Prove that circles which have sides of a convex quadrilateral as diameters cover its interior. (Convex polygon is the one which contains with any two points the whole segment, joining them).

Inscribe a rectangle of base $b$ and height $h$ in a circle of radius one, and inscribe an isosceles triangle in the region of the circle cut off by one base of the rectangle (with that side as the base of the triangle). For what value of $h$ do the rectangle and triangle have the same area?

A set of $n$ points in Euclidean 3-dimensional space, no four of which are coplanar, is partitioned into two subsets $\mathcal{A}$ and $\mathcal{B}$. An $\mathcal{AB}$-tree is a configuration of $n-1$ segments, each of which has an endpoint in $\mathcal{A}$ and an endpoint in $\mathcal{B}$, and such that no segments form a closed polyline. An $\mathcal{AB}$-tree is transformed into another as follows: choose three distinct segments $A_1B_1$, $B_1A_2$, and $A_2B_2$ in the $\mathcal{AB}$-tree such that $A_1$ is in $\mathcal{A}$ and $|A_1B_1|+|A_2B_2|>|A_1B_2|+|A_2B_1|$, and remove the segment $A_1B_1$ to replace it by the segment $A_1B_2$. Given any $\mathcal{AB}$-tree, prove that every sequence of successive transformations comes to an end (no further transformation is possible) after finitely many steps.

Let $f : N \to N$ be a function such that $f(f(1995)) = 95, f(xy) = f(x)f(y)$ and $f(x) \le x$ for all $x,y$. Find all possible values of $f(1995)$.

Let $m$ be a positive integer, and let $T$ denote the set of all subsets of $\{1, 2, \dots, m\}$. Call a subset $S$ of $T$ $\delta$-[I]good[/I] if for all $s_1, s_2\in S$, $s_1\neq s_2$, $|\Delta (s_1, s_2)|\ge \delta m$, where $\Delta$ denotes the symmetric difference (the symmetric difference of two sets is the set of elements that is in exactly one of the two sets). Find the largest possible integer $s$ such that there exists an integer $m$ and $ \frac{1024}{2047}$-good set of size $s$.

It is known that there exists a point $P$ within the interior of $\triangle ABC$ satisfying the following conditions: (i) $\angle PAB \ge 30^\circ$ and $\angle APB \ge \angle PCB + 30^\circ$; (ii) $BP \cdot BC=CP \cdot AB.$ Prove that $\angle BAC \ge 60^\circ$, and that equality holds only when $\triangle ABC$ is equilateral.

Let $\triangle ABC$ be a triangle with $AB =AC$ and $\angle BAC = 20^{\circ}$. Let $D$ be point on the side $AB$ such that $\angle BCD = 70^{\circ}$. Let $E$ be point on the side $AC$ such that $\angle CBE = 60^{\circ}$. Determine the value of angle $\angle CDE$.

The incircle of triangle $ABC$ touches its sides $AB$ and $BC$ at points $P$ and $Q.$ The line $PQ$ meets the circumcircle of triangle $ABC$ at points $X$ and $Y.$ Find $\angle XBY$ if $\angle ABC = 90^\circ.$ [i]Proposed by A. Smirnov[/i]

Hexagon $ARTSCI$ has side lengths $AR=RT=TS=SC=4\sqrt2$ and $CI=IA=10\sqrt2$. Moreover, the vertices $A$, $R$, $T$, $S$, $C$, and $I$ lie on a circle $\mathcal{K}$. Find the area of $\mathcal{K}$.

Positive integers $ a<b$ are given. Prove that among every $ b$ consecutive positive integers there are two numbers whose product is divisible by $ ab$.

Given any $60$ points on a circle of radius $1$, prove that there is a point on the circle the sum of whose distances to these $60$ points is at most $80$.

For each point $X$ in the plane, a real number $r_X > 0$ is assigned such that $2|r_X - r_Y | \le |XY |$, for any two points $X, Y$ . (Here $|XY |$ denotes the distance between $X$ and $Y$) A frog can jump from $X$ to $Y$ if $r_X = |XY |$. Show that for any two points $X$ and $Y$ , the frog can jump from $X$ to $Y$ in a finite number of steps.

Let $x$ and $y$ be real numbers satisfying \[(x^2+x-1)(x^2-x+1)=2(y^3-2\sqrt{5}-1)\] and \[(y^2+y-1)(y^2-y+1)=2(x^3+2\sqrt{5}-1)\] Find $8x^2+4y^3$.

