Given that the polynomial $P(x) = x^5 - x^2 + 1$ has $5$ roots $r_1, r_2, r_3, r_4, r_5$. Find the value of the product $Q(r_1)Q(r_2)Q(r_3)Q(r_4)Q(r_5)$, where $Q(x) = x^2 + 1$.

Let $A$ be a point outside the circle $\omega$. Two points $B, C$ lie on $\omega$ such that $AB, AC$ are tangent to $\omega$. Let $D$ be any point on $\omega$ ($D$ is neither $B$ nor $C$) and $M$ the foot of perpendicular from $B$ to $CD$. The line through $D$ and the midpoint of $BM$ meets $\omega$ again at $P$. Prove that $AP \perp CP$

Show that there are infinitely many positive integers $n$ such that $n$ has at least two prime divisors and $20^n + 16^n$ is divisible by $n^2$.

On a checkered square $10 \times 10$ the cells of the upper left $5 \times 5$ square are black and all the other cells are white. What is the maximal $n$ such that the original square can be dissected (along the borders of the cells) into $n$ polygons such that in each of them the number of black cells is three times less than the number of white cells? (The polygons need not be congruent or even equal in area.)

Let $P_i(x) = x^2 + b_i x + c_i , i = 1,2, ..., n$ be pairwise distinct polynomials of degree $2$ with real coefficients so that for any $0 \le i < j \le n , i, j \in N$, the polynomial $Q_{i,j}(x) = P_i(x) + P_j(x)$ has only one real root. Find the greatest possible value of $n$.

A circle with center $O$ passes through points $A$ and $C$ and intersects the sides $AB$ and $BC$ of triangle $ABC$ at points $K$ and $N$, respectively. The circumcircles of triangles $ABC$ and $KBN$ meet at distinct points $B$ and $M$. Prove that $\angle OMB = 90^o$.

Let $ m $ and $ n $ be odd integers such that $n^2 - 1 $ is divisible by $m^2 + 1 - n^2 $. Prove that $ |m^2 + 1 - n^2 | $ is a perfect square.

How many ways are there to color the vertices of a square with $n$ colors $1,2, .. ., n$. (The colorings must be different so that we can’t get one from the other by a rotation.)

Let $ a > b > c > d $ be positive integers such that \begin{align*} a^2 + ac - c^2 = b^2 + bd - d^2 \end{align*}Prove that $ ab + cd $ is a composite number.

Let $ABC$ be a triangle and $I$ its incenter. The point $D$ is on segment $BC$ and the circle $\omega$ is tangent to the circumcirle of triangle $ABC$ but is also tangent to $DC, DA$ at $E, F$, respectively. Prove that $E, F$ and $I$ are collinear.

Does there exist a polynomial $P(x)$ with integral coefficients such that a) $P(\sqrt[3]{25 }+ \sqrt[3]{5}) = 220\sqrt[3]{25} + 284\sqrt[3]{5}$ ? b) $P(\sqrt[3]{25 }+ \sqrt[3]{5}) = 1184\sqrt[3]{25} + 1210\sqrt[3]{5}$ ?

On a chessboard $5 \times 9$ squares, the following game is played. Initially, a number of frogs are randomly placed on some of the squares, no square containing more than one frog. A turn consists of moving all of the frogs subject to the following rules: $\bullet$ Each frog may be moved one square up, down, left, or right; $\bullet$ If a frog moves up or down on one turn, it must move left or right on the next turn, and vice versa; $\bullet$ At the end of each turn, no square can contain two or more frogs. The game stops if it becomes impossible to complete another turn. Prove that if initially $33$ frogs are placed on the board, the game must eventually stop. Prove also that it is possible to place $32$ frogs on the board so that the game can continue forever.

Given a polynomial $P(x) = a_nx^n + a_{n-1}x^{n-1} + ...+ a_1x + a_0$ of real coefficients. Suppose that $P(x)$ has $n$ real roots (not necessarily distinct), and there exists a positive integer $k$ such that $a_k = a_{k-1} = 0$. Prove that $P(x)$ has a real root of multiplicity $k + 1$.

