- Find the least natural number not divisible by $11$ satisfying the condition: by replaceing an arbitrary digit a of it by the digit $b$ ($b$ equals to $a+1$ or $a-1),$ the obtaining number is divisible by $11$.
- Consider an acute, scalene triangle $A B C$. Let $H$, $I$, $O$ be respectively its orthocenter, incenter and circumcenter. Prove that there is no vertex or there are exactly two vertices of triangle $A B C$ lying on the circle passing through $H$, $I$, $O$.
- Prove that $$x y z+2\left(x^{2}+y^{2}+z^{2}\right)+8 \geq 5(x+y+z)$$ for arbitrary positive real numbers $x, y, z$.
- Consider a board of size $5 \times 5 .$ Can we color $16$ small squares of this board so that in each subboard of size $2 \times 2$ there are at most two small squares which are colored?
- On the plane, let be given some points colored red and some points colored blue; points with distinct colors are joined so that
- each red point is joined with at least one blue point,
- each blue point is joined with one or two red points.

- Let $A B C$ be a right-angled triangle, with right angle at $A$. Find all sets of the six distinct points $(M, N, P, Q, R, S)$ satisfying simultaneously the following conditions
- $N$, $P$ lie on the side $A B$; $Q$, $R$ lie on the side $B C$; $S$, $M$ lie on the side $A C$.
- $M N=P Q=R S$.
- the hexagon $MNPQRS$ is a convex hexagon, inscribable in a circle, with concurrent principal diagonals $M Q$, $N R$, $P S$.
- Let be given a real number $k$ in the interval $(-1 ; 2)$ and three pairwise distinct real numbers $a$, $b$, $c$. Prove that $$\left(a^{2}+b^{2}+c^{2}+k(a b+b c+c a)\right) \left(\frac{1}{(a-b)^{2}}+\frac{1}{(b-c)^{2}}+\frac{1}{(c-a)^{2}}\right) \geq \frac{9(2-k)}{4}.$$ When does equality occur?
- Does there exist a positive integer a such that in the sequence of numbers $\left(a_{n}\right)$ defined by $a_{n}=n^{3}+a^{3}$ for all $n=1,2,3, \ldots$ every two consecutive terms are coprime integers?
- Does there exist a positive integer $n$ such that one can assign to each vertex $A_{1}, A_{2}, \ldots, A_{n}$ of a convex $n$ -polygon an integer (these $n$ integers are not necessarily distinct) so that
- the sum of these $n$ integers is equal to $2007$, and
- for every $i=1,2, \ldots, n,$ the number assigned to $A_{1}$ is equal to the absolute value of the difference of the numbers assigned to $A_{t=1}$ and $A_{1}+2\left(\right.$ with the convention $A_{n}+1 \equiv A_{1}$ and $\left.A_{n}+2 \equiv A_{2}\right) ?$
- Suppose that in the coordinate plane every integral point (i. e. point with integral coordinates) was colored in one of two given colors. Prove that there exists a infinite set of integral points of the same color. forming. a figure admitting a center of symmetry.
- Let $A^{\prime}, B^{\prime}, C^{\prime}$ be the feet of the altitudes of nonright triangle $A B C$ issued from $A$, $B$, $C$ respectively. Let $D$, $E$, $F$ be the incenters of triangles $A B^{\prime} C^{\prime}$, $B C^{\prime} A^{\prime}$, $C A^{\prime} B^{\prime}$ respectively. Calculate the circumradius of triangle $D E F$ in terms of $B C=a$, $C A=b$, $A B=c$.
- Let be given a natural number $n>6$. Consider all natural number belonging to the interval $\left(n(n-1) ; n^{2}\right)$ which are coprime with $n(n-1) .$ Prove that the greates common divisor of these natural numbers is $1$.
- Find all functions $f$ defined on the set of all real number $\mathbb{R},$ with values in $\mathbb{R},$ satisfying the condition $$f(x-y)+f(x y)=f(x)-f(y)+f(x) \cdot f(y)$$ for all real numbers $x$, $y$.
- A group of students consists of $15$ boys and $15$ girls. On The Day of Women, some boys congratulated some girls by telephone. Suppose that we can only partition the group into 15 pairs such that each pair consists of a boy and a girl which had been congratulated by this boy. How many congratulations there were at most?
- Describe all ways to color all natural numbers in blue, red and yellow so that
- each natural number is colored in one color and each color is used to color an infinite set of natural numbers,
- the number 2 is not colored in blue,
- if the natural number $a$ is colored in red and the natural number $b$ is colored in yellow then the number $a+b$ is colored in blue.