S$ satisfying $f(g(f(x))) = f(x).$

Let $f : N \to N$ be a function such that the set $\{k | f(k) < k\}$ is finite. Prove that the set $\{k | g(f(k)) \le k\}$ is infinite for all functions $g : N \to N$.

If $x$ and $y$ are real numbers with $(x+y)^4=x-y$, what is the maximum possible value of $y$?

In each of (a) to (d) you have to find a strictly increasing surjective function from A to B or prove that there doesn't exist any. (a) $A=\{x|x\in \mathbb{Q},x\leq \sqrt{2}\}$ and $B=\{x|x\in \mathbb{Q},x\leq \sqrt{3}\}$ (b) $A=\mathbb{Q}$ and $B=\mathbb{Q}\cup \{\pi \} $ In (c) and (d) we say $(x,y)>(z,t)$ where $x,y,z,t \in \mathbb{R}$ , whenever $x>z$ or $x=z$ and $y>t$. (c) $A=\mathbb{R}$ and $B=\mathbb{R}^2$ (d) $X=\{2^{-x}| x\in \mathbb{N}\}$ , then $A=X \times (X\cup \{0\})$ and $B=(X \cup \{ 0 \}) \times X$ (e) If $A,B \subset \mathbb{R}$ , such that there exists a surjective non-decreasing function from $A$ to $B$ and a surjective non-decreasing function from $B$ to $A$ , does there exist a surjective strictly increasing function from $B$ to $A$? Time allowed for this problem was 2 hours.

Petya and Vasya are playing the following game. Petya chooses a non-negative random value $\xi$ with expectation $\mathbb{E} [\xi ] = 1$, after which Vasya chooses his own value $\eta$ with expectation $\mathbb{E} [\eta ] = 1$ without reference to the value of $\xi$. For which maximal value $p$ can Petya choose a value $\xi$ in such a way that for any choice of Vasya's $\eta$, the inequality $\mathbb{P}[\eta \geq \xi ] \leq p$ holds?

Let $f(x)$ be a continuous function such that $\int_0^1 f(x)\ dx=1.$ Find $f(x)$ for which $\int_0^1 (x^2+x+1)f(x)^2dx$ is minimized.

Let $ f: (0,\infty)\longrightarrow (0,\infty) $ a non-constant function having the property that $ f\left( x^y\right) = \left( f(x)\right)^{f(y)},\quad\forall x,y>0. $ Show that $ f(xy)=f(x)f(y) $ and $ f(x+y)=f(x)+f(y), $ for all $ x,y>0. $