Let $ABC$ be a triangle with $AB \ne AC$. The incirle of triangle $ABC$ is tangent to $BC, CA, AB$ at $D, E, F$, respectively. The perpendicular line from $D$ to $EF$ intersects $AB$ at $X$. The second intersection point of circumcircles of triangles $AEF$ and $ABC$ is $T$. Prove that $TX \perp T F$

For any positive integer $n$, show that there exists a positive integer $m$ such that $n$ divides $2016^m + m$.

There are There are $64$ towns in a country, and some pairs of towns are connected by roads but we do not know these pairs. We may choose any pair of towns and find out whether they are connected by a road. Our aim is to determine whether it is possible to travel between any two towns using roads. Prove that there is no algorithm which would enable us to do this in less than $2016$ questions. but we do not know these pairs. We may choose any pair of towns and find out whether they are connected by a road. Our aim is to determine whether it is possible to travel between any two towns using roads. Prove that there is no algorithm which would enable us to do this in less than $2016$ questions.

Given two non-constant polynomials $P(x),Q(x)$ with real coefficients. For a real number $a$, we define $$P_a= \{z \in C : P(z) = a\}, Q_a =\{z \in C : Q(z) = a\}$$Denote by $K$ the set of real numbers $a$ such that $P_a = Q_a$. Suppose that the set $K$ contains at least two elements, prove that $P(x) = Q(x)$.

Let $I_a$ be the excenter of triangle $ABC$ with respect to $A$. The line $AI_a$ intersects the circumcircle of triangle ABC at $T$. Let $X$ be a point on segment $TI_a$ such that $X I_a^2 = XA \cdot X T$ The perpendicular line from $X$ to $BC$ intersects $BC$ at $A'$. Define $B'$ and $C'$ in the same way. Prove that $AA',BB'$ and $CC'$ are concurrent.

Let $d$ be a positive integer. Show that for every integer $S$, there exist a positive integer $n$ and a sequence $a_1, ..., a_n \in \{-1, 1\}$ such that $S = a_1(1 + d)^2 + a_2(1 + 2d)^2 + ... + a_n(1 + nd)^2$.

Find all natural numbers $n\geq 3$ satisfying one can cut a convex $n$-gon into different triangles along some of the diagonals (None of these diagonals intersects others at any point other than vertices) and the number of diagonals are used at each vertex is even.

Let $ p $ and $ q $ be given primes and the sequence $ \{ p_n \}_{n = 1}^{\infty} $ defined recursively as follows: $ p_1 = p $, $ p_2 = q $, and $ p_{n+2} $ is the largest prime divisor of the number $( p_n + p_{n + 1} + 2016) $ for all $ n \geq 1 $. Prove that this sequence is bounded. That is, there exists a positive real number $ M $ such that $ p_n < M $ for all positive integers $ n $.

Let $I$ be the incenter of an acute triangle $ABC$. Assume that $K_1$ is the point such that $AK_1 \perp BC$ and the circle with center $K_1$ of radius $K_1A$ is internally tangent to the incircle of triangle $ABC$ at $A_1$. The points $B_1, C_1$ are defined similarly. a) Prove that $AA_1, BB_1, CC_1$ are concurrent at a point $P$. b) Let $\omega_1,\omega_2,\omega_3$ be the excircles of triangle $ABC$ with respect to $A, B, C$, respectively. The circles $\gamma_1,\gamma_2\gamma_3$ are the reflections of $\omega_1,\omega_2,\omega_3$ with respect to the midpoints of $BC, CA, AB$, respectively. Prove that P is the radical center of $\gamma_1,\gamma_2,\gamma_3$.

Find all integers $n$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying $$P(\sqrt[3]{n^2} + \sqrt[3]{ n}) = 2016n + 20\sqrt[3]{n^2} + 16\sqrt[3]{n}$$

Given six three-element subsets of the set $X$ with at least $5$ elements, show that it is possible to color the elements of $X$ in two colors such that none of the given subsets is all in one color.