In an isosceles triangle ABC with the base AB given a point M $\in$ BC: Let O be the center of its circumscribed circle and S be the center of the inscribed circle in
ABC and SM // AC: Prove that OM perpendicular BS.

For which integers $n\ge 2$ can the numbers $1$ to $16$ be written each in one square of a squared $4\times 4$ paper such that the $8$ sums of the numbers in rows and columns are all different and divisible by $n$?

Find all real numbers $x, y,z,t \in [0, \infty)$ so that $$x + y + z \le t, \,\,\, x^2 + y^2 + z^2 \ge t \,\,\, and \,\,\,x^3 + y^3 + z^3 \le t.$$

Suppose $a_1,a_2,a_3,b_1,b_2,b_3$ are distinct positive integers such that \[(n \plus{} 1)a_1^n \plus{} na_2^n \plus{} (n \minus{} 1)a_3^n|(n \plus{} 1)b_1^n \plus{} nb_2^n \plus{} (n \minus{} 1)b_3^n\] holds for all positive integers $n$. Prove that there exists $k\in N$ such that $ b_i \equal{} ka_i$ for $ i \equal{} 1,2,3$.

Find the minimum value of $\int_0^{\pi} (x-y)^2 (\sin x)|\cos x|dx$.

Given is a grid with $2$ rows and $120$ columns, such that each cell has a number from the set $1, 2, ..., 120$. It is known that in each column, the upper number in it is smaller than the lower number, and in each row, the numbers are in non-strict increasing order from left to right. Prove that the number of these tables is multiple of $239$.

If $p,q\in\mathbb{R}[x]$ satisfy $p(p(x))=q(x)^2$, does it follow that $p(x)=r(x)^2$ for some $r\in\mathbb{R}[x]$?

Let $(a_n)_{n \ge 1}$ be a sequence of real numbers satisfying the properties: [list=1] [*] the sequence $x_n=\sum\limits_{k=1}^n a_k^2$ is convergent; [*] the sequence $y_n=\sum\limits_{k=1}^n a_k$ is unbounded. [/list] Prove that the sequence $(b_n)_{n \ge 1}$ given by $b_n=\{y_n\}$ is divergent. Note: $\{ x \}$ denotes the fractional part of $x.$

Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

Two kids $A$ and $B$ play a game as follows: from a box containing $n$ marbles ($n > 1$), they alternately take some marbles for themselves, such that: 1. $A$ goes first. 2. The number of marbles taken by $A$ in his first turn, denoted by $k$, must be between $1$ and $n - 1$, inclusive. 3. The number of marbles taken in a turn by any player must be between $1$ and $k$, inclusive. The winner is the one who takes the last marble. Determine all natural numbers $n$ for which $A$ has a winning strategy

The numbers $1, 2, 3, ... , n$ are written on a blackboard (where $n \ge 3$). A move is to replace two numbers by their sum and non-negative difference. A series of moves makes all the numbers equal $k$. Find all possible $k$

Determine the least real number $M$ such that the inequality \[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \leq M(a^{2}+b^{2}+c^{2})^{2}\] holds for all real numbers $a$, $b$ and $c$.

Anja has to write $2014$ integers on the board such that arithmetic mean of any of the three numbers is among those $2014$ numbers. Show that this is possible only when she writes nothing but $2014$ equal integers.

A quadrilateral $ABCD$ is inscribed in a circle with center $O$. Let $A_1 B_1 C_1 D_1$ be the image of $ABCD$ after rotation with center $O$ and angle $\alpha \in (0,90^\circ)$. The points $P,Q,R$ and $S$ are intersections of $AB$ and $A_1 B_1$, $BC$ and $B_1 C_1$, $CD$ and $C_1 D_1$, and $DA$ and $D_1 A_1$. Prove that $PQRS$ is a parallelogram.

Find the locus of all numbers $z\in\mathbb C$ in complex plane satisfying $$z+\bar z=a\cdot|z|,$$ where $a\in\mathbb R$ is given.

Let $a, b, c$ be the side lengths of a triangle. Prove that $2 (a^3 + b^3 + c^3) < (a + b + c) (a^2 + b^2 + c^2) \le 3 (a^3 + b^3 + c^3)$

Let $O$ be the center of the inscribed circle of $\Delta ABC$ and point $D$ be the middle point of $AB$. If $\angle AOD=90^\circ$, prove that $AB+BC=3AC$.

Let \[f = X^{n}+a_{n-1}X^{n-1}+\ldots+a_{1}X+a_{0}\] be an integer polynomial of degree $n \geq 3$ such that $a_{k}+a_{n-k}$ is even for all $k \in \overline{1,n-1}$ and $a_{0}$ is even. Suppose that $f = gh$, where $g,h$ are integer polynomials and $\deg g \leq \deg h$ and all the coefficients of $h$ are odd. Prove that $f$ has an integer root.

Let $n$ be a natural number. A tiling of a $2n \times 2n$ board is a placing of $2n^2$ dominos (of size $2 \times 1$ or $1 \times 2$) such that each of them covers exactly two squares of the board and they cover all the board.Consider now two [i]sepearate tilings[/i] of a $2n \times 2n$ board: one with red dominos and the other with blue dominos. We say two squares are red neighbours if they are covered by the same red domino in the red tiling; similarly define blue neighbours. Suppose we can assign a non-zero integer to each of the squares such that the number on any square equals the difference between the numbers on it's red and blue neighbours i.e the number on it's red neigbhbour minus the number on its blue neighbour. Show that $n$ is divisible by $3$ [i] Proposed by Tejaswi Navilarekallu [/i]

For fixed positive integers $s, t$, define $a_n$ as the following. $a_1 = s, a_2 = t$, and $\forall n \ge 1$, $a_{n+2} = \lfloor \sqrt{a_n+(n+2)a_{n+1}+2008} \rfloor$. Prove that the solution set of $a_n \not= n$, $n \in \mathbb{N}$ is finite.

Let $f(N) = N \left( \frac{9}{10} \right)^N$ , and let $\frac{m}{n}$ denote the maximum value of $f(N)$, as $N$ ranges over the positive integers. If $m$ and $n$ are relatively prime positive integers, find the remainder when $m + n$ is divided by $1000$.

Solve in positive integers: $(x^{2}+2)(y^{2}+3)(z^{2}+4)=60xyz$.

A number $ x$ is uniformly chosen on the interval $ [0,1]$, and $ y$ is uniformly randomly chosen on $ [\minus{}1,1]$. Find the probability that $ x>y$.

Let $A, B,$ and $C$ denote the angles of a triangle. If $\sin^2 A + \sin^2 B + \sin^2 C = 2$, prove that the triangle is right-angled.

How many zeros does $101^{100} - 1$ end with?