Find all pairs of non-negative integers $(x,y)$ such that \[\sqrt{x+y}-\sqrt{x}-\sqrt{y}+2=0.\]

Let $ABC$ be an acute-angled scalene triangle. Let $P$ be a point on the extension of $AB$ past $B$, and $Q$ a point on the extension of $AC$ past $C$ such that $BPQC$ is a cyclic quadrilateral. Let $N$ be the foot of the perpendicular from $A$ to $BC$. If $NP = NQ$ then prove that $N$ is also the centre of the circumcircle of $APQ$.

In an acute-angled isosceles triangle $ABC$, altitudes $CC_1$ and $BB_1$ intersect the line passing through the vertex $A$ and parallel to the line $BC$, at points $P$ and $Q$. Let $A_0$ be the midpoint of side $BC$, and $AA_1$ the altitude. Lines $A_0C_1$ and $A_0B_1$ intersect line $PQ$ at points $K$ and $L$. Prove that the circles circumscribed around triangles $PQA_1, KLA_0, A_1B_1C_1$ and a circle with a diameter $AA_1$ intersect at one point.

In the plane, three distinct non-collinear points $A,B,C$ are marked. In each step, Ege can do one of the following: [list] [*] For marked points $X,Y$, mark the reflection of $X$ across $Y$. [*]For distinct marked points $X,Y,Z,T$ which do not form a parallelogram, mark the center of spiral similarity which takes segment $XY$ to $ZT$. [*] For distinct marked points $X,Y,Z,T$, mark the intersection of lines $XY$ and $ZT$. [/list] No matter how the points $A,B,C$ are marked in the beginning, can Ege always mark, after finitely many moves, a) The circumcenter of $\triangle ABC$. b) The incenter of $\triangle ABC$. Proposed by [i]Deniz Can Karaçelebi[/i]

Given a positive integer $n$, let $a_1,a_2,..,a_n$ be a sequence of nonnegative integers. A sequence of one or more consecutive terms of $a_1,a_2,..,a_n$ is called $dragon$ if their aritmetic mean is larger than 1. If a sequence is a $dragon$, then its first term is the $head$ and the last term is the $tail$. Suppose $a_1,a_2,..,a_n$ is the $head$ or/and $tail$ of some $dragon$ sequence; determine the minimum value of $a_1+a_2+\cdots +a_n$ in terms of $n$.

Given $2009 \times 4018$ rectangular board. Frame is a rectangle $n \times n$ or $n \times(n + 2)$ for $ ( n \geq 3 )$ without all cells which don’t have any common points with boundary of rectangle. Rectangles $1\times1,1\times 2,1\times 3$ and $ 2\times 4$ are also frames. Two players by turn paint all cells of some frame that has no painted cells yet. Player that can't make such move loses. Who has a winning strategy?

Point $O$ is the center of the regular octagon $ABCDEFGH$, and $X$ is the midpoint of the side $\overline{AB}.$ What fraction of the area of the octagon is shaded? $\textbf{(A) }\frac{11}{32} \qquad\textbf{(B) }\frac{3}{8} \qquad\textbf{(C) }\frac{13}{32} \qquad\textbf{(D) }\frac{7}{16}\qquad \textbf{(E) }\frac{15}{32}$ [asy] pair A,B,C,D,E,F,G,H,O,X; A=dir(45); B=dir(90); C=dir(135); D=dir(180); E=dir(-135); F=dir(-90); G=dir(-45); H=dir(0); O=(0,0); X=midpoint(A--B); fill(X--B--C--D--E--O--cycle,rgb(0.75,0.75,0.75)); draw(A--B--C--D--E--F--G--H--cycle); dot("$A$",A,dir(45)); dot("$B$",B,dir(90)); dot("$C$",C,dir(135)); dot("$D$",D,dir(180)); dot("$E$",E,dir(-135)); dot("$F$",F,dir(-90)); dot("$G$",G,dir(-45)); dot("$H$",H,dir(0)); dot("$X$",X,dir(135/2)); dot("$O$",O,dir(0)); draw(E--O--X); [/asy]

Suppose that \[\operatorname{lcm}(1024,2016)=\operatorname{lcm}(1024,2016,x_1,x_2,\ldots,x_n),\] with $x_1$, $x_2$, $\cdots$, $x_n$ are distinct postive integers. Find the maximum value of $n$. [i]Proposed by Le Duc Minh[/i]

Alireza is currently standing at the point $(0, 0)$ in the $x-y$ plane. At any given time, Alireza can move from the point $(x, y)$ to the point $(x + 1, y)$ or the point $(x, y + 1)$. However, he cannot move to any point of the form $(x, y)$ where $y \equiv 2x\,\, (\mod \,\,5)$. Let $p_k$ be the number of paths Alireza can take starting from the point $(0, 0)$ to the point $(k + 1, 2k + 1)$. Evaluate the sum $$\sum^{\infty}_{k=1} \frac{p_k}{5^k}.$$.

