Let $ n > 1$ be an integer. Find all sequences $ a_1, a_2, \ldots a_{n^2 \plus{} n}$ satisfying the following conditions: \[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 \plus{} n; \] \[ \text{ (b) } a_{i \plus{} 1} \plus{} a_{i \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} n} < a_{i \plus{} n \plus{} 1} \plus{} a_{i \plus{} n \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} 2n} \text{ for all } 0 \leq i \leq n^2 \minus{} n. \] [i]Author: Dusan Dukic, Serbia[/i]

Let $ABC$ be an acute triangle such that $AB\not=AC.$Let $M$ be the midpoint $BC,H$ the orthocenter of $\triangle ABC$$,O_1$ the midpoint of $AH$ and $O_2$ the circumcenter of $\triangle BCH$$.$ Prove that $O_1AMO_2$ is a parallelogram.

Let $A$ be a finite set of pairs of real numbers such that for any pairs $(a,b)$ in $A$ we have $a>0$. Let $X_0=(x_0, y_0)$ be a pair of real numbers(not necessarily from $A$). We define $X_{j+1}=(x_{j+1}, y_{j+1})$ for all $j\ge 0$ as follows: for all $(a,b)\in A$, if $ax_j+by_j>0$ we let $X_{j+1}=X_j$; otherwise we choose a pair $(a,b)$ in $A$ for which $ax_j+by_j\le 0$ and set $X_{j+1}=(x_j+a, y_j+b)$. Show that there exists an integer $N\ge 0$ such that $X_{N+1}=X_N$.

Let $a, b$ and $c$ denote the side lengths and $m_a, m_b$ and $m_c$ of the median's lengths in an arbitrary triangle. Show that $$\frac34 < \frac{m_a + m_b + m_c}{a + b + c}<1$$ Also show that there is no narrower range that for each triangle that contains the fraction $$\frac{m_a + m_b + m_c}{a + b + c}$$

Let $g(x)$ be a polynomial of degree at least $2$ with all of its coefficients positive. Find all functions $f:\mathbb R^+ \longrightarrow \mathbb R^+$ such that \[f(f(x)+g(x)+2y)=f(x)+g(x)+2f(y) \quad \forall x,y\in \mathbb R^+.\] [i]Proposed by Mohammad Jafari[/i]

The integer numbers from $1$ to $2002$ are written in a blackboard in increasing order $1,2,\ldots, 2001,2002$. After that, somebody erases the numbers in the $ (3k+1)-th$ places i.e. $(1,4,7,\dots)$. After that, the same person erases the numbers in the $(3k+1)-th$ positions of the new list (in this case, $2,5,9,\ldots$). This process is repeated until one number remains. What is this number?

Let $G$ be a planar graph all of whose vertices are of degree $4$. Vasya and Petya walk along its edges. The first time each of them goes as he pleases, and then each of them goes straight (from the three roads they have to choose the middle one). As the result, each vertex was visited by exactly one of them and exactly once. Prove that this graph has an even number of vertices.

Let \( L \), \( M \) and \( N \) be the midpoints on the sides \( BC \), \( AC \) and \( AB\) of a triangle \( ABC \). Points \( D \), \( E \) and \( F \) are taken on the circle circumscribed to the triangle \( LMN \) so that the segments \( LD \), \( ME \) and \( NF \) are diameters of said circumference. Prove that the area of the hexagon \( LENDMF \) is equal to half the area of the triangle \( ABC \)

If $\tan x+\tan y=25$ and $\cot x + \cot y=30$, what is $\tan(x+y)$?

Let $x \ne 1$ be a fixed positive number and $a_1, a_2, a_3,...$ some kind of number sequence. Prove that $x^{a_1},x^{a_2},x^{a_3},...$ is a non-constant geometric sequence if and only if $a_1, a_2, a_3,...$. is a non-constant arithmetic sequence.

We are given a fixed point on the circle of radius $1$, and going from this point along the circumference in the positive direction on curved distances $0, 1, 2, \ldots $ from it we obtain points with abscisas $n = 0, 1, 2, .\ldots$ respectively. How many points among them should we take to ensure that some two of them are less than the distance $\frac 15$ apart ?

