Prove that a triangle with angles $\alpha, \beta, \gamma$, circumradius $R$, and area $A$ satisfies \[\tan \frac{ \alpha}{2}+\tan \frac{ \beta}{2}+\tan \frac{ \gamma}{2} \leq \frac{9R^2}{4A}.\] [hide="Remark."]Remark. Can we determine [i]all[/i] of equality cases ?[/hide]

Call a polynomial $P$ [i]prime-covering[/i] if for every prime $p$, there exists an integer $n$ for which $p$ divides $P(n)$. Determine the number of ordered triples of integers $(a,b,c)$, with $1\leq a < b < c \leq 25$, for which $P(x)=(x^2-a)(x^2-b)(x^2-c)$ is prime-covering.

For a positive integer $N>1$ with unique factorization $N=p_1^{\alpha_1}p_2^{\alpha_2}\dotsb p_k^{\alpha_k}$, we define \[\Omega(N)=\alpha_1+\alpha_2+\dotsb+\alpha_k.\] Let $a_1,a_2,\dotsc, a_n$ be positive integers and $p(x)=(x+a_1)(x+a_2)\dotsb (x+a_n)$ such that for all positive integers $k$, $\Omega(P(k))$ is even. Show that $n$ is an even number.

Let $a_1,a_2,\dots,a_m$ be a finite sequence of positive integers. Prove that there exist nonnegative integers $b,c,$ and $N$ such that $$\left\lfloor \sum_{i=1}^m \sqrt{n+a_i} \right\rfloor =\left\lfloor \sqrt{bn+c} \right\rfloor$$ holds for all integers $n>N.$ [i]Proposed by Carl Schildkraut[/i]

Points $A_1, A_2,\ldots, A_n$ are found on a circle in this order. Each point $A_i$ has exactly $i$ coins. A move consists in taking two coins from two points (may be the same point) and moving them to adjacent points (one move clockwise and another counter-clockwise). Find all possible values of $n$ for which it is possible after a finite number of moves to obtain a configuration with each point $A_i$ having $n+1-i$ coins.

Do there exist infinitely many triplets of positive integers $(a, b, c)$ such that the following two conditions hold: 1. $\gcd(a, b, c) = 1$; 2. $a+b+c, a^2+b^2+c^2$ and $abc$ are all perfect squares? [i](Proposed by Ivan Chan Guan Yu)[/i]

$p$ is a prime number that is greater than $2$. Let $\{ a_{n}\}$ be a sequence such that $ na_{n+1}= (n+1) a_{n}-\left( \frac{p}{2}\right)^{4}$. Show that if $a_{1}=5$, the $16 \mid a_{81}$.

The circles $\omega_1$ and $\omega_2$ intersect at two points $A$ and $B$. On the circle $\omega_2$, point $C$ is taken in such a way that $CA$ is tangent to the circle $\omega_1$. Through point $A$, a straight line is drawn that intersects the circles $\omega_1$, and $\omega_2$ at points $M$ and $N$, respectively , different from point $A$. Point $P$ is the midpoint of the segment $AC$, $Q$ is the midpoint of $MN$, and $S$ is the intersection point of the line $BQ$ with the circle $\omega_1$, different from point $B$. Prove that the lines $AS$ and $PQ$ are parallel.

Let $f(n)$ and $g(n)$ be polynomials of degree $2014$ such that $f(n)+(-1)^ng(n)=2^n$ for $n=1,2,\ldots,4030$. Find the coefficient of $x^{2014}$ in $g(x)$.

Let $ABC$ be a triangle such that $\vert AB\vert \ne \vert AC\vert$. Prove that there exists a point $D \ne A$ on its circumcircle satisfying the following property: For any points $M, N$ outside the circumcircle on the rays $AB$ and $AC$, respectively, satisfying $\vert BM\vert=\vert CN\vert$, the circumcircle of $AMN$ passes through $D$.

Compute $$\sum^{\infty}_{k=1} \frac{(-1)^k}{(2k - 1)(2k + 1)}$$

In triangle $ABC$ the angle bisector $AK$ is perpendicular on the median is $CL$. Prove that in the triangle $BKL$ also one of angle bisectors are perpendicular to one of the medians.

