Školjka

%V0 Initially, only the integer $44$ is written on a board. An integer a on the board can be re- placed with four pairwise different integers $a_1, a_2, a_3, a_4$ such that the arithmetic mean $\frac 14 (a_1 + a_2 + a_3 + a_4)$ of the four new integers is equal to the number $a$. In a step we simultaneously replace all the integers on the board in the above way. After $30$ steps we end up with $n = 4^{30}$ integers $b_1, b2,\ldots, b_n$ on the board. Prove that $$\frac{b_1^2 + b_2^2+b_3^2+\cdots+b_n^2}{n}\geq 2011.$$

%V0 Let $n \geq 3$ be an integer. John and Mary play the following game: First John labels the sides of a regular $n$-gon with the numbers $1, 2,\ldots, n$ in whatever order he wants, using each number exactly once. Then Mary divides this $n$-gon into triangles by drawing $n-3$ diagonals which do not intersect each other inside the $n$-gon. All these diagonals are labeled with number $1$. Into each of the triangles the product of the numbers on its sides is written. Let S be the sum of those $n - 2$ products. Determine the value of $S$ if Mary wants the number $S$ to be as small as possible and John wants $S$ to be as large as possible and if they both make the best possible choices.

%V0 In a plane the circles $\mathcal K_1$ and $\mathcal K_2$ with centers $I_1$ and $I_2$, respectively, intersect in two points $A$ and $B$. Assume that $\angle I_1AI_2$ is obtuse. The tangent to $\mathcal K_1$ in $A$ intersects $\mathcal K_2$ again in $C$ and the tangent to $\mathcal K_2$ in $A$ intersects $\mathcal K_1$ again in $D$. Let $\mathcal K_3$ be the circumcircle of the triangle $BCD$. Let $E$ be the midpoint of that arc $CD$ of $\mathcal K_3$ that contains $B$. The lines $AC$ and $AD$ intersect $\mathcal K_3$ again in $K$ and $L$, respectively. Prove that the line $AE$ is perpendicular to $KL$.

%V0 Let $k$ and $m$, with $k > m$, be positive integers such that the number $km(k^2 - m^2)$ is divisible by $k^3 - m^3$. Prove that $(k - m)^3 > 3km$.

%V0 Find all functions $f \colon \mathbb R \to \mathbb R$ such that the equality $$y^2f(x) + x^2f(y) + xy = xyf(x + y) + x^2 + y^2$$ holds for all $x, y \in \Bbb R$, where $\Bbb R$ is the set of real numbers.

%V0 Let $a, b, c$ be positive real numbers such that $$\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=2\text{.}$$ Prove that $$\frac{\sqrt a + \sqrt b+\sqrt c}{2} \geq \frac{1}{\sqrt a}+\frac{1}{\sqrt b}+\frac{1}{\sqrt c}\text{.}$$

%V0 For an integer $n \geq 3$, let $\mathcal M$ be the set $\{(x, y) | x, y \in \mathbb Z, 1 \leq x \leq n, 1 \leq y \leq n\}$ of points in the plane. What is the maximum possible number of points in a subset $S \subseteq \mathcal M$ which does not contain three distinct points being the vertices of a right triangle?

%V0 Let $n \geq 3$ be an integer. At a MEMO-like competition, there are $3n$ participants, there are n languages spoken, and each participant speaks exactly three different languages. Prove that at least $\left\lceil\frac{2n}{9}\right\rceil$ of the spoken languages can be chosen in such a way that no participant speaks more than two of the chosen languages. Note. $\lceil x\rceil$ is the smallest integer which is greater than or equal to $x$.

%V0 Let $ABCDE$ be a convex pentagon with all five sides equal in length. The diagonals $AD$ and $EC$ meet in $S$ with $\angle ASE = 60^\circ$. Prove that $ABCDE$ has a pair of parallel sides.

%V0 Let $ABC$ be an acute triangle. Denote by $B_0$ and $C_0$ the feet of the altitudes from vertices $B$ and $C$, respectively. Let $X$ be a point inside the triangle $ABC$ such that the line $BX$ is tangent to the circumcircle of the triangle $AXC_0$ and the line $CX$ is tangent to the circumcircle of the triangle $AXB_0$. Show that the line $AX$ is perpendicular to $BC$.

%V0 Let $A$ and $B$ be disjoint nonempty sets with $A \cup B = \{1, 2,3, \ldots, 10\}$. Show that there exist elements $a \in A$ and $b \in B$ such that the number $a^3 + ab^2 + b^3$ is divisible by $11$.

%V0 We call a positive integer $n$ amazing if there exist positive integers $a, b, c$ such that the equality $$n = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab)$$ holds. Prove that there exist $2011$ consecutive positive integers which are amazing. Note. By $(m, n)$ we denote the greatest common divisor of positive integers $m$ and $n$.