Let $a$, $b$ and $c$ be positive real numbers such that $a^2+b^2+c^2=3$. Prove the following inequality: $\frac{a}{3c(a^2-ab+b^2)} + \frac{b}{3a(b^2-bc+c^2)} + \frac{c}{3b(c^2-ca+a^2)} \leq \frac{1}{abc}$

Find all continuous functions $f: R \to R$ such that $g(g(x)) = g(x)+2x$ for all real $x$.

Let $ x,y\in\mathbb{R}$ , and $ x,y \in $ $ \left(0,\frac{\pi}{2}\right) $, and $ m \in \left(2,+\infty\right) $ such that $ \tan x * \tan y = m $ . Find the minimum value of the expression $ E(x,y) = \cos x + \cos y $.

Determine all $n \in \mathbb{N}$ for which [list][*] $n$ is not the square of any integer, [*] $\lfloor \sqrt{n}\rfloor ^3$ divides $n^2$. [/list]

Two sequences $(x_n)_{n\in N}$ and $(y_n)_{n\in N}$ are defined recursively as follows: $x_0 = 2015$ and $x_{n+1} =\left \lfloor x_n \cdot \frac{y_{n+1}}{y_{n-1}} \right \rfloor$ for all $n \ge 0$, $y_0 = 307$ and $y_{n+1} = y_n + 1$ for all $n \ge 0$. Compute $\lim_{n\to \infty} \frac{x_n}{(y_n)^2}$.

Let $M$ be the point of intersection of the diagonals of the inscribed quadrilateral $ABCD$, and let the angle $\angle AMB$ be an acute angle. On the side $BC$, as a base, an isosceles triangle $BCK$ is constructed in the direction external to the quadrilateral such that $\angle KBC+\angle AMB= 90^o$. Prove that line $KM$ is perpendicular to $AD$.

Determine all non-negative integers $n$ for which there exist two relatively prime non-negative integers $x$ and $y$ and a positive integer $k\geqslant 2$ such that $3^n=x^k+y^k$.

A tetrahedron is called a [i]MEMO-tetrahedron[/i] if all six sidelengths are different positive integers where one of them is $ 2$ and one of them is $ 3$. Let $ l(T)$ be the sum of the sidelengths of the tetrahedron $ T$. (a) Find all positive integers $ n$ so that there exists a MEMO-Tetrahedron $ T$ with $ l(T)\equal{}n$. (b) How many pairwise non-congruent MEMO-tetrahedrons $ T$ satisfying $ l(T)\equal{}2007$ exist? Two tetrahedrons are said to be non-congruent if one cannot be obtained from the other by a composition of reflections in planes, translations and rotations. (It is not neccessary to prove that the tetrahedrons are not degenerate, i.e. that they have a positive volume).

Let $a_1, a_2, \ldots, a_{2024}$ be positive integers such that $a_{i+1}+1$ is a multiple of $a_i$ for all $i = 1, 2, \ldots , 2024$, with indices taken modulo $2024$. Find the maximum possible value of $a_1 + a_2 + \ldots + a_{2024}$. [i](Proposed by Ivan Chan Guan Yu)[/i]

A soccer coach named $C$ does a header drill with two players $A$ and $B$, but they all forgot to put sunscreen on their foreheads. They solve this issue by dunking the ball into a vat of sunscreen before starting the drill. Coach $C$ heads the ball to $A$, who heads the ball back to $C$, who then heads the ball to $B$, who heads the ball back to $C$; this pattern $CACBCACB\ldots\,$ repeats ad infinitum. Each time a person heads the ball, $1/10$ of the sunscreen left on the ball ends up on the person's forehead. In the limit, what fraction of the sunscreen originally on the ball will end up on the coach's forehead?

What is the area of a circle with a circumference of $8$?

Let $P(x)= x^n+a_{1}x^{n-1}+...+a_{n-1}x+(-1)^{n}$ , $a_{i} \in C$ , $n\geq 2$ with all roots having same modulo. Prove that $P(-1) \in R$

Given two convex polygons, $A_1A_2...A_n$ and $B_1B_2...B_n$ such that $A_1A_2 = B_1B_2$, $A_2A_3 = B_2B_3$,$ ...$, $A_nA_1 = B_nB_1$ and $n - 3$ angles of one polygon are equal to the respective angles of the other. Find whether these polygons are equal.

Let $a$ and $b$ be two distinct positive integers with the same parity. Prove that the fraction $\frac{a!+b!}{2^a}$ is not an integer.