Let $J_A$ and $J_B$ be the $A$-excenter and $B$-excenter of $\triangle ABC$. Consider a chord $\overline{PQ}$ of circle $ABC$ which is parallel to $AB$ and intersects segments $\overline{AC}$ and $\overline{BC}$. If lines $AB$ and $CP$ intersect at $R$, prove that $$\angle J_AQJ_B+\angle J_ARJ_B=180^\circ.$$

Given a positive integer $k,$ prove that for any integer $n \geq 20k,$ there exist $n - k$ pairwise distinct positive integers whose squares add up to $n(n + 1)(2n + 1)/6.$ [i]The Problem Selection Committee[/i]

Find all positive integers $n$ such that $n!$ can be written as the product of two Fibonacci numbers.

Let $x_i,$ $1 \leq i \leq n$ be real numbers with $n \geq 3.$ Let $p$ and $q$ be their symmetric sum of degree $1$ and $2$ respectively. Prove that: i) $p^2 \cdot \frac{n-1}{n}-2q \geq 0$ ii) $\left|x_i - \frac{p}{n}\right| \leq \sqrt{p^2 - \frac{2nq}{n-1}} \cdot \frac{n-1}{n}$ for every meaningful $i$.

Given the polynomial sequence $(p_{n}(x))$ satisfying $p_{1}(x)=x^{2}-1$, $p_{2}(x)=2x(x^{2}-1)$, and $p_{n+1}(x)p_{n-1}(x)=(p_{n}(x)^{2}-(x^{2}-1)^{2}$, for $n\geq 2$, let $s_{n}$ be the sum of the absolute values of the coefficients of $p_{n}(x)$. For each $n$, find a non-negative integer $k_{n}$ such that $2^{-k_{n}}s_n$ is odd.

Let $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. A point $D$ is chosen on the internal bisector of $\angle ACB$ so that the points $D$ and $C$ are separated by $AB$. A circle $\omega$ centered at $D$ is tangent to the segment $AB$ at $E$. The tangents to $\omega$ through $C$ meet the segment $AB$ at $K$ and $L$, where $K$ lies on the segment $AL$. A circle $\Omega_1$ is tangent to the segments $AL, CL$, and also to $\Omega$ at point $M$. Similarly, a circle $\Omega_2$ is tangent to the segments $BK, CK$, and also to $\Omega$ at point $N$. The lines $LM$ and $KN$ meet at $P$. Prove that $\angle KCE = \angle LCP$. Poland

Let $P$ be a non-zero polynomial with non-negative real coefficients, let $N$ be a positive integer, and let $\sigma$ be a permutation of the set $\{1,2,...,n\}$. Determine the least value the sum \[\sum_{i=1}^{n}\frac{P(x_i^2)}{P(x_ix_{\sigma(i)})}\] may achieve, as $x_1,x_2,...,x_n$ run through the set of positive real numbers. [i]Fedor Petrov[/i]

Let $g(t)$ and $h(t)$ be real, continuous functions for $t\geq 0.$ Show that any function $v(t)$ satisfying the differential inequality $$\frac{dv}{dt}+g(t)v \geq h(t),\;\; v(t)=c,$$ satisfies the further inequality $v(t)\geq u(t),$ where $$\frac{du}{dt}+g(t)u = h(t),\;\; u(t)=c.$$ From this, conclude that for sufficiently small $t>0,$ the solution of $$\frac{dv}{dt}+g(t)v = v^2 ,\;\; v(t)=c$$ may be written $$v=\max_{w(t)} \left( c e^{- \int_{0}^{t} |g(s)-2w(s)| \, ds} -\int_{0}^{t} e^{-\int_{0}^{t} |g(s')-2w(s')| \, ds'} w(s)^{2} ds \right),$$ where the maximum is over all continuous functions $w(t)$ defined over some $t$-interval $[0,t_0 ].$

The integer sequence $(s_i)$ "having pattern 2016'" is defined as follows:\\ $\circ$ The first member $s_1$ is 2.\\ $\circ$ The second member $s_2$ is the least positive integer exceeding $s_1$ and having digit 0 in its decimal notation.\\ $\circ$ The third member $s_3$ is the least positive integer exceeding $s_2$ and having digit 1 in its decimal notation.\\ $\circ$ The third member $s_3$ is the least positive integer exceeding $s_2$ and having digit 6 in its decimal notation.\\ The following members are defined in the same way. The required digits change periodically: $2 \rightarrow 0 \rightarrow 1 \rightarrow 6 \rightarrow 2 \rightarrow 0 \rightarrow \ldots$. The first members of this sequence are the following: $2; 10; 11; 16; 20; 30; 31; 36; 42; 50$. What are the 4 numbers that immediately follow $s_k = 2016$ in this sequence?

One morning, each member of Manjul’s family drank an $8$-ounce mixture of coffee and milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Manjul drank $1/7$-th of the total amount of milk and $2/17$-th of the total amount of coffee. How many people are there in Manjul’s family?

