A society has to elect a board of governors. Each member of the society has chosen $10$ candidates for the board, but he will be happy if at least one of them will be on the board. For each six members of the society there exists a board consisting of two persons making all of these six members happy. Prove that a board consisting of $10$ persons can be elected making every member of the society happy.

On the coordinate plane, draw a set of points whose coordinates $(x, y)$ satisfy the equation $y=x+|y-3x-2x^2|$.

Let $ABC$ be a right triangle, with the right angle at $A$. The altitude from $A$ meets $BC$ at $H$ and $M$ is the midpoint of the hypotenuse $[BC]$. On the legs, in the exterior of the triangle, equilateral triangles $BAP$ and $ACQ$ are constructed. If $N$ is the intersection point of the lines $AM$ and $PQ$, prove that the angles $\angle NHP$ and $\angle AHQ$ are equal. Miguel Ochoa Sanchez and Leonard Giugiuc

Let $C$ be a circle with center $O$ and radius $R$. From point $A$ of circle $C$ we construct a tangent $t$ on circle $C$. We construct line $d$ through point $O$ whch intersects tangent $t$ in point $M$ and circle $C$ in points $B$ and $D$ ($B$ lies between points $O$ and $M$). If $AM=R\sqrt{3}$, prove: $a)$ Triangle $AMD$ is isosceles $b)$ Circumcenter of $AMD$ lies on circle $C$

Let $\Gamma$ be a circle with centre $I$, and $A B C D$ a convex quadrilateral such that each of the segments $A B, B C, C D$ and $D A$ is tangent to $\Gamma$. Let $\Omega$ be the circumcircle of the triangle $A I C$. The extension of $B A$ beyond $A$ meets $\Omega$ at $X$, and the extension of $B C$ beyond $C$ meets $\Omega$ at $Z$. The extensions of $A D$ and $C D$ beyond $D$ meet $\Omega$ at $Y$ and $T$, respectively. Prove that \[A D+D T+T X+X A=C D+D Y+Y Z+Z C.\] [i]Proposed by Dominik Burek, Poland and Tomasz Ciesla, Poland[/i]

Let $S_1$ be a square with side $s$ and $C_1$ be the circle inscribed in it. Let $C_2$ be a circle with radius $r$ and $S_2$ be a square inscribed in it. We are told that the area of $S_1 - C_1$ is the same as the area of $C_2 - S_2.$ Which of the following numbers is closest to $s/r?$ $$ \mathrm a. ~ 1\qquad \mathrm b.~2\qquad \mathrm c. ~3 \qquad \mathrm d. ~4 \qquad \mathrm e. ~5 $$

Numbers $0$ and $2013$ are written at two opposite vertices of a cube. Some real numbers are to be written at the remaining $6$ vertices of the cube. On each edge of the cube the difference between the numbers at its endpoints is written. When is the sum of squares of the numbers written on the edges minimal?

Three semicircles of radius $ 1$ are constructed on diameter $ AB$ of a semicircle of radius $ 2$. The centers of the small semicircles divide $ \overline{AB}$ into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles? [asy]import graph; unitsize(14mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dashed=linetype("4 4"); dotfactor=3; pair A=(-2,0), B=(2,0); fill(Arc((0,0),2,0,180)--cycle,mediumgray); fill(Arc((-1,0),1,0,180)--cycle,white); fill(Arc((0,0),1,0,180)--cycle,white); fill(Arc((1,0),1,0,180)--cycle,white); draw(Arc((-1,0),1,60,180)); draw(Arc((0,0),1,0,60),dashed); draw(Arc((0,0),1,60,120)); draw(Arc((0,0),1,120,180),dashed); draw(Arc((1,0),1,0,120)); draw(Arc((0,0),2,0,180)--cycle); dot((0,0)); dot((-1,0)); dot((1,0)); draw((-2,-0.1)--(-2,-0.3),gray); draw((-1,-0.1)--(-1,-0.3),gray); draw((1,-0.1)--(1,-0.3),gray); draw((2,-0.1)--(2,-0.3),gray); label("$A$",A,W); label("$B$",B,E); label("1",(-1.5,-0.1),S); label("2",(0,-0.1),S); label("1",(1.5,-0.1),S);[/asy]$ \textbf{(A)}\ \pi\minus{}\sqrt3 \qquad \textbf{(B)}\ \pi\minus{}\sqrt2 \qquad \textbf{(C)}\ \frac{\pi\plus{}\sqrt2}{2} \qquad \textbf{(D)}\ \frac{\pi\plus{}\sqrt3}{2}$ $ \textbf{(E)}\ \frac{7}{6}\pi\minus{}\frac{\sqrt3}{2}$

Let the numbers $\alpha ,\beta $ satisfy $0<\alpha <\beta <\frac{\pi}{2}$ and let $\gamma $ and $\delta $ be the numbers defined by the conditions: $(\text{i})\ 0<\gamma<\frac{\pi}{2}$, and $\tan\gamma$ is the arithmetic mean of $\tan\alpha$ and $\tan\beta$; $(\text{ii})\ 0<\delta<\frac{\pi}{2}$, and $\frac{1}{\cos\delta}$ is the arithmetic mean of $\frac{1}{\cos\alpha}$ and $\frac{1}{\cos\beta}$. Prove that $\gamma <\delta $.

