Školjka

%V0 Let $S$ be the set of all pairs $(m,n)$ of relatively prime positive integers $m,n$ with $n$ even and $m < n.$ For $s = (m,n) \in S$ write $n = 2^k \cdot n_o$ where $k, n_0$ are positive integers with $n_0$ odd and define $$f(s) = (n_0, m + n - n_0).$$ Prove that $f$ is a function from $S$ to $S$ and that for each $s = (m,n) \in S,$ there exists a positive integer $t \leq \frac{m+n+1}{4}$ such that $$f^t(s) = s,$$ where $$f^t(s) = \underbrace{ (f \circ f \circ \cdots \circ f) }_{t \text{ times}}(s).$$ If $m+n$ is a prime number which does not divide $2^k - 1$ for $k = 1,2, \ldots, m+n-2,$ prove that the smallest value $t$ which satisfies the above conditions is $\left [\frac{m+n+1}{4} \right ]$ where $\left[ x \right]$ denotes the greatest integer $\leq x.$

%V0 The function $F$ is defined on the set of nonnegative integers and takes nonnegative integer values satisfying the following conditions: for every $n \geq 0,$ (i) $F(4n) = F(2n) + F(n),$ (ii) $F(4n + 2) = F(4n) + 1,$ (iii) $F(2n + 1) = F(2n) + 1.$ Prove that for each positive integer $m,$ the number of integers $n$ with $0 \leq n < 2^m$ and $F(4n) = F(3n)$ is $F(2^{m + 1}).$

%V0 Find all functions $f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying $$\left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}$$ for any two positive integers $m$ and $n$. Remark. The abbreviation $\mathbb{N^{*}}$ stands for the set of all positive integers: $\mathbb{N^{*}}=\left\{1,2,3,...\right\}$. By $f^{2}\left(m\right)$, we mean $\left(f\left(m\right)\right)^{2}$ (and not $f\left(f\left(m\right)\right)$).

%V0 For an integer $m$, denote by $t(m)$ the unique number in $\{1, 2, 3\}$ such that $m + t(m)$ is a multiple of $3$. A function $f: \mathbb{Z}\to\mathbb{Z}$ satisfies $f( - 1) = 0$, $f(0) = 1$, $f(1) = - 1$ and $f\left(2^{n} + m\right) = f\left(2^n - t(m)\right) - f(m)$ for all integers $m$, $n\ge 0$ with $2^n > m$. Prove that $f(3p)\ge 0$ holds for all integers $p\ge 0$. Proposed by Gerhard Woeginger, Austria

%V0 Determine all functions $f$ from the set of positive integers to the set of positive integers such that, for all positive integers $a$ and $b$, there exists a non-degenerate triangle with sides of lengths $$a, f(b) \text{ and } f(b + f(a) - 1).$$ (A triangle is non-degenerate if its vertices are not collinear.) Proposed by Bruno Le Floch, France

%V0 Let $P\!\left(x\right)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n\!\left(x\right) = x$ is equal to $P\!\left(n\right)$ for every $n \geqslant 1$, where $T^n$ denotes the $n$-fold application of $T$. Proposed by Jozsef Pelikan, Hungary