Let $BC$ be a chord of a circle $\Gamma$ and $A$ be a point inside $\Gamma$ such that $\angle BAC$ is acute. Outside $\triangle ABC$, construct two isosceles triangles $\triangle ACP$ and $\triangle ABR$ such that $\angle ACP$ and $\angle ABR$ are right angles. Let lines $BA$ and $CA$ meet $\Gamma$ again at points $E$ and $F$, respectively. Let lines $EP$ and $FR$ meet $\Gamma$ again at points $X$ and $Y$, respectively. Prove that $BX=CY$.

Given an integer number $n \geq 3$, consider $n$ distinct points on a circle, labelled $1$ through $n$. Determine the maximum number of closed chords $[ij]$, $i \neq j$, having pairwise non-empty intersections. [i]János Pach[/i]

An infinite sequence of distinct real numbers is given. Prove that it contains a subsequence of $1999$ terms which is either monotonically increasing or monotonically decreasing.

Find all real-valued functions $f$ defined on the set of real numbers such that $$f(f(x)+y)+f(x+f(y))=2f(xf(y))$$ for any real numbers $x$ and $y$.

Al and Barb start their new jobs on the same day. Al's schedule is $3$ work-days followed by $1$ rest-day. Barb's schedule is $7$ work-days followed by $3$ rest-days. On how many of their first $1000$ days do both have rest-days on the same day? $ \textbf{(A)}\ 48 \qquad\textbf{(B)}\ 50 \qquad\textbf{(C)}\ 72 \qquad\textbf{(D)}\ 75 \qquad\textbf{(E)}\ 100 $

The product of the digits of the natural number $N$ is denoted by $P(N)$ whereas the sum of these digits is denoted by $S(N)$. How many solutions does the equation $P(P(N)) + P(S(N)) + S(P(N)) + S(S(N)) = 1984$ have?

There are $101$ chess players who participated in several tournaments. There was no tournament in which all of them participated. Each pair of these $101$ players met exactly once during these tournaments. Prove that one of them participated in no less than $11$ tournaments. (Assume that each pair of participants in each tournament plays each other once in that tournament). (A Andjans, Riga)

Find the least positive integer $n$ greater than $1$ such that $n^3 -n^2$ is divisible by $7^2 \times 11$. [i]Proposed by Jacob Xu[/i]

Consider a natural number $ n $ for which it exist a natural number $ k $ and $ k $ distinct primes so that $ n=p_1\cdot p_2\cdots p_k. $ [b]a)[/b] Find the number of functions $ f:\{ 1, 2,\ldots , n\}\longrightarrow\{ 1,2,\ldots ,n\} $ that have the property that $ f(1)\cdot f(2)\cdots f\left( n \right) $ divides $ n. $ [b]b)[/b] If $ n=6, $ find the number of functions $ f:\{ 1, 2,3,4,5,6\}\longrightarrow\{ 1,2,3,4,5,6\} $ that have the property that $ f(1)\cdot f(2)\cdot f(3)\cdot f(4)\cdot f(5)\cdot f(6) $ divides $ 36. $

A copsychus and a sparrow, each initially located at one of the vertex of a regular polygon with $103$ edges, fly clockwise to another vertex each. The copsychus moves across $\ell$ edges each time while the sparrow moves through$ d$ edges of the polygon, where $\ell \ne d$ are both integers less than $103$. Assume that, during their journeys, the copsychus has stopped at $m$ vertices while sparrow has stopped at $n$ vertices of the polygon, for $m \ge n \ge 3$. Determine the value of $m, n$ given that there is only one common single vertex of the polygon that both of birds have stopped at, and there is only one vertex that neither of the birds have reached. Vu Thi Khoi, Topo University, Hanoi Mathematics Institute, Vietnam, Hoang Qu6c Vietnam, Hanoi.

A regular hexagon $ABCDEF$ has area $36$. Find the area of the region which lies in the overlap of the triangles $ACE$ and $BDF$. $\text{(A) }3\qquad\text{(B) }9\qquad\text{(C) }12\qquad\text{(D) }18\qquad\text{(E) }24$

Given such positive real numbers $a, b$ and $c$, that the system of equations: $ \{\begin{matrix}a^2x+b^2y+c^2z=1&&\\xy+yz+zx=1&&\end{matrix} $ has exactly one solution of real numbers $(x, y, z)$. Prove, that there is a triangle, which borders lengths are equal to $a, b$ and $c$.

For a positive integer $ n$, let $ S(n)$ denote the sum of its digits. Find the largest possible value of the expression $ \frac {S(n)}{S(16n)}$.

The sequence ($a_n$) of natural numbers is defined by $a_1 = 2$ and $a_{n+1}$ equals the sum of tenth powers of the decimal digits of $a_n$ for all $n \ge 1$. Are there numbers which appear twice in the sequence ($a_n$)?

The expression $*3^5*3^4*3^3*3^2*3*1$ is given. Ana and Branka alternately change the signs $*$ to $+$ or $-$ (one time each turn). Can Branka, who plays second, do this so as to obtain an expression whose value is divisible by $7$?

There are $1978$ points in the flat plane. Each point has a circular disk with that point as its center and the radius is the distance to a fixed point. Prove that there are five of these circular disks, which together cover all $1978$ points (circular disk means: the circle and its inner area).