Školjka

%V0 $a > 0$ and $b$, $c$ are integers such that $ac$ – $b^2$ is a square-free positive integer P. For example P could be 3*5, but not 3^2*5. Let $f(n)$ be the number of pairs of integers $d, e$ such that $ad^2 + 2bde + ce^2= n$. Show that$f(n)$ is finite and that $f(n) = f(P^{k}n)$ for every positive integer $k$. Original Statement: Let $a,b,c$ be given integers $a > 0,$ $ac-b^2 = P = P_1 \cdots P_n$ where $P_1 \cdots P_n$ are (distinct) prime numbers. Let $M(n)$ denote the number of pairs of integers $(x,y)$ for which $$ax^2 + 2bxy + cy^2 = n.$$ Prove that $M(n)$ is finite and $M(n) = M(P_k \cdot n)$ for every integer $k \geq 0.$ Note that the "$n$" in $P_N$ and the "$n$" in $M(n)$ do not have to be the same.

%V0 Let $c_1, \ldots, c_n \in \mathbb{R}$ with $n \geq 2$ such that $$0 \leq \sum^n_{i=1} c_i \leq n.$$ Show that we can find integers $k_1, \ldots, k_n$ such that $$\sum^n_{i=1} k_i = 0$$ and $$1-n \leq c_i + n \cdot k_i \leq n$$ for every $i = 1, \ldots, n.$ Another formulation:Let $x_1, \ldots, x_n,$ with $n \geq 2$ be real numbers such that $$|x_1 + \ldots + x_n| \leq n.$$ Show that there exist integers $k_1, \ldots, k_n$ such that $$|k_1 + \ldots + k_n| = 0.$$ and $$|x_i + 2 \cdot n \cdot k_i| \leq 2 \cdot n -1$$ for every $i = 1, \ldots, n.$ In order to prove this, denote $c_i = \frac{1+x_i}{2}$ for $i = 1, \ldots, n,$ etc.

%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 Does there exist a function $s\colon \mathbf{Q} \rightarrow \{-1,1\}$ such that if $x$ and $y$ are distinct rational numbers satisfying ${xy=1}$ or ${x+y\in \{0,1\}}$, then ${s(x)s(y)=-1}$? Justify your answer.

%V0 Find all functions $f\colon\mathbb{R} \rightarrow\mathbb{R}$ satisfying the equation $$f\left(x^2 + y^2 + 2f\left(xy\right)\right) = \left(f\left(x + y\right)\right)^2$$ for all $x,y\in \mathbb{R}$.

%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 $f\left(f\left(m\right)\right)$).