A real number is assigned to each vertex of a triangular prism so that the number on any vertex is the arithmetic mean of the numbers on the three adjacent vertices. Prove that all six numbers are equal.

In this exercise we only consider convex quadrilaterals. (a) For such a quadrilateral $ABCD$, determine the set of points $P$ contained within that quadrilateral for which $PA$ and $PC$ divide the quadrilateral into two pieces of equal areas. (b) Prove that there is a point $P$ inside such a quadrilateral, such that the triangles $PAB$ and $PCD$ have equal areas, as well as the triangles $PBC$ and $PAD$. (c) Find out which quadrilaterals $ABCD$ contains a point $P$, so that the triangles $PAB$, $PBC$, $PCD$ and $PDA$ have equal areas.

Let $(a_1, a_2, ..., a_8)$ be a permutation of $(1, 2, ... , 8)$. Find, with proof, the maximum possible number of elements of the set $$\{a_1, a_1 + a_2, ... , a_1 + a_2 + ... + a_8\}$$ that can be perfect squares.

Let $S$ be the total area of a tetrahedron whose edges have lengths $a,b,c,d, e, f$ . Prove that $S \le \frac{\sqrt3}{6} (a^2 +b^2 +...+ f^2)$

A grasshopper jumps either $364$ or $715$ units on the real number line. If it starts from the point $0$, what is the smallest distance that the grasshoper can be away from the point $2010$? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 34 \qquad\textbf{(E)}\ 164 $

All the fields of a square $n\times n$ (n>2) table are filled with $+1$ or $-1$ according to the rules: [i]At the beginning $-1$ are put in all the boundary fields. The number put in the field in turn (the field is chosen arbitrarily) equals to the product of the closest, from the different sides, numbers in its row or in its column. [/i] a) What is the minimal b) What is the maximal possible number of $+1$ in the obtained table?

For any natural number $m \ge 1$ and any real number $x \ge 0$ we define expression $$E (x, m) = \frac{(1^4 + x) (3^4 + x) (5^4 + x) ... [(2m -1)^ 4 + x]}{(2^4 + x) (4^4 + x) (6^4 + x) ... [(2m )^ 4 + x]}.$$ It is known that $E\left(\frac{1}{4},m\right)=\frac{1}{1013}.$ . Determine the value of $m$

Let m be a positive integer. Prove that there exist infinitely many prime numbers p such that m+p^3 is composite.

Let $I$ be the incenter of a triangle $ABC$, $D$ be an arbitrary point of segment $AC$, and $A_{1}, A_{2}$ be the common points of the perpendicular from $D$ to the bisector $CI$ with $BC$ and $AI$ respectively. Define similarly the points $C_{1}$, $C_{2}$. Prove that $B$, $A_{1}$, $A_{2}$, $I$, $C_{1},$ $C_{2}$ are concyclic. Proposed by:D.Shvetsov

For what pairs of positive real numbers $(a,b)$ does the improper integral $(1)$ converge? \begin{align}\int_{b}^{\infty}\left(\sqrt{\sqrt{x+a}-\sqrt{x}}-\sqrt{\sqrt{x}-\sqrt{x-b}}\right)\,\mathrm{d}x \end{align}

For which natural numbers $ n $ the floor of the number $ \frac{n^3+8n^2+1}{3n} $ is prime? [i]Gabriel Popa[/i]

The game of pool includes $15$ balls that fit within a triangular rack as shown: [asy] // thanks Ritwin for this diagram :D unitsize(0.6cm); pair pos(real i, real j) { return i*dir(60) + (j,0); } for (int i = 0; i <= 4; ++i) { for (int j = 0; j <= 4-i; ++j) { draw(circle(pos(i,j), .5)); } } pair A = pos(0,0); pair B = pos(0,4); pair C = pos(4,0); pair dd = dir(270) * .5; pair ul = dir(150) * .5; pair ur = dir( 30) * .5; real S = 1.75; draw(A+dd -- B+dd ^^ B+ur -- C+ur ^^ C+ul -- A+ul ); draw(A+dd*S -- B+dd*S ^^ B+ur*S -- C+ur*S ^^ C+ul*S -- A+ul*S); draw(arc(A, A+ul*S, A+dd*S)); draw(arc(B, B+dd*S, B+ur*S)); draw(arc(C, C+ur*S, C+ul*S)); [/asy] Seven of the balls are "striped" (not colored with a single color) and eight are "solid" (colored with a single color). Prove that no matter how the $15$ balls are arranged in the rack, there must always be a pair of striped balls adjacent to each other.

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

Let $S=2^3+3^4+5^4+7^4+\cdots+17497^4$ be the sum of the fourth powers of the first $2014$ prime numbers. Find the remainder when $S$ is divided by $240$.

(a)The game of "super- chess" is played on a $30 \times 30$ board and involves $20$ different pieces. Each piece moves according to its own rules , but cannot move from any square to more than $20$ other squares . A piece "captures" another piece which is on a square to which it has moved. A permitted move (e.g. $m$ squares forward and $n$ squares to the right) does not depend on the piece 's starting square . Prove that (i) A piece cannot cap ture a piece on a given square from more than $20$ starting squares. (ii) It is possible to arrange all $20$ pieces on the board in such a way that not one of them can capture any of the others in one move. (b) The game of "super-chess" is played on a $100 \times 100$ board and involves $20$ different pieces. Each piece moves according to its own rules , but cannot move from any square to more than $20$ other squares. A piece "captures" another piece which is on a square to which it has moved. It is possible that a permitted move (e.g. $m$ squares forward and $n$ squares to the right) may vary, depending on the piece's position . Prove that one can arrange all $20$ pieces on the board in such a way that not one of them can capture any of the others in one move. ( A . K . Tolpygo, Kiev) PS. (a) for Juniors , (b) for Seniors

Let $O$ be the circumcenter of the triangle $ABC$. The segment $XY$ is the diameter of the circumcircle perpendicular to $BC$ and it meets $BC$ at $M$. The point $X$ is closer to $M$ than $Y$ and $Z$ is the point on $MY$ such that $MZ = MX$. The point $W$ is the midpoint of $AZ$. a) Show that $W$ lies on the circle through the midpoints of the sides of $ABC$; b) Show that $MW$ is perpendicular to $AY$.