Školjka

%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 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 $i = 1,\cdots ,1995$.

%V0 Let $a, b, c$ be positive real numbers so that $abc = 1$. Prove that $$\left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1.$$

%V0 Find all polynomials $f$ with real coefficients such that for all reals $a,b,c$ such that $ab+bc+ca = 0$ we have the following relations $$f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c).$$

%V0 Find all functions $f: (0, \infty) \mapsto (0, \infty)$ (so $f$ is a function from the positive real numbers) such that $$\frac {\left( f(w) \right)^2 + \left( f(x) \right)^2}{f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2}$$ for all positive real numbes $w,x,y,z,$ satisfying $wx = yz.$ Author: Hojoo Lee, South Korea