Školjka

%V0 Let $a$ and $b$ be non-negative integers such that $ab \geq c^2,$ where $c$ is an integer. Prove that there is a number $n$ and integers $x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n$ such that $$\sum^n_{i=1} x^2_i = a, \sum^n_{i=1} y^2_i = b, \text{ and } \sum^n_{i=1} x_iy_i = c.$$

%V0 Let $A,B,C,D$ be four distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at $X$ and $Y$. The line $XY$ meets $BC$ at $Z$. Let $P$ be a point on the line $XY$ other than $Z$. The line $CP$ intersects the circle with diameter $AC$ at $C$ and $M$, and the line $BP$ intersects the circle with diameter $BD$ at $B$ and $N$. Prove that the lines $AM,DN,XY$ are concurrent.

%V0 Let $A, B$ and $C$ be non-collinear points. Prove that there is a unique point $X$ in the plane of $ABC$ such that $$XA^2 + XB^2 + AB^2 = XB^2 + XC^2 + BC^2 = XC^2 + XA^2 + CA^2.$$

%V0 Let $\mathbb{Z}$ denote the set of all integers. Prove that for any integers $A$ and $B,$ one can find an integer $C$ for which $M_1 = \{x^2 + Ax + B : x \in \mathbb{Z}\}$ and $M_2 = {2x^2 + 2x + C : x \in \mathbb{Z}}$ do not intersect.

%V0 Does there exist a sequence $F(1), F(2), F(3), \ldots$ of non-negative integers that simultaneously satisfies the following three conditions? (a) Each of the integers $0, 1, 2, \ldots$ occurs in the sequence. (b) Each positive integer occurs in the sequence infinitely often. (c) For any $n \geq 2,$ $$F(F(n^{163})) = F(F(n)) + F(F(361)).$$

%V0 Find the maximum value of $x_{0}$ for which there exists a sequence $x_{0},x_{1}\cdots ,x_{1995}$ of positive reals with $x_{0} = x_{1995}$, such that $$x_{i - 1} + \frac {2}{x_{i - 1}} = 2x_{i} + \frac {1}{x_{i}},$$ for all $i = 1,\cdots ,1995$.