Let $ W$ be a dense, open subset of the real line $ \mathbb{R}$. Show that the following two statements are equivalent: (1) Every function $ f : \mathbb{R} \rightarrow \mathbb{R}$ continuous at all points of $ \mathbb{R} \setminus W$ and nondecreasing on every open interval contained in $ W$ is nondecreasing on the whole $ \mathbb{R}$. (2) $ \mathbb{R} \setminus W$ is countable. [i]E. Gesztelyi[/i]

Show that for any finite set $S$ of distinct positive integers, we can find a set $T \supseteq S$ such that every member of $T$ divides the sum of all the members of $T$. [b]Original Statement:[/b] A finite set of (distinct) positive integers is called a [b]DS-set[/b] if each of the integers divides the sum of them all. Prove that every finite set of positive integers is a subset of some [b]DS-set[/b].

Let $ABCDEFGHIJ$ be a regular $10$-gon. Let $T$ be a point inside the $10$-gon, such that the $DTE$ is isosceles: $DT = ET$ , and its angle at the apex is $72^\circ$. Prove that there exists a point $S$ such that $FTS$ and $HIS$ are both isosceles, and for both of them the angle at the apex is $72^\circ$.

Let $a_n$ be a sequence with $a_0=1$ and defined recursively by $$a_{n+1}=\begin{cases}a_n+2&\text{if }n\text{ is even},\\2a_n&\text{if }n\text{ is odd.}\end{cases}$$ What are the last two digits of $a_{2015}$?

Given an infinite sheet of square ruled paper. Some of the squares contain a piece. A move consists of a piece jumping over a piece on a neighbouring square (which shares a side) onto an empty square and removing the piece jumped over. Initially, there are no pieces except in an $m x n$ rectangle ($m, n > 1$) which has a piece on each square. What is the smallest number of pieces that can be left after a series of moves?

a) Let $a\in \mathbb{R}$ and $f \colon \mathbb{R} \to \mathbb{R}$ be a continuous function for which there exists an antiderivative $F \colon \mathbb{R} \to \mathbb{R} $, such that $F(x)+a\cdot f(x) \geq 0$, for any $x \in \mathbb{R}$, and$ \lim_{|x| \to \infty} \frac{F(x)}{e^{|\alpha \cdot x|}}=0$ holds for any $\alpha \in \mathbb{R}^*$. Prove that $F(x) \geq 0$ for all $x \in \mathbb{R}$. b) Let $n\geq 2$ be a positive integer, $g \in \mathbb{R}[X]$, $g = X^n + a_1X^{n-1}+ \dots + a_{n-1}X+a_n$ be a polynomial with all of its roots being real, and $f \colon \mathbb{R} \to \mathbb{R}$ a polynomial function such that $f(x)+a_1\cdot f'(x)+a_2\cdot f^{(2)}(x)+\dots+a_n\cdot f^{(n)}(x) \geq 0$ for any $x \in \mathbb{R}$. Prove that $f(x) \geq 0$ for all $x \in \mathbb{R}$.

Prove that the six sides of any tetrahedron can be the sides of a convex hexagon.

Does there exist a natural number $n$, for which $n.2^{2^{2014}}-81-n$ is a perfect square?

The table below gives the percent of students in each grade at Annville and Cleona elementary schools: \[\begin{tabular}{rccccccc} & \textbf{\underline{K}} & \textbf{\underline{1}} & \textbf{\underline{2}} & \textbf{\underline{3}} & \textbf{\underline{4}} & \textbf{\underline{5}} & \textbf{\underline{6}} \\ \textbf{Annville:} & 16\% & 15\% & 15\% & 14\% & 13\% & 16\% & 11\% \\ \textbf{Cleona:} & 12\% & 15\% & 14\% & 13\% & 15\% & 14\% & 17\% \end{tabular}\] Annville has 100 students and Cleona has 200 students. In the two schools combined, what percent of the students are in grade 6? $\text{(A)}\ 12\% \qquad \text{(B)}\ 13\% \qquad \text{(C)}\ 14\% \qquad \text{(D)}\ 15\% \qquad \text{(E)}\ 28\%$

Opposite sides of a convex hexagon are parallel. Let's call the "height" of such a hexagon a segment with ends on straight lines containing opposite sides and perpendicular to them. Prove that a circle can be circumscribed around this hexagon if and only if its "heights" can be parallelly moved so that they form a triangle. (A. Zaslavsky)

Let $a,b$ and $c$ be positive real numbers satisfying $abc=1.$ Prove that$$\frac{a^2-b^2}{a+bc}+\frac{b^2-c^2}{b+ca}+\frac{c^2-a^2}{c+ab}\leq a+b+c-3.$$

Given a regular $n$-gon $A_{1}A_{2}...A_{n}$ (with $n\geq 3$) in a plane. How many triangles of the kind $A_{i}A_{j}A_{k}$ are obtuse ?

Express the sum of $99$ terms$$\frac{1\cdot 4}{2\cdot 5}+\frac{2\cdot 7}{5\cdot 8}+\ldots +\frac{k(3k+1 )}{(3k-1)(3k+2)}+\ldots +\frac{99\cdot 298}{296\cdot 299}$$ as an irreducible fraction.

Compute \[\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}}}\]

$(SWE 6)$ Prove that there are infinitely many positive integers that cannot be expressed as the sum of squares of three positive integers.

Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations: [list=1] [*] Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell. [*] Choose two (not necessarily adjacent) cells with the same number in them; denote that number by $2^j$. Replace the number in one of the cells with $2^{j+1}$ and erase the number in the other cell. [/list] At the end of the game, one cell contains $2^n$, where $n$ is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of $n$. [i]Proposed by Warut Suksompong, Thailand[/i]