Given is a triangle $ABC$. Let $X$ be the reflection of $B$ in $AC$ and $Y$ is the reflection of $C$ in $AB$. The tangent to $(XAY)$ at $A$ meets $XY$ and $BC$ at $E, F$. Show that $AE=AF$.

Let $a$ and $b$ be fixed positive numbers. Find the smallest possible depending on $a$ and $b$ value of the sum $$\frac{x^2}{(ay + bz)(az + by)}+\frac{y^2}{(az + bx)(ax + bz)}+\frac{z^2}{(ax + by)(ay + bx)},$$ where $x, y, z$ are positive real numbers.

Find all integers $n \geq 2$ for which there exists a sequence of $2n$ pairwise distinct points $(P_1, \dots, P_n, Q_1, \dots, Q_n)$ in the plane satisfying the following four conditions: [list=i] [*]no three of the $2n$ points are collinear; [*] $P_iP_{i+1} \ge 1$ for all $i = 1, 2, \dots ,n$, where $P_{n+1}=P_1$; [*] $Q_iQ_{i+1} \ge 1$ for all $i = 1, 2, \dots, n$, where $Q_{n+1} = Q_1$; and [*] $P_iQ_j \le 1$ for all $i = 1, 2, \dots, n$ and $j = 1, 2, \dots, n$.[/list] [i]Ray Li[/i]

In a simple graph $G$, an operation is defined as taking two neighbor vertices $u,v$ which have a common neighbor, deleting the edge between $u,v$ and adding a new vertex $w$ whose neighbors are exactly the common neighbors of $u$ and $v$. Starting with the complete graph $G=K_n$ where $n\ge 3$ is a positive integer, find the maximum number of operations that can be applied. Proposed by[i] Deniz Can Karaçelebi[/i]

When askes: "What time is it?", father said to a son: "Quarter of time that passed and half of the remaining time gives the exact time". What time was it?

Let $n$ be a fixed natural number. The maze is a grid of dimensions $n \times n$, with a gate to the sky on one of the squares and some adjacent squares with partitions separated from each other so that it is still possible to move from one square to another. The program is in the UP, DOWN, RIGHT, LEFT final sequence, With each command, the Creature moves from its current square to the corresponding neighboring square, unless the partition or the outer boundary of the labyrinth prevents execution of the command (otherwise it does nothing), upon entering the gate, the Creature moves on to heaven. God creates a program, then Satan creates a labyrinth and places it on a square. Prove that God can make such a program that, independently of Satan's labyrinth and selected from the source square, the Creature always reaches heaven by following this program.

A tree has $10$ pounds of apples at dawn. Every afternoon, a bird comes and eats $x$ pounds of apples. Overnight, the amount of food on the tree increases by $10\%$. What is the maximum value of $x$ such that the bird can sustain itself indefinitely on the tree without the tree running out of food?

Let $\mathbb{R}^+$ be the set of positive real numbers. Find all functions $f \colon \mathbb{R}^+ \to \mathbb{R}^+$ such that, for all $x,y \in \mathbb{R}^+$, $$f(xy+f(x))=xf(y)+2.$$

In right $ \triangle ABC$ with hypotenuse $ \overline{AB}$, $ AC \equal{} 12$, $ BC \equal{} 35$, and $ \overline{CD}$ is the altitude to $ \overline{AB}$. Let $ \omega$ be the circle having $ \overline{CD}$ as a diameter. Let $ I$ be a point outside $ \triangle ABC$ such that $ \overline{AI}$ and $ \overline{BI}$ are both tangent to circle $ \omega$. The ratio of the perimeter of $ \triangle ABI$ to the length $ AB$ can be expressed in the form $ \displaystyle\frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.

Benny the Bear has $100$ rabbits in his rabbit farm. He observes that $53$ rabbits are spotted, and $73$ rabbits are blue-eyed. Compute the minimum number of rabbits that are both spotted and blue-eyed.

