Prove that $$\left( \frac{1}{\sqrt1+\sqrt2}+\frac{1}{\sqrt2+\sqrt3}+\frac{1}{\sqrt3+\sqrt4}+...+\frac{1}{\sqrt{2015}+\sqrt{2016}}\right)^2(2017+24\sqrt{14})=2015^2$$

Find all integer solutions of the equation $p (x + y) = xy$, where $p$ is a prime number.

Let $x$ and $y$ be two positive real numbers, such that $x + y = 1$. Prove that $$\left(1 +\frac{1}{x}\right)\left(1 +\frac{1}{y}\right) \ge 9$$

Let $\vartriangle ABC$ be isosceles with $AB = AC$. Let $\omega$ be its circumscribed circle and $O$ its circumcenter. Let $D$ be the second intersection of $CO$ with $\omega$. Take a point $E$ in $AB$ such that $DE \parallel AC$ and suppose that $AE: BE = 2: 1$. Show that $\vartriangle ABC$ is equilateral.

Consider $\vartriangle ABC$ right at $B, F$ a point such that $B - F - C$ and $AF$ bisects $\angle BAC$, $I$ a point such that $A - I - F$ and CI bisect $\angle ACB$, and $E$ a point such that $A- E - C$ and $AF \perp EI$. If $AB = 4$ and $\frac{AI}{IF}={4}{3}$ , determine $AE$. Notation: $A-B-C$ means than points $A,B,C$ are collinear in that order i.e. $ B$ lies between $ A$ and $C$.

Let the $JHIZ$ be a rectangle and let $A$ and $C$ be points on the sides $ZI$ and $ZJ$, respectively. The perpendicular from $A$ on $CH$ intersects line $HI$ at point $X$ and perpendicular from $C$ on $AH$ intersects line $HJ$ at point $Y$. Show that points $X, Y$, and $Z$ are collinear.

With $21$ tiles, some white and some black, a $3 \times 7$ rectangle is formed. Show that there are always four tokens of the same color located at the vertices of a rectangle.

There are $2016$ participants in the Olcotournament of chess. It is known that in any set of four participants, there is one of them who faced the other three. Prove there is at least $2013$ participants who faced everyone else.

Consider an arithmetic progression made up of $100$ terms. If the sum of all the terms of the progression is $150$ and the sum of the even terms is $50$, find the sum of the squares of the $100$ terms of the progression.

Find all $x \in R$ such that $$ x - \left[ \frac{x}{2016} \right]= 2016$$, where $[k]$ represents the largest smallest integer or equal to $k$.

Determine all positive integers $a$ and $b$ for which $a^4 + 4b^4$ be a prime number.

Find all nonnegative integers $a$ and $b$ that satisfy the equation $$3 \cdot 2^a + 1 = b^2.$$

Find all solutions of the system $\sqrt[3]{\frac{yz^4}{x^2}}+2wx=0 $ $\sqrt[3]{\frac{xz^4}{y}}+5wy=0 $ $\sqrt[3]{\frac{xy}{x}}+7wz^{-1/3}=0$ $x^{12}+\frac{125}{4}y^5+\frac{343}{2}z^4=16$ where $x, y, z \ge 0$ and $w \in R$ PSI attached the system, in case I have any typos

The initial number of inhabitants of a city of more than $150$ inhabitants is a perfect square. With an increase of $1000$ inhabitants it becomes a perfect square plus a unit. After from another increase of $1000$ inhabitants it is again a perfect square. Determine the quantity of inhabitants that are initially in the city.

Let $a, b$ and $c$ be real numbers, and let $f (x) = ax^2 + bx + c$ and $g (x) = cx^2 + bx + a$ functions such that $| f (-1) | \le 1$, $| f (0) | \le 1$ and $| f (1) | \le 1$. Show that if $-1 \le x \le 1$, then $| f (x) | \le \frac54$ and $| g (x) | \le 2$.

Sea $f: R^+ \to R$ defined as $$f (x) = \frac{1}{\sqrt[3]{x^2 + 6x + 9} + \sqrt[3]{x^2 + 4x + 3} + \sqrt[3]{x^2 + 2x + 1}}$$Calculate $$f (1) + f (2) + f (3) + ... + f (2016).$$

Let $f: Z^+ \to Z^+ \cup \{0\}$ a function that meets the following conditions: a) $f (a b) = f (a) + f (b)$, b) $f (a) = 0$ provided that the digits of the unit of $a$ are $7$, c) $f (10) = 0$. Find $f (2016).$

Let $\vartriangle ABC$ be acute with orthocenter $H$. Let $X$ be a point on $BC$ such that $B-X-C$. Let $\Gamma$ be the circumscribed circle of $\vartriangle BHX$ and $\Gamma_2$ be the circumscribed circle of $\vartriangle CHX$. Let $E$ be the intersection of $AB$ with $\Gamma$ , and $D$ be the intersection of $AC$ with $\Gamma_2$. Let $L$ be the intersection of line $HD$ with $\Gamma$ and $J$ be the intersection of line $EH$ with $\Gamma_2$. Prove that points $L$, $X$, and $J$ are collinear. Notation: $A-B-C$ means than points $A,B,C$ are collinear in that order i.e. $ B$ lies between $ A$ and $C$.

Let $ABCD$ be a convex quadrilateral, such that $ A$, $ B$, $C$, and $D$ lie on a circle, with $\angle DAB < \angle ABC$. Let $I$ be the intersection of the bisector of $\angle ABC$ with the bisector of $\angle BAD$. Let $\ell$ be the parallel line to $CD$ passing through point $I$. Suppose $\ell$ cuts segments $DA$ and $BC$ at $ L$ and $J$, respectively. Prove that $AL + JB = LJ$.

Let $\vartriangle ABC$ be acute, with incircle $\Gamma$ and incenter $ I$. $\Gamma$ touches sides $AB$, $BC$ and $AC$ at $Z$, $X$ and $Y$, respectively. Let $D$ be the intersection of $XZ$ with $CI$ and $L$ the intersection of $BI$ with $XY$. Suppose $D$ and $L$ are outside of $\vartriangle ABC$. Prove that $A$, $D$, $Z$, $I$, $Y$, and $ L$ lie on a circle.

Let $p> 5$ be a prime such that none of its digits is divisible by $3$ or $7$. Prove that the equation $x^4 + p = 3y^4$ does not have integer solutions.

Let $x, y, z$ be positive integers and $p$ a prime such that $x <y <z <p$. Also $x^3, y^3, z^3$ leave the same remainder when divided by $p$. Prove that $x + y + z$ divides $x^2 + y^2 + z^2$.

Find all natural values of $n$ and $m$, such that $(n -1)2^{n - 1} + 5 = m^2 + 4m$.