Triangle $ABC$ is inscribed in circle $\omega$. Point $P$ lies inside triangle $ABC$.Lines $AP,BP$ and $CP$ intersect $\omega$ again at points $A_1$, $B_1$ and $C_1$ (other than $A, B, C$), respectively. The tangent lines to $\omega$ at $A_1$ and $B_1$ intersect at $C_2$.The tangent lines to $\omega$ at $B_1$ and $C_1$ intersect at $A_2$. The tangent lines to $\omega$ at $C_1$ and $A_1$ intersect at $B_2$. Prove that the lines $AA_2,BB_2$ and $CC_2$ are concurrent.

Let $S = f\{0.1. 2.3,...\}$ be the set of the non-negative integers. Find all strictly increasing functions $f : S \to S$ such that $n + f(f(n)) \le 2f(n)$ for every $n$ in $S$

A Saudi company has two offices. One office is located in Riyadh and the other in Jeddah. To insure the connection between the two offices, the company has designated from each office a number of correspondents so that : (a) each pair of correspondents from the same office share exactly one common correspondent from the other office. (b) there are at least $10$ correspondents from Riyadh. (c) Zayd, one of the correspondents from Jeddah, is in contact with exactly $8$ correspondents from Riyadh. What is the minimum number of correspondents from Jeddah who are in contact with the correspondent Amr from Riyadh?

Determine whether it is possible to place the integers $1, 2,...,2012$ in a circle in such a way that the $2012$ products of adjacent pairs of numbers leave pairwise distinct remainders when divided by $2013$.

Find the maximum and the minimum values of $S = (1 - x_1)(1 -y_1) + (1 - x_2)(1 - y_2)$ for real numbers $x_1, x_2, y_1,y_2$ with $x_1^2 + x_2^2 = y_1^2 + y_2^2 = 2013$.

Let $ABC$ be an acute triangle, and let $AA_1, BB_1$, and $CC_1$ be its altitudes. Segments $AA_1$ and $B_1C_1$ meet at point $K$. The perpendicular bisector of segment $A_1K$ intersects sides $AB$ and $AC$ at $L$ and $M$, respectively. Prove that points $A,A_1, L$, and $M$ lie on a circle.

For a positive integer $n$, we consider all its divisors (including $1$ and itself). Suppose that $p\%$ of these divisors have their unit digit equal to $3$. (For example $n = 117$, has six divisors, namely $1,3,9,13,39,117$. Two of these divisors namely $3$ and $13$, have unit digits equal to $3$. Hence for $n = 117$, $p =33.33...$). Find, when $n$ is any positive integer, the maximum possible value of $p$.

Determine if there exists an infinite sequence of positive integers $a_1,a_2, a_3, ...$ such that (i) each positive integer occurs exactly once in the sequence, and (ii) each positive integer occurs exactly once in the sequence $ |a_1 - a_2|, |a_2 - a_3|, ..., |a+k - a_{k+1}|, ...$

Adel draws an $m \times n$ grid of dots on the coordinate plane, at the points of integer coordinates $(a,b)$ where $1 \le a \le m$ and $1 \le b \le n$. He proceeds to draw a closed path along $k$ of these dots, $(a_1, b_1)$,$(a_2,b_2)$,...,$(a_k,b_k)$, such that $(a_i,b_i)$ and $(a_{i+1}, b_{i+1})$ (where $(a_{k+1}, b_{k+1}) = (a_1, b_1)$) are $1$ unit apart for each $1 \le i \le k$. Adel makes sure his path does not cross itself, that is, the $k$ dots are distinct. Find, with proof, the maximum possible value of $k$ in terms of $m$ and $n$.

Given an integer $n \ge 2$, determine the number of ordered $n$-tuples of integers $(a_1, a_2,...,a_n)$ such that (a) $a_1 + a_2 + .. + a_n \ge n^2$ and (b) $a_1^2 + a_2^2 + ... + a_n^2 \le n^3 + 1$

Let $ABC$ be an acute triangle, $M$ be the midpoint of $BC$ and $P$ be a point on line segment $AM$. Lines $BP$ and $CP$ meet the circumcircle of $ABC$ again at $X$ and $Y$ , respectively, and sides $AC$ at $D$ and $AB$ at $E$, respectively. Prove that the circumcircles of $AXD$ and $AYE$ have a common point $T \ne A$ on line $AM$.

Find all polynomials $p(x)$ with integer coefficients such that for each positive integer $n$, the number $2^n - 1$ is divisible by $p(n)$.