Determine all real numbers $q$ for which the equation $x^4 -40x^2 +q = 0$ has four real solutions which form an arithmetic progression

In rectangle $ ABCD$, $ AB\equal{}5$ and $ BC\equal{}3$. Points $ F$ and $ G$ are on $ \overline{CD}$ so that $ DF\equal{}1$ and $ GC\equal{}2$. Lines $ AF$ and $ BG$ intersect at $ E$. Find the area of $ \triangle{AEB}$. [asy]unitsize(6mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair A=(0,0), B=(5,0), C=(5,3), D=(0,3), F=(1,3), G=(3,3); pair E=extension(A,F,B,G); draw(A--B--C--D--A--E--B); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); label("$D$",D,NW); label("$E$",E,N); label("$F$",F,SE); label("$G$",G,SW); label("$B$",B,SE); label("1",midpoint(D--F),N); label("2",midpoint(G--C),N); label("3",midpoint(B--C),E); label("3",midpoint(A--D),W); label("5",midpoint(A--B),S);[/asy]$ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ \frac{21}{2} \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ \frac{25}{2} \qquad \textbf{(E)}\ 15$

Consider the points $P$ =$(0,0)$,$Q$ = $(1,0)$, $R$= $(2,0)$, $S$ =$(3,0)$ in the $xy$-plane. Let $A,B,C,D$ be four finite sets of colours(not necessarily distinct nor disjoint). In how many ways can $P,Q,R$ be coloured bu colours in $A,B,C$ respectively if adjacent points have to get different colours? In how many ways can $P,Q,R,S$ be coloured by colours in $A,B,C,D$ respectively if adjacent points have to get different colors?

Let $(a_i)_{i\in \mathbb{N}}$ a sequence of positive integers, such that for any finite, non-empty subset $S$ of $\mathbb{N}$, the integer$$\Pi_{k\in S} a_k -1$$is prime. Prove that the number of $a_i$'s with $i\in \mathbb{N}$ such that $a_i$ has less than $m$ distincts prime factors is finite.

An equilateral triangle $ABC$ is given. On the line through $B$ parallel to $AC$ there is a point $D$, such that $D$ and $C$ are on the same side of the line $AB$. The perpendicular bisector of $CD$ intersects the line $AB$ in $E$. Prove that triangle $CDE$ is equilateral.

In triangle $ABC$ ,$M,N,K$ are midpoints of sides $BC,AC,AB$,respectively.Construct two semicircles with diameter $AB,AC$ outside of triangle $ABC$.$MK,MN$ intersect with semicircles in $X,Y$.The tangents to semicircles at $X,Y$ intersect at point $Z$.Prove that $AZ \perp BC$.(Mehdi E'tesami Fard)

Given the pyramid $ABCD$. Let $O$ be the midpoint of edge $AC$. Given that $DO$ is the height of the pyramid, $AB=BC=2DO$ and the angle $ABC$ is right. Cut this pyramid into $8$ equal and similar to it pyramids.

A positive number $a$ is given, such that $a$ could be expressed as difference of two inverses of perfect squares ($a=\frac1{n^2}-\frac1{m^2}$). Is it possible for $2a$ to be expressed as difference of two perfect squares?

Let $ABC$ be a triangle. Let $A_1$, $A_2$ be points on $AB$ and $AC$ respectively such that $A_1A_2 \parallel BC$ and the circumcircle of $\triangle AA_1A_2$ is tangent to $BC$ at $A_3$. Define $B_3$, $C_3$ similarly. Prove that $AA_3$, $BB_3$, and $CC_3$ are concurrent. [i]Carl Lian.[/i]

$ABC$ is a triangle, with sides $a$, $b$, $c$ , circumradius $R$, and exradii $r_a$, $r_b$, $r_c$If $2R\leq r_a$, show that $a > b$, $a > c$, $2R > r_b$, and $2R > r_c$.

Josie and Kevin are each thinking of a two digit positive integer. Josie’s number is twice as big as Kevin’s. One digit of Kevin’s number is equal to the sum of digits of Josie’s number. The other digit of Kevin’s number is equal to the difference between the digits of Josie’s number. What is the sum of Kevin and Josie’s numbers?

Suppose $xy-5x+2y=30$, where $x$ and $y$ are positive integers. Find the sum of all possible values of $x$

Given is a triangle $ABC$ and on its sides the triangles $ABM, BCN$ and $CAP$ are build such that $\angle AMB = 150^o$, $AM = MB$, $\angle CAP = \angle CBN = 30^o$, $\angle ACP = \angle BCN = 45^o$. Prove that the triangle $MNP$ is an equilateral triangle.

Evaluate the sum $$1 + 2 + 3 - 4 - 5 + 6 + 7 + 8 - 9 - 1 0 + . . . - 2010$$ , where each three consecutive signs $+$ are followed by two signs $-$.

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfy the equality, \[f(x+f(y))=f(x-f(y))+4xf(y)\] for any $x,y\in\mathbb{R}$.