Find all finite sets $S$ of positive integers with at least $2$ elements, such that if $m>n$ are two elements of $S$, then $$ \frac{n^2}{m-n} $$ is also an element of $S$.

Let $ABC$ be a triangle. An arbitrary circle which passes through the points $B,C$ intersects the sides $AC,AB$ for the second time in $D,E$ respectively. The line $BD$ intersects the circumcircle of the triangle $AEC$ at $P{}$ and $Q{}$ and the line $CE$ intersects the circumcircle of the triangle $ABD$ at $R{}$ and $S{}$ such that $P{}$ is situated on the segment $BD{}$ and $R{}$ lies on the segment $CE.$ Prove that: [list=a] [*]The points $P,Q,R$ and $S{}$ are concyclic. [*]The triangle $APQ$ is isosceles. [/list] [i]Petru Braica[/i]

In every row of a grid $100 \times n$ is written a permutation of the numbers $1,2 \ldots, 100$. In one move you can choose a row and swap two non-adjacent numbers with difference $1$. Find the largest possible $n$, such that at any moment, no matter the operations made, no two rows may have the same permutations.

Let $\triangle ABC$ be an acute triangle with circumcenter $O$ such that $AB<AC$, let $Q$ be the intersection of the external bisector of $\angle A$ with $BC$, and let $P$ be a point in the interior of $\triangle ABC$ such that $\triangle BPA$ is similar to $\triangle APC$. Show that $\angle QPA + \angle OQB = 90^{\circ}$. [i]Alex Zhu.[/i]

Let $ p_1,p_2,\ldots,p_k $ be the distinct prime divisors of $ n $ and let $ a_n=\frac {1}{p_1}+\frac {1}{p_2}+\cdots+\frac {1}{p_k} $ for $ n\geq 2 $. Show that for every positive integer $ N\geq 2 $ the following inequality holds: $ \sum_{k=2}^{N} a_2a_3 \cdots a_k <1 $ [i]Laurentiu Panaitopol[/i]

Calculate the mean value of the solid angle by which the disc $x^2 + y^2 \le 1$ lying in the plane $z = 0$ is seen from points of the sphere $x^2 + y^2 + (z-2)^2 = 1$.

Find the relation of the black part length and the white part length for the main diagonal of the a) $100\times 99$ chess-board; b) $101\times 99$ chess-board.

Let triangle $ABC$ have $\angle A = 70^\circ, \angle B = 60^\circ$, and $\angle C = 50^\circ$. Extend altitude $BH$ past $H$ to point $D$ so that $BD = BC$. Find $\angle BDA$ in degrees.

Evan's bike lock has been stolen by Jonathan, and he has changed the passcode. Jonathan is refusing to tell Evan the passcode. All Evan knows is it is a five-digit number with following properties: (a) It can be written as $a\cdot \overline{ab}\cdot\overline{abc}$ where $a, b, c$ are pairwise different digits and $a$, $\overline{ab}$, $\overline{abc}$ are prime. (b) The sum of its digits is 21. (c) The passcode's last digit is $c$. Find the bike passcode. [i]Team #1[/i]

Prove that the set $ \left\{ \left. \begin{pmatrix} \frac{1-2x^3}{3x^2} & \frac{1+x^3}{3x^2} & \frac{1+x^3}{3x^2} \\ \frac{1+x^3}{3x^2} & \frac{1-2x^3}{3x^2} & \frac{1+x^3}{3x^2} \\ \frac{1+x^3}{3x^2} & \frac{1+x^3}{3x^2} & \frac{1-2x^3}{3x^2}\end{pmatrix}\right| x\in\mathbb{R}^{*} \right\} $ along with the usual multiplication of matrices form a group, determine an isomorphism between this group and the group of multiplicative real numbers.

On a chess board, the boundaries of the squares are assumed to be black. Draw a circle of the greatest possible radius lying entirely on the black squares.

Variable points $M$ and $N$ are considered on the arms $\left[ OX \right.$ and $\left[ OY \right.$ , respectively, of an angle $XOY$ so that $\left| OM \right|+\left| ON \right|$ is constant. Determine the locus of the midpoint of $\left[ MN \right]$.

