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 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 $\left(\frac {1}{3}, \frac {2}{3} \right].$

%V0 Let $P$ be a cubic polynomial given by $P(x)=ax^3+bx^2+cx+d$, where $a,b,c,d$ are integers and $a\ne0$. Suppose that $xP(x)=yP(y)$ for infinitely many pairs $x,y$ of integers with $x\ne y$. Prove that the equation $P(x)=0$ has an integer root.

%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 Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f\!\left(a\right)-f\!\left(b\right)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f\!\left(c\right)$ for some positive integer $c$. Proposed by Juhan Aru, Estonia