An equilateral triangle has been drawn inside the circle. Split the triangle to two parts with equal area by a line segment parallel to the triangle side. Draw an inscribed circle inside this smaller triangle. What is the ratio of the area of this circle compared to the area of original circle.

Let $\, a,b \,$ be odd positive integers. Define the sequence $\, (f_n ) \,$ by putting $\, f_1 = a,$ $f_2 = b, \,$ and by letting $\, f_n \,$ for $\, n \geq 3 \,$ be the greatest odd divisor of $\, f_{n-1} + f_{n-2}$. Show that $\, f_n \,$ is constant for $\, n \,$ sufficiently large and determine the eventual value as a function of $\, a \,$ and $\, b$.

Let $n \geq 3$ be a positive integer. In an election debate, we have $n$ seats arranged in a circle and these seats are numbered from $1$ to $n$, clockwise. In each of these chairs sits a politician, who can be a liar or an honest one. Lying politicians always tell lies, and honest politicians always tell the truth. At one heated moment in the debate, they accused each other of being liars, with the politician in chair $1$ saying that the politician immediately to his left is a liar, the politician in chair $2$ saying that all the $2$ politicians immediately to his left are liars, the politician in the char $3$ saying that all the $3$ politicians immediately to his left are liars, and so on. Note that the politician in chair $n$ accuses all $n$ politicians (including himself) of being liars. For what values of $n$ is this situation possible to happen?

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!$