In a triangle $ABC$ angle $\angle BAC = 60^{\circ}$. Point $M$ is the midpoint of $BC$, and $D$ is the foot of altitude from point $A$. Points $T$ and $P$ are marked such that $TBC$ is equilateral, and $\angle BPD=\angle DPC = 30^{\circ}$ and this points lie in the same half-plane with respect to $BC$, not in the same as $A$. Prove that the circumcircles of $ADP$ and $AMT$ are tangent. [i]Ivan Korshunau[/i]

Let $L$ be the midpoint of the minor arc $AC$ of the circumcircle of an acute-angled triangle $ABC$. A point $P$ is the projection of $B$ to the tangent at $L$ to the circumcircle. Prove that $P$, $L$, and the midpoints of sides $AB$, $BC$ are concyclic.

Assume $A=\{a_{1},a_{2},...,a_{12}\}$ is a set of positive integers such that for each positive integer $n \leq 2500$ there is a subset $S$ of $A$ whose sum of elements is $n$. If $a_{1}<a_{2}<...<a_{12}$ , what is the smallest possible value of $a_{1}$?

For a positive number $n$, let $g(n)$ be the product of all $1 \le k \le n$ such that gcd $(k, n) =1$, and say that $n > 1$ is reckless if $n$ is odd and $g(n) \equiv -1$ (mod $n$). Find the number of reckless numbers less than $50$.

Observe that, in the usual chessboard colouring of the two-dimensional grid, each square has 4 of its 8 neighbours black and 4 white. Does there exist a way to colour the three-dimensional grid such that each cube has half of its 26 neighbours black and half white? Is this possible in four dimensions?

On a board there is a regular polygon $A_1A_2\ldots A_{99}.$ Ana and Barbu alternatively occupy empty vertices of the polygon and write down triangles on a list: Ana only writes obtuse triangles, while Barbu only writes acute ones. At the first turn, Ana chooses three vertices $X,Y$ and $Z$ and writes down $\triangle XYZ.$ Then, Barbu chooses two of $X,Y$ and $Z,$ for example $X$ and $Y$, and an unchosen vertex $T$, and writes down $\triangle XYT.$ The game goes on and at each turn, the player must choose a new vertex $R$ and write down $\triangle PQR$, where $P$ is the last vertex chosen by the other player, and $Q$ is one of the other vertices of the last triangle written down by the other player. If one player cannot perform a move, then the other one wins. If both people play optimally, determine who has a winning strategy.

Prove that if $a_1 , a_2 ,... , a_n$ are positive real numbers, then $$(a_1 + a_2 + ... + a_n) \left( \frac{1}{a_1}+ \frac{1}{a_1}+...+\frac{1}{a_n}\right)\ge n^2$$. When is equality valid?

Let $r\geq 2$ be a positive integer, and let each positive integer be painted in one of $r$ different colors. For every positive integer $n$ and every pair of colors $a$ and $b$, if the difference between the number of divisors of $n$ that are painted in color $a$ and the number of divisors of $n$ that are painted in color $b$ is at most $1$, find all possible values of $r$.

Given a positive integer $n \ge 2$, we define $f(n)$ to be the sum of all remainders obtained by dividing $n$ by all positive integers less than $n$. For example dividing $5$ with $1, 2, 3$ and $4$ we have remainders equal to $0, 1, 2$ and $1$ respectively. Therefore $f(5) = 0 + 1 + 2 + 1 = 4$. Find all positive integers $n \ge 3$ such that $f(n) = f(n - 1) + (n - 2)$.

Let $ a>0$, and let $ P(x)$ be a polynomial with integer coefficients such that \[ P(1)\equal{}P(3)\equal{}P(5)\equal{}P(7)\equal{}a\text{, and}\] \[ P(2)\equal{}P(4)\equal{}P(6)\equal{}P(8)\equal{}\minus{}a\text{.}\] What is the smallest possible value of $ a$? $ \textbf{(A)}\ 105 \qquad \textbf{(B)}\ 315 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 7! \qquad \textbf{(E)}\ 8!$

For each real number $p > 1$, find the minimum possible value of the sum $x+y$, where the numbers $x$ and $y$ satisfy the equation $(x+\sqrt{1+x^2})(y+\sqrt{1+y^2}) = p$.

Find all polynomials $f$ such that $f$ has non-negative integer coefficients, $f(1)=7$ and $f(2)=2017$.

Determine the number of real roots of the equation ${x^8 -x^7 + 2x^6- 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x + \frac{5}{2}= 0}$