Determine all real numbers $x$ which satisfy \[ x = \sqrt{a - \sqrt{a+x}} \]where $a > 0$ is a parameter.

Given a convex polygon with $n$ sides and perimeter $S$, which has an incircle $\omega$ with radius $R$. A regular polygon with $n$ sides, whose vertices lie on $\omega$, has a perimeter $s$. Determine whether the following inequality holds: \[ S \ge \frac{2sRn}{\sqrt{4n^2R^2-s^2}}. \]

Let $\{E_1, E_2, \dots, E_m\}$ be a collection of sets such that $E_i \subseteq X = \{1, 2, \dots, 100\}$, $E_i \neq X$, $i = 1, 2, \dots, m$. It is known that every two elements of $X$ is contained together in exactly one $E_i$ for some $i$. Determine the minimum value of $m$.

We call a subset $B$ of natural numbers loyal if there exists natural numbers $i\le j$ such that $B=\{i,i+1,\ldots,j\}$. Let $Q$ be the set of all loyal sets. For every subset $A=\{a_1<a_2<\ldots<a_k\}$ of $\{1,2,\ldots,n\}$ we set \[f(A)=\max_{1\le i \le k-1}{a_{i+1}-a_i}\qquad\text{and}\qquad g(A)=\max_{B\subseteq A, B\in Q} |B|.\]Furthermore, we define \[F(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} f(A)\qquad\text{and}\qquad G(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} g(A).\]Prove that there exists $m\in \mathbb N$ such that for each natural number $n>m$ we have $F(n)>G(n)$. (By $|A|$ we mean the number of elements of $A$, and if $|A|\le 1$, we define $f(A)$ to be zero). Proposed by Javad Abedi

Let $k$ and $n$ be positive integers. Determine the smallest integer $N \ge k$ such that the following holds: If a set of $N$ integers contains a complete residue modulo $k$, then it has a non-empty subset whose sum of elements is divisible by $n$.

Determine all triples of real numbers $(x, y, z)$ which satisfy the following system of equations: \[ \begin{cases} x+y+z=0 \\ x^3+y^3+z^3 = 90 \\ x^5+y^5+z^5=2850. \end{cases} \]

Circles $\Omega $ and $\omega $ are tangent at a point $P$ ($\omega $ lies inside $\Omega $). A chord $AB$ of $\Omega $ is tangent to $\omega $ at $C;$ the line $PC$ meets $\Omega $ again at $Q.$ Chords $QR$ and $QS$ of $ \Omega $ are tangent to $\omega .$ Let $I,X,$ and $Y$ be the incenters of the triangles $APB,$ $ARB,$ and $ASB,$ respectively. Prove that $\angle PXI+\angle PYI=90^{\circ }.$

The Hawking Space Agency operates $n-1$ space flights between the $n$ habitable planets of the Local Galaxy Cluster. Each flight has a fixed price which is the same in both directions, and we know that using these flights, we can travel from any habitable planet to any habitable planet. In the headquarters of the Agency, there is a clearly visible board on a wall, with a portrait, containing all the pairs of different habitable planets with the total price of the cheapest possible sequence of flights connecting them. Suppose that these prices are precisely $1,2, ... , \binom{n}{2}$ monetary units in some order. prove that $n$ or $n-2$ is a square number.

Let $n \ge 3$ be a positive integer. We call a $3 \times 3$ grid beautiful if the cell located at the center is colored white and all other cells are colored black, or if it is colored black and all other cells are colored white. Determine the minimum value of $a+b$ such that there exist positive integers $a$, $b$ and a coloring of an $a \times b$ grid with black and white, so that it contains $n^2 - n$ beautiful subgrids.

Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]

Let $n$ be a positive integer greater than $1$. Evaluate the following summation: \[ \sum_{k=0}^{n-1} \frac{1}{1 + 8 \sin^2 \left( \frac{k \pi}{n} \right)}. \]

In a non-isosceles triangle $ABC$, let $I$ be its incenter. The incircle of $ABC$ touches $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. A line passing through $D$ and perpendicular to $AD$ intersects $IB$ and $IC$ at $A_b$ and $A_c$, respectively. Define the points $B_c$, $B_a$, $C_a$, and $C_b$ similarly. Let $G$ be the intersection of the cevians $AD$, $BE$, and $CF$. The points $O_1$ and $O_2$ are the circumcenter of the triangles $A_bB_cC_a$ and $A_cB_aC_b$, respectively. Prove that $IG$ is the perpendicular bisector of $O_1O_2$.

Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are: (i) A player cannot choose a number that has been chosen by either player on any previous turn. (ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. (iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies. Proposed by Finland

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\]for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.