Does there exist a $6$-digit number $A$ such that none of its $500 000$ multiples $A$, $2A$, $3A$, ..., $500 000A$ ends in $6$ identical digits? (S Tokarev)

In a convex quadrilateral $ABCD$, let $AB + CD = BC + AD$. Prove that the circle inscribed in $ABC$ is tangent to the circle inscribed in $ACD$.

We call a monic polynomial $P(x) \in \mathbb{Z}[x]$ [i]square-free mod n[/i] if there [u]dose not[/u] exist polynomials $Q(x),R(x) \in \mathbb{Z}[x]$ with $Q$ being non-constant and $P(x) \equiv Q(x)^2 R(x) \mod n$. Given a prime $p$ and integer $m \geq 2$. Find the number of monic [i]square-free mod p[/i] $P(x)$ with degree $m$ and coeeficients in $\{0,1,2,3,...,p-1\}$. [i]Proposed by Masud Shafaie[/i]

The numbers, in order, of each row and the numbers, in order, of each column of a $5 \times 5$ array of integers form an arithmetic progression of length $5{.}$ The numbers in positions $(5, 5), \,(2,4),\,(4,3),$ and $(3, 1)$ are $0, 48, 16,$ and $12{,}$ respectively. What number is in position $(1, 2)?$ \[ \begin{bmatrix} . & ? &.&.&. \\ .&.&.&48&.\\ 12&.&.&.&.\\ .&.&16&.&.\\ .&.&.&.&0\end{bmatrix}\] $\textbf{(A) } 19 \qquad \textbf{(B) } 24 \qquad \textbf{(C) } 29 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 39$

In making Euclidean constructions in geometry it is permitted to use a ruler and a pair of compasses. In the constructions considered in this question no compasses are permitted, but the ruler is assumed to have two parallel edges, which can be used for constructing two parallel lines through two given points whose distance is at least equal to the breadth of the rule. Then the distance between the parallel lines is equal to the breadth of the ruler. Carry through the following constructions with such a ruler. Construct: [b]a)[/b] The bisector of a given angle. [b]b)[/b] The midpoint of a given rectilinear line segment. [b]c)[/b] The center of a circle through three given non-collinear points. [b]d)[/b] A line through a given point parallel to a given line.

There are $n$ students $A_1,A_2,...,A_n$ and some of them shaked hands with each other. ($A_i$ and $A-j$ can shake hands more than one time.) Let the student $A_i$ shaked hands $d_i$ times. Suppose $d_1 + d_2 +... + d_n > 0$. Prove that there exist $1 \le i < j \le n$ satisfying the following conditions: (a) Two students $A_i$ and $A_j$ shaked hands each other. (b) $\frac{(d_1 + d_2 +... + d_n)^2}{n^2}\le d_id_j$

Let $n \geq 2$ be an integer. Define $x_i =1$ or $-1$ for every $i=1,2,3,\cdots, n$. Call an operation [i]adhesion[/i], if it changes the string $(x_1,x_2,\cdots,x_n)$ to $(x_1x_2, x_2x_3, \cdots ,x_{n-1}x_n, x_nx_1)$ . Find all integers $n \geq 2$ such that the string $(x_1,x_2,\cdots, x_n)$ changes to $(1,1,\cdots,1)$ after finitely [i]adhesion[/i] operations.

Define $f$ on the positive integers by $f(n) = k^2 + k + 1$, where $n=2^k(2l+1)$ for some $k,l$ nonnegative integers. Find the smallest $n$ such that $f(1) + f(2) + ... + f(n) \geq 123456$.

Evaluate the infinite sum $ \sum_{n\equal{}1}^{\infty}\frac{n}{n^4\plus{}4}$.

Find the greatest and smallest value of $f(x, y) = y-2x$, if x, y are distinct non-negative real numbers with $\frac{x^2+y^2}{x+y}\leq 4$.

Denote by $S(k)$ the sum of the digits in the decimal representation of $k$. Prove that there are infinitely many $n\in \mathbb{Z_{+}}$ such that: ${S(2^{n}+n})<S(2^{n})$.

Let $p_n$ be the probability that $c+d$ is a perfect square when the integers $c$ and $d$ are selected independently at random from the set $\{1,2,\ldots,n\}$. Show that $\lim_{n\to\infty}p_n\sqrt n$ exists and express this limit in the form $r(\sqrt s-t)$, where $s$ and $t$ are integers and $r$ is a rational number.