Let $x,y,z$ be positive integers such that $\frac{x+1}{y}+\frac{y+1}{z}+\frac{z+1}{x}$ is an integer. Let $d$ be the greatest common divisor of $x,y$ and $z$. Prove that $d\le \sqrt[3]{xy+yz+zx}$.

Which of the following statements are true? (A) $X$ implies $Y$, or $Y$ implies $X$, where $X$ is the statement, the lines $L_1, L_2, L_3$ lie in a plane, and $Y$ is the statement, each pair of the lines $L_1, L_2, L_3$ intersect. (B) Every sufficiently large integer $n$ satisfies $n = a^4 + b^4$ for some integers a, b. (C) There are real numbers $a_1, a_2,... , a_n$ such that $a_1 \cos x + a_2 \cos 2x +... + a_n \cos nx > 0$ for all real $x$.

Let $f(x)$ be a continuous function such that $\int_0^1 f(x)\ dx=1.$ Find $f(x)$ for which $\int_0^1 (x^2+x+1)f(x)^2dx$ is minimized.

Let $M$ be a $2014\times 2014$ invertible matrix, and let $\mathcal{F}(M)$ denote the set of matrices whose rows are a permutation of the rows of $M$. Find the number of matrices $F\in\mathcal{F}(M)$ such that $\det(M + F) \ne 0$.

Calculate the following indefinite integrals. [1] $\int (4-5\tan x)\cos x dx$ [2] $\int \frac{dx}{\sqrt[3]{(1-3x)^2}}dx$ [3] $\int x^3\sqrt{4-x^2}dx$ [4] $\int e^{-x}\sin \left(x+\frac{\pi}{4}\right)dx$ [5] $\int (3x-4)^2 dx$

An acute triangle $\triangle ABC$ has incenter $I$, and the incircle hits $BC, CA, AB$ at $D, E, F$. Lines $BI, CI, BC, DI$ hits $EF$ at $K, L, M, Q$ and the line connecting the midpoint of segment $CL$ and $M$ hits the line segment $CK$ at $P$. Prove that $$PQ=\frac{AB \cdot KQ}{BI}$$

Let $ b_1<b_2<b_3<\dots $ be the sequence of all natural numbers which are sum of squares of two natural numbers. Prove that there exists infinite natural numbers like $m$ which $b_{m+1}-b_m=2015$ .

Evaluate $ \int_0^{\pi} \frac{x\sin ^ 3 x}{\sin ^ 2 x\plus{}8}dx$.

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.

Find all positive integers $a$ so that for any $\left \lfloor \frac{a+1}{2} \right \rfloor$-digit number that is composed of only digits $0$ and $2$ (where $0$ cannot be the first digit) is not a multiple of $a$.

Let $ a,b,c$ be three pairwise distinct real numbers such that $ a\plus{}b\plus{}c\equal{}6\equal{}ab\plus{}bc\plus{}ca\minus{}3$. Prove that $ 0<abc<4$.

A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\tan \theta$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

All angles of a cyclic pentagon $ABCDE$ are obtuse. The sidelines $AB$ and $CD$ meet at point $E_1$, the sidelines $BC$ and $DE$ meet at point $A_1$. The tangent at $B$ to the circumcircle of the triangle $BE_1C$ meets the circumcircle $\omega$ of the pentagon for the second time at point $B_1$. The tangent at $D$ to the circumcircle of the triangle $DA_1C$ meets $\omega$ for the second time at point $D_1$. Prove that $B_1D_1 // AE$

There is a point inside a regular triangle located at distances $5$, $6$ and $7$ from its vertices. Find the area of this regular triangle.

Let $ABC$ be an acute-angled triangle with $AB \ne BC$ and radius $k$. Let $P$ and $Q$ be the points of intersection of $k$ with the internal bisector and the external bisector of $\angle CBA$ respectively. Let $D$ be the intersection of $AC$ and $PQ$. Find the ratio $AD: DC$.