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 Find all of the positive real numbers like $x,y,z,$ such that : 1.) $x + y + z = a + b + c$ 2.) $4xyz = a^2x + b^2y + c^2z + abc$ Proposed to Gazeta Matematica in the 80s by VASILE CÎRTOAJE and then by Titu Andreescu to IMO 1995.

%V0 Let $k$ be a positive integer. Show that there are infinitely many perfect squares of the form $n \cdot 2^k - 7$ where $n$ is a positive integer.

%V0 Find all $x,y$ and $z$ in positive integer: $z + y^{2} + x^{3} = xyz$ and $x = \gcd(y,z)$.

%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 For positive integers $n,$ the numbers $f(n)$ are defined inductively as follows: $f(1) = 1,$ and for every positive integer $n,$ $f(n+1)$ is the greatest integer $m$ such that there is an arithmetic progression of positive integers $a_1 < a_2 < \ldots < a_m = n$ for which $$f(a_1) = f(a_2) = \ldots = f(a_m).$$ Prove that there are positive integers $a$ and $b$ such that $f(an+b) = n+2$ for every positive integer $n.$