Let $m<n$ be two positive integers and $x_m<x_{m+1}<\cdots<x_n$ be a sequence of rational numbers. Suppose that $kx_k$ is an integer for all integers $k$ which $m\leq k\leq n$. Prove that $$x_n-x_m\geq \frac{1}{m}-\frac{1}{n}.$$

There are $1001$ stacks of coins $S_1, S_2, \dots, S_{1001}$. Initially, stack $S_k$ has $k$ coins for each $k = 1,2,\dots,1001$. In an operation, one selects an ordered pair $(i,j)$ of indices $i$ and $j$ satisfying $1 \le i < j \le 1001$ subject to two conditions: [list] [*]The stacks $S_i$ and $S_j$ must each have at least $1$ coin. [*]The ordered pair $(i,j)$ must [i]not[/i] have been selected before. [/list] Then, if $S_i$ and $S_j$ have $a$ coins and $b$ coins respectively, one removes $\gcd(a,b)$ coins from each stack. What is the maximum number of times this operation could be performed? [i]Galin Totev[/i]

Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that \[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\] [i]Proposed by Nazar Serdyuk, Ukraine[/i]

The year $1978$ was “peculiar” in that the sum of the numbers formed with the first two digits and the last two digits is equal to the number formed with the middle two digits, i.e., $19+78=97$. What was the last previous peculiar year, and when will the next one occur?

Let $ABC$ be a triangle. The ex-circles touch sides $BC, CA$ and $AB$ at points $U, V$ and $W$, respectively. Be $r_u$ a straight line that passes through $U$ and is perpendicular to $BC$, $r_v$ the straight line that passes through $V$ and is perpendicular to $AC$ and $r_w$ the straight line that passes through W and is perpendicular to $AB$. Prove that the lines $r_u$, $r_v$ and $r_w$ pass through the same point.

Let $m \neq -1$ be a real number. Consider the quadratic equation $$ (m + 1)x^2 + 4mx + m - 3 =0. $$ Which of the following must be true? $\quad\rm(I)$ Both roots of this equation must be real. $\quad\rm(II)$ If both roots are real, then one of the roots must be less than $-1.$ $\quad\rm(III)$ If both roots are real, then one of the roots must be larger than $1.$ $$ \mathrm a. ~ \text{Only} ~(\mathrm I)\rm \qquad \mathrm b. ~(I)~and~(II)\qquad \mathrm c. ~Only~(III) \qquad \mathrm d. ~Both~(I)~and~(III) \qquad \mathrm e. ~(I), (II),~and~(III) $$

For positive real numbers $x$ and $y$, define their special mean to be average of their arithmetic and geometric means. Find the total number of pairs of integers $(x, y)$, with $x \le y$, from the set of numbers $\{1,2,...,2016\}$, such that the special mean of $x$ and $y$ is a perfect square.

Let $n$ and $k$ be natural numbers and $p$ a prime number. Prove that if $k$ is the exact exponent of $p$ in $2^{2^n}+1$ (i.e. $p^k$ divides $2^{2^n}+1$, but $p^{k+1}$ does not), then $k$ is also the exact exponent of $p$ in $2^{p-1}-1$.

A point is chosen inside a regular $k$-gon in such a way that its orthogonal projections on to the sides all meet the respective sides at interior points. These points divide the sides into $2k$ segments. Let these segments be enumerated consecutively by the numbers $1,2, 3, ... ,2k$. Prove that the sum of the lengths of the segments having even numbers equals the sum of the segments having odd numbers. (A Andjans, Riga)

In an island, there exist 7 towns and a railway system which connected some of the towns. Every railway segment connects 2 towns, and in every town there exists at least 3 railway segments that connects the town to another towns. Prove that there exists a route that visits 4 different towns once and go back to the original town. (Example: $ A\minus{}B\minus{}C\minus{}D\minus{}A$)

Circles ${{w} _ {1}}$ and ${{w} _ {2}}$ with centers ${{O} _ {1}}$ and ${{O} _ {2}}$ intersect at points $A$ and $B$, respectively. The line ${{O} _ {1}} {{O} _ {2}}$ intersects ${{w} _ {1}}$ at the point $Q$, which does not lie inside the circle ${{w} _ {2}}$, and ${{w} _ {2}}$ at the point $X$ lying inside the circle ${{w} _ {1} }$. Around the triangle ${{O} _ {1}} AX$ circumscribe a circle ${{w} _ {3}}$ intersecting the circle ${{w} _ {1}}$ for the second time in point $T$. The line $QT$ intersects the circle ${{w} _ {3}}$ at the point $K$, and the line $QB$ intersects ${{w} _ {2}}$ the second time at the point $H$. Prove that a) points $T, \, \, X, \, \, B$ lie on one line; b) points $K, \, \, X, \, \, H$ lie on one line. (Vadym Mitrofanov)

Two circles, having their centers in A and B, intersect at points M and N. The radii AP and BQ are parallel and are in different semi-planes determined of the line AB. If the external common tangent intersect AB in D, and PQ intersects AB at C, prove that the <CND is right.

Let $a$, $b$ and $c$ be positive real numbers such that $abc=1$. Prove the inequality: $\frac{ab}{a^5+b^5+ab} +\frac{bc}{b^5+c^5+bc}+\frac{ac}{c^5+a^5+ac}\leq 1$