Let $ABCD$ and $A_1B_1C_1D_1$ be convex quadrangles in a plane, such that $AB=A_1B_1$, $BC=B_1C_1$, $CD=C_1D_1$ and $DA=D_1A_1$. Given that diagonals $AC$ and $BD$ are perpendicular to each other, prove that the same holds for diagonals $A_1C_1$ and $B_1D_1$.

Given are $5$ segments, such that from any three of them one can form a triangle. Prove that from some three of them one can form an acute-angled triangle.

Let $p_{1}, p_{2},...,p_{n}$, where $n>2$, be the first $n$ prime numbers. Prove that $\frac{1}{p_{1}^2}+\frac{1}{p_{2}^2}+...+\frac{1}{p_{n}^2}+\frac{1}{p_{1}p_{2}...p_{n}}<\frac{1}{2}$

There are $n$ coins in the pile. Two players play a game by alternately performing a move. A move consists of taking $5,7$ or $11$ coins away from the pile. The player unable to perform a move loses the game. Which player - the one playing first or second - has the winning strategy if: (a) $n=2001$; (b) $n=5000$?

Let $S=\{x^2+2y^2\mid x,y\in\mathbb Z\}$. If $a$ is an integer with the property that $3a$ belongs to $S$, prove that then $a$ belongs to $S$ as well.

Vertices of a square $ABCD$ of side $\frac{25}4$ lie on a sphere. Parallel lines passing through points $A,B,C$ and $D$ intersect the sphere at points $A_1,B_1,C_1$ and $D_1$, respectively. Given that $AA_1=2$, $BB_1=10$, $CC_1=6$, determine the length of the segment $DD_1$.

Determine all positive integers $ n$ for which there is a coloring of all points in space so that each of the following conditions is satisfied: (i) Each point is painted in exactly one color. (ii) Exactly $ n$ colors are used. (iii) Each line is painted in at most two different colors.

Let $S$ be the set of all $n$-tuples of real numbers, with the property that among the numbers $x_1,\frac{x_1+x_2}2,\ldots,\frac{x_1+x_2+\ldots+x_n}n$ the least is equal to $0$, and the greatest is equal to $1$. Determine $$\max_{(x_1,x_2,\ldots,x_n)\in S}\max_{1\le i,j\le n}(x_i-x_j)\qquad\text{and}\min_{(x_1,x_2,\ldots,x_n)\in S}\max_{1\le i,j\le n}(x_i-x_j).$$

Solve in positive integers \[ x^y + y = y^x + x \]

Let $x_1,x_2,\ldots,x_{2001}$ be positive numbers such that $$x_i^2\ge x_1^2+\frac{x_2^2}{2^3}+\frac{x_3^2}{3^3}+\ldots+\frac{x_{i-1}^2}{(i-1)^3}\enspace\text{for }2\le i\le2001.$$Prove that $\sum_{i=2}^{2001}\frac{x_i}{x_1+x_2+\ldots+x_{i-1}}>1.999$.

Let $k$ be a positive integer and $N_k$ be the number of sequences of length $2001$, all members of which are elements of the set $\{0,1,2,\ldots,2k+1\}$, and the number of zeroes among these is odd. Find the greatest power of $2$ which divides $N_k$.

Parallelogram $ABCD$ is the base of a pyramid $SABCD$. Planes determined by triangles $ASC$ and $BSD$ are mutually perpendicular. Find the area of the side $ASD$, if areas of sides $ASB,BSC$ and $CSD$ are equal to $x,y$ and $z$, respectively.