Školjka

%V0 Show that there exists a bijective function $f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $m,n\in \mathbb{N}_{0}$: $$f(3mn + m + n) = 4f(m)f(n) + f(m) + f(n).$$

%V0 Let $n$ be an even positive integer. Prove that there exists a positive integer $k$ such that $$k = f(x) \cdot (x+1)^n + g(x) \cdot (x^n + 1)$$ for some polynomials $f(x), g(x)$ having integer coefficients. If $k_0$ denotes the least such $k,$ determine $k_0$ as a function of $n,$ i.e. show that $k_0 = 2^q$ where $q$ is the odd integer determined by $n = q \cdot 2^r, r \in \mathbb{N}.$ Note: This is variant A6' of the three variants given for this problem.

%V0 Let $P(x)$ be the real polynomial function, $P(x) = ax^3 + bx^2 + cx + d.$ Prove that if $|P(x)| \leq 1$ for all $x$ such that $|x| \leq 1,$ then $$|a| + |b| + |c| + |d| \leq 7.$$

%V0 Let $a > 2$ be given, and starting $a_0 = 1, a_1 = a$ define recursively: $$a_{n+1} = \left(\frac{a^2_n}{a^2_{n-1}} - 2 \right) \cdot a_n.$$ Show that for all integers $k > 0,$ we have: $\sum^k_{i = 0} \frac{1}{a_i} < \frac12 \cdot (2 + a - \sqrt{a^2-4}).$

%V0 Let $a_1 \geq a_2 \geq \ldots \geq a_n$ be real numbers such that for all integers $k > 0,$ $$a^k_1 + a^k_2 + \ldots + a^k_n \geq 0.$$ Let $p = max\{|a_1|, \ldots, |a_n|\}.$ Prove that $p = a_1$ and that $$(x - a_1) \cdot (x - a_2) \cdots (x - a_n) \leq x^n - a^n_1$$ for all $x > a_1.$

%V0 Suppose that $a, b, c > 0$ such that $abc = 1$. Prove that $$\frac{ab}{ab + a^5 + b^5} + \frac{bc}{bc + b^5 + c^5} + \frac{ca}{ca + c^5 + a^5} \leq 1.$$