Let $ABC$ be a triangle and $D$ a point on the side $BC$. Point $E$ is the symmetric of $D$ with respect to $AB$. Point $F$ is the symmetric of $E$ with respect to $AC$. Point $P$ is the intersection of line $DF$ with line $AC$. Prove that the quadrilateral $AEDP$ is cyclic. (Malik Talbi)

How many polynomials $P$ of integer coefficients and degree at most $4$ satisfy $0 \le P(x) < 72$ for all $x\in \{0, 1, 2, 3, 4\}$? Harvard-MIT Mathematics Tournament 2011

Find all integer solutions of the equation $x^2y^5 - 2^x5^y = 2015 + 4xy$. (Malik Talbi)

We color each unit square of a $8\times 8$ table into green or blue such that there are $a$ green unit squares in each $3 \times 3$ square and $b$ green unit squares in each $2 \times 4$ rectangle. Find all possible values of $(a, b)$. (Le Anh Vinh)

Let $ABC$ be a triangle and $D$ a point on the side $BC$. The tangent line to the circumcircle of the triangle $ABD$ at the point $D$ intersects the side $AC$ at $E$. The tangent line to the circumcircle of the triangle $ACD$ at the the point $D$ intersects the side $AB$ at $F$. Prove that the point $A$ and the circumcenters of the triangles $ABC$ and $DEF$ are collinear. (Malik Talbi)

Find all functions $f : R \to R$ that satisfy $f(x + y^2 - f(y)) = f(x)$ for all $x,y \in R$. (Vo Quoc Ba Can)

Find all integer solutions of the equation $14^x - 3^y = 2015$. (Malik Talbi)

How many sequences of integers $1 \le a_1 \le a_2\le ... \le a_{11 }\le 2015$ that satisfy $a_i \equiv i^2$ (mod $12$) for all $1 \le i \le 11$ are there? (Le Anh Vinh)

Let $ABC$ be a triangle, $I$ its incenter, and $D$ a point on the arc $BC$ of the circumcircle of $ABC$ not containing $A$. The bisector of the angle $\angle ADB$ intesects the segment $AB$ at $E$. The bisector of the angle $\angle CDA$ intesects the segment $AC$ at $F$. Prove that the points $E, F,I$ are collinear. (Malik Talbi)

Prove that the polynomial $P(X) = (X^2-12X +11)^4+23$ can not be written as the product of three non-constant polynomials with integer coefficients. (Le Anh Vinh)

Let $(a_n)_{n\ge0}$ be a sequence of positive integers such that $a^2_n$ divides $a_{n-1}a_{n+1}$, for all $n \ge 1$. Prove that if there exists an integer $k \ge 2$ such that $a_k$ and $a_1$ are relatively prime, then $a_1$ divides $a_0$. (Malik Talbi)

There are $22$ chairs in a round table. Find the minimum n such that for any group of $n$ people sitting in the table, we always can find two people with exactly $2$ or $8$ chairs between them. (Le Anh Vinh)