Find the largest positive integer $n$, such that there exists a finite set $A$ of $n$ reals, such that for any two distinct elements of $A$, there exists another element from $A$, so that the arithmetic mean of two of these three elements equals the third one.

All members of the senate were firstly divided into $S$ senate commissions . According to the rules, no commission has less that $5$ senators and every two commissions have different number of senators. After the first session the commissions were closed and new commissions were opened. Some of the senators now are not a part of any commission. It resulted also that every two senators that were in the same commission in the first session , are not any more in the same commission. [b](a)[/b]Prove that at least $4S+10$ senators were left outside the commissions. [b](b)[/b]Prove that this number is achievable. Albanian National Mathematical Olympiad 2010---12 GRADE Question 5.

Let $\{m, n, k\}$ be positive integers. $\{k\}$ coins are placed in the squares of an $m \times n$ grid. A square may contain any number of coins, including zero. Label the $\{k\}$ coins $C_1, C_2, · · · C_k$. Let $r_i$ be the number of coins in the same row as $C_i$, including $C_i$ itself. Let $s_i$ be the number of coins in the same column as $C_i$, including $C_i$ itself. Prove that \[\sum_{i=1}^k \frac{1}{r_i+s_i} \leq \frac{m+n}{4}\]

In the triangle $A,B,C$, let $D$ be the middle of $BC$ and $E$ the projection of $C$ on $AD$. Suppose $\angle ACE = \angle ABC$. Show that the triangle $ABC$ is isosceles or rectangle.

Find all positive integers that can be written in the following way $\frac{m^2 + 20mn + n^2}{m^3 + n^3}$ Also, $m,n$ are relatively prime positive integers.

The least common multiple of $a$ and $b$ is $12$, and the least common multiple of $b$ and $c$ is $15$. What is the least possible value of the least common multiple of $a$ and $c$? $\textbf{(A) }20\qquad\textbf{(B) }30\qquad\textbf{(C) }60\qquad\textbf{(D) }120\qquad \textbf{(E) }180$

Let $p , q, r , s$ be four integers such that $s$ is not divisible by $5$. If there is an integer $a$ such that $pa^3 + qa^2+ ra +s$ is divisible be 5, prove that there is an integer $b$ such that $sb^3 + rb^2 + qb + p$ is also divisible by 5.

The triangle ABC has sides AB = 137, AC = 241, and BC =200. There is a point D, on BC, such that both incircles of triangles ABD and ACD touch AD at the same point E. Determine the length of CD. [asy] pair A = (2,6); pair B = (0,0); pair C = (10,0); pair D = (3.5,0) ; pair E = (3.1,2); draw(A--B); draw(B--C); draw(C--A); draw (A--D); dot ((3.1,1.7)); label ("E", E, dir(45)); label ("A", A, dir(45)); label ("B", B, dir(45)); label ("C", C, dir(45)); label ("D", D, dir(45)); draw(circle((1.8,1.3),1.3)); draw(circle((4.9,1.7),1.75)); [/asy]

Let $N \geq 2$ be an integer, and let $\mathbf a$ $= (a_1, \ldots, a_N)$ and $\mathbf b$ $= (b_1, \ldots b_N)$ be sequences of non-negative integers. For each integer $i \not \in \{1, \ldots, N\}$, let $a_i = a_k$ and $b_i = b_k$, where $k \in \{1, \ldots, N\}$ is the integer such that $i-k$ is divisible by $n$. We say $\mathbf a$ is $\mathbf b$-[i]harmonic[/i] if each $a_i$ equals the following arithmetic mean: \[a_i = \frac{1}{2b_i+1} \sum_{s=-b_i}^{b_i} a_{i+s}.\] Suppose that neither $\mathbf a $ nor $\mathbf b$ is a constant sequence, and that both $\mathbf a$ is $\mathbf b$-[i]harmonic[/i] and $\mathbf b$ is $\mathbf a$-[i]harmonic[/i]. Prove that at least $N+1$ of the numbers $a_1, \ldots, a_N,b_1, \ldots, b_N$ are zero.

Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. Proposed by United Kingdom

Given positive integers $m$ and $n$ so there is a chessboard with $mn$ $1 \times 1$ grids. Colour the grids into red and blue (Grids that have a common side are not the same colour and the grid in the left corner at the bottom is red). Now the diagnol that goes from the left corner at the bottom to the top right corner is coloured into red and blue segments (Every segment has the same colour with the grid that contains it). Find the sum of the length of all the red segments.

On a map, a 12-centimeter length represents $72$ kilometers. How many kilometers does a 17-centimeter length represent? $\textbf{(A)}\ 6\qquad \textbf{(B)}\ 102\qquad \textbf{(C)}\ 204\qquad \textbf{(D)}\ 864\qquad \textbf{(E)}\ 1224$

Let $M=(0,1)\cap \mathbb Q$. Determine, with proof, whether there exists a subset $A\subset M$ with the property that every number in $M$ can be uniquely written as the sum of finitely many distinct elements of $A$.

In an acute-angled triangle $ABC$ with orthocenter $H$, the line $AH$ cuts $BC$ at point $A_1$. Let $\Gamma$ be a circle centered on side $AB$ tangent to $AA_1$ at point $H$. Prove that $\Gamma$ is tangent to the circumscribed circle of triangle $AMA_1$, where $M$ is the midpoint of $AC$.

Find all family $\mathcal{F}$ of subsets of $[n]$ such that for any nonempty subset $X\subseteq [n]$, exactly half of the elements $A\in \mathcal{F}$ satisfies that $|A\cap X|$ is even.

Six distinguishable players are participating in a tennis tournament. Each player plays one match of tennis against every other player. The outcome of each tennis match is a win for one player and a loss for the other players; there are no ties. Suppose that whenever $A$ and $B$ are players in the tournament for which $A$ won (strictly) more matches than $B$ over the course of the tournament, it is also the case that $A$ won the match against $B$ during the tournament. In how many ways could the tournament have gone?