Let $\mathbb R$ be the set of real numbers. We denote by $\mathcal F$ the set of all functions $f\colon\mathbb R\to\mathbb R$ such that $$f(x + f(y)) = f(x) + f(y)$$ for every $x,y\in\mathbb R$ Find all rational numbers $q$ such that for every function $f\in\mathcal F$, there exists some $z\in\mathbb R$ satisfying $f(z)=qz$.

$a_{0}=2,a_{1}=1$ and for $n\geq 1$ we know that : $a_{n+1}=a_{n}+a_{n-1}$ $m$ is an even number and $p$ is prime number such that $p$ divides $a_{m}-2$. Prove that $p$ divides $a_{m+1}-1$.

Let $f(x)=x^2-2$ and let $f^{(n)}(x)$ denote the $n$-th iteration of $f$. Let $H=\{x:f^{(100)}(x)\leq -1\}$. Find the length of $H$ (the sum of the lengths of the intervals of $H$).

Let $x_1$, $x_2$, $\cdots$, $x_n$ be positive real numbers, and let \[ S = x_1 + x_2 + \cdots + x_n. \] Prove that \[ (1 + x_1)(1 + x_2) \cdots (1 + x_n) \leq 1 + S + \frac{S^2}{2!} + \frac{S^3}{3!} + \cdots + \frac{S^n}{n!} \]

Determine all positive integers$ n$ such that $f_n(x,y,z) = x^{2n} + y^{2n} + z^{2n} - xy - yz - zx$ divides $g_n(x,y, z) = (x - y)^{5n} + (y -z)^{5n} + (z - x)^{5n}$, as polynomials in $x, y, z$ with integer coefficients.

[color=darkred]Find all differentiable functions $f\colon [0,\infty)\to [0,\infty)$ for which $f(0)=0$ and $f^{\prime}(x^2)=f(x)$ for any $x\in [0,\infty)$ .[/color]

Prove the inequality for all positive integer $n$ : \[ \left(\frac{2n-1}{e}\right)^{\frac{2n-1}{2}}<1\cdot 3\cdot 5\cdots (2n-1)<\left(\frac{2n+1}{e}\right)^{\frac{2n+1}{2}} \]

Let $a$ and $b$ be positive integers. The cells of an $(a+b+1)\times (a+b+1)$ grid are colored amber and bronze such that there are at least $a^2+ab-b$ amber cells and at least $b^2+ab-a$ bronze cells. Prove that it is possible to choose $a$ amber cells and $b$ bronze cells such that no two of the $a+b$ chosen cells lie in the same row or column.

Let $S_1, S_2, \dots, S_{2^n - 1}$ be the nonempty subsets of $\{1, 2, \dots, n\}$ in some order, and let $M$ be the $(2^n - 1) \times (2^n - 1)$ matrix whose $(i, j)$ entry is \[m_{ij} = \left\{ \begin{array}{cl} 0 & \text{if $S_i \cap S_j = \emptyset$}, \\ 1 & \text{otherwise}. \end{array} \right.\] Calculate the determinant of $M$.

Let $n > 1$ be an integer. Find, with proof, all sequences $x_1 , x_2 , \ldots , x_{n-1}$ of positive integers with the following three properties: (a). $x_1 < x_2 < \cdots < x_{n-1}$ ; (b). $x_i + x_{n-i} = 2n$ for all $i = 1, 2, \ldots , n - 1$; (c). given any two indices $i$ and $j$ (not necessarily distinct) for which $x_i + x_j < 2n$, there is an index $k$ such that $x_i + x_j = x_k$.

For each integer $C>1$ decide whether there exist pairwise distinct positive integers $a_1,a_2,a_3,...$ such that for every $k\ge 1$, $a_{k+1}^k$ divides $C^ka_1a_2...a_k$. [i]Proposed by Daniel Liu

Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$ [i]Michael Ren and Ankan Bhattacharya, USA[/i]

Let $V$ be a convex polygon. (a) Show that if $V$ has $3k$ vertices, then $V$ can be triangulated such that each vertex is in an odd number of triangles. (b) Show that if the number of vertices is not divisible with 3, then $V$ can be triangulated such that exactly 2 vertices have an even number of triangles.

Let $n$ be a positive integer. A [i]Nordic[/i] square is an $n \times n$ board containing all the integers from $1$ to $n^2$ so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only to cells containing larger numbers is called a [i]valley[/i]. An [i]uphill path[/i] is a sequence of one or more cells such that: (i) the first cell in the sequence is a valley, (ii) each subsequent cell in the sequence is adjacent to the previous cell, and (iii) the numbers written in the cells in the sequence are in increasing order. Find, as a function of $n$, the smallest possible total number of uphill paths in a Nordic square. Author: Nikola Petrović