Let $\omega= \cos \frac{2\pi}{727} + i \sin \frac{2\pi}{727}$. The imaginary part of the complex number $$\prod^{13}_{k=8} \left(1 + \omega^{3^{k-1}}+ \omega^{2\cdot 3^{k-1}}\right)$$ is equal to $\sin a$ for some angle $a$ between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ , inclusive. Find $a$.

The two points $P(8, 2)$ and $Q(5, 11)$ are considered in the plane. A mobile moves from $P$ to $Q$ according to a path that has to fulfill the following conditions: The moving part of $ P$ and arrives at a point on the $x$-axis, along which it travels a segment of length $1$, then it departs from this axis and goes towards a point on the $y$ axis, on which travels a segment of length $2$, separates from the $y$ axis finally and goes towards the point $Q$. Among all the possible paths, determine the one with the minimum length, thus like this same length.

Denote by S the set of all primes such the decimal representation of $\frac{1}{p}$ has the fundamental period divisible by 3. For every $p \in S$ such that $\frac{1}{p}$ has the fundamental period $3r$ one may write \[\frac{1}{p}=0,a_{1}a_{2}\ldots a_{3r}a_{1}a_{2} \ldots a_{3r} \ldots , \] where $r=r(p)$; for every $p \in S$ and every integer $k \geq 1$ define $f(k,p)$ by \[ f(k,p)= a_{k}+a_{k+r(p)}+a_{k+2.r(p)}\] a) Prove that $S$ is infinite. b) Find the highest value of $f(k,p)$ for $k \geq 1$ and $p \in S$

For a unit circle $O$, arrange points $A,B,C,D$ and $E$ in that order evenly along $O$'s circumference. For each of those points, draw the arc centered at that point inside O from the point to its left to the point to its right. Denote the outermost intersections of these arcs as $A', B', C', D'$ and $E'$, where the prime of any point is opposite the point. The length of $AC'$ can be written as an expression $f(x)$, where $f$ is a trigonometric function. Find this expression.

Given positive integers $a,b$, let the integers $q,r$, with $0 \leq r < ab$, be such that $a^2 + b^2 = abq + r$. Prove that $q + r \leq ab + 1$ and find all equality cases.

An even number of politicians are sitting at a round table. After a break, they come back and sit down again in arbitrary places. Show that there must be two people with the same number of people sitting between them as before the break.. [b]Additional problem:[/b] Solve the problem when the number of people is in a form $6k+3$.

Let $O$ be the center of an equilateral triangle $ABC$ of area $1/\pi$. As shown in the diagram below, a circle centered at $O$ meets the triangle at points $D$, $E$, $F$, $G$, $H$, and $I$, which trisect each of the triangle's sides. Compute the total area of all six shaded regions. [asy] unitsize(90); pair A = dir(0); pair B = dir(120); pair C = dir(240); draw(A -- B -- C -- cycle); pair D = (2*A + B)/3; pair E = (A + 2*B)/3; pair F = (2*B + C)/3; pair G = (B + 2*C)/3; pair H = (2*C + A)/3; pair I = (C + 2*A)/3; draw(E -- F); draw(G -- H); draw(I -- D); draw(D -- G); draw(E -- H); draw(F -- I); pair O = (0, 0); real r = 1/sqrt(3); draw(circle(O, r)); fill(O -- D -- E -- cycle, gray); fill(O -- F -- G -- cycle, gray); fill(O -- H -- I -- cycle, gray); fill(arc(O, r, -30, 30) -- cycle, gray); fill(arc(0, r, 90, 150) -- cycle, gray); fill(arc(0, r, 210, 270) -- cycle, gray); label("$A$", A, A); label("$B$", B, B); label("$C$", C, C); label("$D$", D, unit(D)); label("$E$", E, unit(E)); label("$F$", F, unit(F)); label("$G$", G, unit(G)); label("$H$", H, unit(H)); label("$I$", I, unit(I)); label("$O$", O, C); [/asy]

The surface of a soccer ball is made up of black pentagons and white hexagons together. On the sides of each pentagon are nothing but hexagons, while on the sides of each border of hexagons alternately pentagons and hexagons. Determine from this information about the soccer ball , the number of its pentagons and its hexagons.

Find all $(x,y,z) \in {\mathbb{Z}}^3$ such that $x^{3}+y^{3}+z^{3}=x+y+z=3$.

What is the smallest positive integer $n$ such that $n=x^3+y^3$ for two different positive integer tuples $(x,y)$?

A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What is the probability that a reader guessing at random will match all three correctly? $\textbf{(A) }\frac{1}{9}\qquad\textbf{(B) }\frac{1}{6}\qquad\textbf{(C) }\frac{1}{4}\qquad\textbf{(D) }\frac{1}{3}\qquad \textbf{(E) }\frac{1}{2}$

In a triangle $ABC$ it is given $\tan A,\tan B,\tan C$ are integers. Find their values.

(a) Prove or find a counter-example: For every two complex numbers $z,w$ the following inequality holds: $$|z|+|w|\le|z+w|+|z-w|.$$(b) Prove that for all $z_1,z_2,z_3,z_4\in\mathbb C$: $$\sum_{k=1}^4|z_k|\le\sum_{1\le i<j\le4}|z_i+z_j|.$$

Roy's cat eats $\frac{1}{3}$ of a can of cat food every morning and $\frac{1}{4}$ of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing $6$ cans of cat food. On what day of the week did the cat finish eating all the cat food in the box? ${ \textbf{(A)}\ \text{Tuesday}\qquad\textbf{(B)}\ \text{Wednesday}\qquad\textbf{(C)}\ \text{Thursday}\qquad\textbf{(D)}}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}$

Let $ \left( x_n\right)_{n\ge 1} $ be a sequence of real numbers of the interval $ [1,\infty) . $ Suppose that the sequence $ \left( \left[ x_n^k\right]\right)_{n\ge 1} $ is convergent for all natural numbers $ k. $ Prove that $ \left( x_n\right)_{n\ge 1} $ is convergent. Here, $ [\beta ] $ means the greatest integer smaller than $ \beta . $

Let $ a_1, a_2, \ldots, a_{100}$ be nonnegative real numbers such that $ a^2_1 \plus{} a^2_2 \plus{} \ldots \plus{} a^2_{100} \equal{} 1.$ Prove that \[ a^2_1 \cdot a_2 \plus{} a^2_2 \cdot a_3 \plus{} \ldots \plus{} a^2_{100} \cdot a_1 < \frac {12}{25}. \] [i]Author: Marcin Kuzma, Poland[/i]