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 For an integer $x \geq 1,$ let $p(x)$ be the least prime that does not divide $x,$ and define $q(x)$ to be the product of all primes less than $p(x).$ In particular, $p(1) = 2.$ For $x$ having $p(x) = 2,$ define $q(x) = 1.$ Consider the sequence $x_0, x_1, x_2, \ldots$ defined by $x_0 = 1$ and $$x_{n+1} = \frac{x_n p(x_n)}{q(x_n)}$$ for $n \geq 0.$ Find all $n$ such that $x^n = 1995.$

%V0 Let $a, b, c$ be positive integers satisfying the conditions $b > 2a$ and $c > 2b.$ Show that there exists a real number $\lambda$ with the property that all the three numbers $\lambda a, \lambda b, \lambda c$ have their fractional parts lying in the interval $\left(\frac {1}{3}, \frac {2}{3} \right].$