Find all $n$ for which there are real numbers $0 < a_1 \le a_2 \le ... \le a_n$ with $$\begin{cases} \sum_{k=1}^{n}a_k = 96 \\ \\ \sum_{k=1}^{n}a_k^2 = 144 \\ \\ \sum_{k=1}^{n}a_k^3 = 216 \end{cases}$$

Find all pairs of function $ f: \mathbb{N} \rightarrow \mathbb{N}$ and polynomial with integer coefficients $ p$ such that: (i) $ p(mn) \equal{} p(m)p(n)$ for all positive integers $ m,n > 1$ with $ \gcd(m,n) \equal{} 1$, and (ii) $ \sum_{d|n}f(d) \equal{} p(n)$ for all positive integers $ n$.

(a) Let $ABC$ be any triangle. Let $X, Y, Z$ be points on the sides $BC, CA, AB$ respectively. Suppose that $BX \leq XC, CY \leq YA, AZ \leq ZB$. Show that the area of the triangle $XYZ$ $\geq 1\slash 4$ times the area of $ABC.$ (b) Let $ABC$ be any triangle, and let $X, Y, Z$ be points on the sides $BC, CA, AB$ respectively. Using (a) or by any other method, show: One of the three corner triangles $AZY, BXZ, CYX$ has an area $\leq$ area of the triangle $XYZ.$

Find all real solutions of the equation $x + cos x = 1$.

Prove: If $x, y, z$ are the lengths of the angle bisectors of a triangle with perimeter 6, than we have: \[\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2} \geq 1.\]

Projector emits rays in space. Consider all acute angles between the rays. It is known that no matter what ray we remove, the number of acute angles decreases by exactly $2$ What is the maximal number of rays the projector can emit? [i]M. Karpuk, E. Barabanov[/i]

Six points in general position are given in the space. For each two of them color red the common points (if they exist) of the segment between these points and the surface of the tetrahedron formed by four remaining points. Prove that the number of red points is even.

Let $0<t<\frac{\pi}{2}$. Evaluate \[\lim_{t\rightarrow \frac{\pi}{2}} \int_0^t \tan \theta \sqrt{\cos \theta}\ln (\cos \theta)d\theta\]

A man with mass $m$ jumps off of a high bridge with a bungee cord attached to his ankles. The man falls through a maximum distance $H$ at which point the bungee cord brings him to a momentary rest before he bounces back up. The bungee cord is perfectly elastic, obeying Hooke's force law with a spring constant $k$, and stretches from an original length of $L_0$ to a final length $L = L_0 + h$. The maximum tension in the Bungee cord is four times the weight of the man. Find the maximum extension of the bungee cord $h$. $\textbf{(A) } h = \frac{1}{2}H \\ \\ \textbf{(B) } h = \frac{1}{4}H\\ \\ \textbf{(C) } h = \frac{1}{5}H\\ \\ \textbf{(D) } h = \frac{2}{5}H\\ \\ \textbf{(E) } h = \frac{1}{8}H$

[b]1)[/b] Show that there exist quadratic polynoms $ P\in\mathbb{R}[X] $ whose composition with themselves have $ 1,2 $ and $ 3 $ as their fixed points. [b]2)[/b] Prove that the polynoms referred to at [b]1)[/b] are not integer. [i]Gheorghe Iurea[/i]

Does there exist an increasing arithmetic progression of (a) $11$ (b) $10000$ (c) infinitely many positive integers such that the sums of their digits in base $10$ also form an increasing arithmetic progression? (A Shapovalov)

Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, $CA = 15$. Let $H$ be the orthocenter of $ABC$. Find the radius of the circle with nonzero radius tangent to the circumcircles of $AHB$, $BHC$, $CHA$.

In a trapezoid $ABCD$ with $AB\parallel CD$, the diagonals $AC$ and $BD$ meet at $O$. Let $M$ and $N$ be the centers of the regular hexagons constructed on the sides $AB$ and $CD$ in the exterior of the trapezoid. Prove that $M,O$ and $N$ are collinear.

Let $ ABC$ be an acute-angled triangle. The lines $ L_{A}$, $ L_{B}$ and $ L_{C}$ are constructed through the vertices $ A$, $ B$ and $ C$ respectively according the following prescription: Let $ H$ be the foot of the altitude drawn from the vertex $ A$ to the side $ BC$; let $ S_{A}$ be the circle with diameter $ AH$; let $ S_{A}$ meet the sides $ AB$ and $ AC$ at $ M$ and $ N$ respectively, where $ M$ and $ N$ are distinct from $ A$; then let $ L_{A}$ be the line through $ A$ perpendicular to $ MN$. The lines $ L_{B}$ and $ L_{C}$ are constructed similarly. Prove that the lines $ L_{A}$, $ L_{B}$ and $ L_{C}$ are concurrent.

Compute the number of ordered triples $(a, b, c)$ of integers between $-100$ and $100$ inclusive satisfying the simultaneous equations $$a^3 - 2a = abc - b - c$$ $$b^3 - 2b = bca - c - a$$ $$c^3 - 2c = cab - a - b.$$

Fifty points are chosen inside a convex polygon having eighty sides such that no three of the fifty points lie on the same straight line. The polygon is cut into triangles such that the vertices of the triangles are just the fifty points and the eighty vertices of the polygon. How many triangles are there?