Let $A', B', C', D', E'$ and $F'$ be the midpoints of the sides $AB$, $BC$, $CD$, $DE$, $EF$ and $FA$ of an arbitrary convex hexagon $ABCDEF$ respectively. Express the area of $ABCDEF$ in terms of the areas of the triangles $ABC$, $BCD'$, $CDS'$, $DEF'$, $EFA'$ and $FAB'$. (A Lopshi tz, NB Vassiliev)

In triangle $ABC$, let $D$ be on side $BC$. The line through $D$ parallel to $AB,AC$ meet $AC,AB$ at $E,F$, respectively. (a) Show that if $D$ varies on line $BC$, the circumcircle of $AEF$ passes through a fixed point $T$. (b) Show that if $D$ lies on line $AT$, then the circumcircle of $AEF$ is tangent to the circumcircle of $BTC$.

Let $ABCD$ be a convex quadrilateral with side lengths satisfying the equality: $$ AB \cdot CD = AD \cdot BC = AC \cdot BD.$$ Determine the sum of the acute angles of quadrilateral $ABCD$. [i]Proposed by Zaza Meliqidze, Georgia[/i]

We define the [i]Fibonacci sequence[/i] $\{F_n\}_{n\ge0}$ by $F_0=0$, $F_1=1$, and for $n\ge2$, $F_n=F_{n-1}+F_{n-2}$; we define the [i]Stirling number of the second kind[/i] $S(n,k)$ as the number of ways to partition a set of $n\ge1$ distinguishable elements into $k\ge1$ indistinguishable nonempty subsets. For every positive integer $n$, let $t_n = \sum_{k=1}^{n} S(n,k) F_k$. Let $p\ge7$ be a prime. Prove that \[ t_{n+p^{2p}-1} \equiv t_n \pmod{p} \] for all $n\ge1$. [i]Proposed by Victor Wang[/i]

Eight students were asked to solve $8$ problems (the same set of problems for each of the students). (a) Each problem was solved by $5$ students. Prove that one canfind two students so that each of the problems was solved by at least one of them. (b) If each problem was solved by $4$ students, then it is possible that no such pair of students exists. Prove this. (S Tokarev)

$p(x)$ is an irreducible polynomial with integer coefficients, and $q$ is a fixed prime number. Let $a_n$ be a number of solutions of the equation $p(x)\equiv 0\mod q^n$. Prove that we can find $M$ such that $\{a_n\}_{n\ge M}$ is constant.

Let $ABC$ be a triangle and let $D$ be a point in the interior of the triangle which lies on the angle bisector of $\angle BAC$. Suppose that lines $BD$ and $AC$ meet at $E$, and that lines $CD$ and $AB$ meet at $F$. The circumcircle of $ABC$ intersects line $EF$ at points $P$ and $Q$. Show that if $O$ is the circumcenter of $DPQ$, then $OD$ is perpendicular to $BC$. [i]Michael Ren[/i]

In the acute triangle $ABC$ point $M$ is the midpoint of $AC$ and $N$ is the foot of the height of $A$ on $BC$. Point $D$ is on the circumcircle of triangle $BMN$ such that $AD$ and $BM$ are parallel and $AC$ is between the points $B$ and $D$. Prove that $BD=BC$.

Find the maximal possible numbers one can choose from $1,\ldots,100$ such that none of the products of non-empty subset of this numbers was a perfect square.

Let $a, b,c$ be positive real numbers satisfying the condition $$(x + y + z) \left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right)= 10$$ Find the greatest value and the least value of $$T = (x^2 + y^2 + z^2) \left(\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2}\right)$$ Trần Nam Dũng

Let $p=1601$. Prove that if \[\dfrac {1} {0^2+1}+\dfrac{1}{1^2+1}+\cdots+\dfrac{1}{(p-1)^2+1}=\dfrac{m} {n},\] where we only sum over terms with denominators not divisible by $p$ (and the fraction $\dfrac {m} {n}$ is in reduced terms) then $p \mid 2m+n$. [i]Proposed by A. Golovanov[/i]

Let $ p>5$ be a prime number. Prove that every algebraic integer of the $ p$th cyclotomic field can be represented as a sum of (finitely many) distinct units of the ring of algebraic integers of the field. [i]K. Gyory[/i]

Determine the number of pairs of positive integers $(a, b)$, with $a \le b$, for which lcm $(a, b) = 2004$. lcm ($a, b$) means the least common multiple of $a$ and $b$. Example: lcm $(18, 24) = 72$.

In tetrahedron $ABCD,$ as shown below, compute the number of ways to start at $A,$ walk along some path of edges, and arrive back at $A$ without walking over the same edge twice. [Insert Diagram] [i]Proposed by Richard Chen[/i]

Positive integers 1 to 9 are written in each square of a $ 3 \times 3 $ table. Let us define an operation as follows: Take an arbitrary row or column and replace these numbers $ a, b, c$ with either non-negative numbers $ a-x, b-x, c+x $ or $ a+x, b-x, c-x$, where $ x $ is a positive number and can vary in each operation. (1) Does there exist a series of operations such that all 9 numbers turn out to be equal from the following initial arrangement a)? b)? \[ a) \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} )\] \[ b) \begin{array}{ccc} 2 & 8 & 5 \\ 9 & 3 & 4 \\ 6 & 7 & 1 \end{array} )\] (2) Determine the maximum value which all 9 numbers turn out to be equal to after some steps.

You are given a word consisting of letters $a;b;c$ You can apply 3 operations on this word: [b]1)[/b] You can write any $3$ letter long $\text{subword}$ in reverse. ($\text{xyz}\rightarrow \text{zyx}$) [b]2)[/b] You can add same $2$ letters between any $2$ consecutive letters. ($\text{xyxy}\rightarrow \text{xy}$[b]zz[/b]$\text{xy}$) [b]3)[/b] You can remove any $\text{subword}$ in the given form $\text{xyyx}$ ($\text{x}$[b]yzzy[/b]$\text{xy}\rightarrow\text{xxy}$) Given these $3$ operations, can you go from $\text{abccab}$ to $\text{baccba}$? (Note: A $\text{subword}$ is a string of consecutive letters from the given word)

Let $P=A_1A_2\cdots A_k$ be a convex polygon in the plane. The vertices $A_1, A_2, \ldots, A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $n$ is given such that the squares of the side lengths of $P$ are integers divisible by $n$. Prove that $2S$ is an integer divisible by $n$.

An isosceles triangle $ABC$ with base $AC$ is given. On the rays $CA$, $AB$ and $BC$, the points $D, E$ and $F$ were marked, respectively, in such a way that $AD = AC$, $BE = BA$ and $CF = CB$. Find the sum of the angles $\angle ADB$, $\angle BEC$ and $\angle CFA$.

Let $S$ be the set of all points in the plane whose coordinates are positive integers less than or equal to $100$ (so $S$ has $100^2$ elements), and let $L$ be the set of all lines $\ell$ such that $\ell$ passes through at least two points in $S$. Find, with proof, the largest integer $N \geq 2$ for which it is possible to choose $N$ distinct lines in $L$ such that every two of the chosen lines are parallel.

$20$ football teams take part in a tournament . On the first day all the teams play one match . On the second day all the teams play a further match . Prove that after the second day it is possible to select $10$ teams, so that no two of them have yet played each other. ( S . A . Genkin)

Determine all pairs of non-negative integers $(m, n)$ with m ≥n, such that $(m+n)^3$ divides $2n(3m^2+n^2)+8$

Let $ \{\xi_{k \ell} \}_{k,\ell=1}^{\infty}$ be a double sequence of random variables such that \[ \Bbb{E}( \xi_{ij} \xi_{k\ell})= \mathcal{O} \left(\frac{ \log(2|i-k|+2)}{ \log(2|j-\ell|+2)^{2}}\right) \;\;\;(i,j,k,\ell =1,2, \ldots ) \\\ .\] Prove that with probability one, \[ \frac{1}{mn} \sum_{k=1}^m \sum_{\ell=1}^n \xi_{k\ell} \rightarrow 0 \;\;\textrm{as} \; \max (m,n)\rightarrow \infty\ \\ .\] [i]F. Moricz[/i]

Find the maximum of $x+y$ given that $x$ and $y$ are positive real numbers that satisfy \[x^3+y^3+(x+y)^3+36xy=3456.\]

Find all real values of a for which the equation $ (a \minus{} 3x^2 \plus{} \cos \frac {9\pi x}{2})\sqrt {3 \minus{} ax} \equal{} 0$ has an odd number of solutions in the interval $ [ \minus{} 1,5]$

Let $c>2$ and $a_0,a_1, \ldots$ be a sequence of real numbers such that \begin{align*} a_n = a_{n-1}^2 - a_{n-1} < \frac{1}{\sqrt{cn}} \end{align*} for any $n$ $\in$ $\mathbb{N}$. Prove, $a_1=0$