PL-OM 2025

Find all $(a, b, c, d)\in \mathbb{R}$ satisfying $$\begin{aligned} \begin{cases} a+b+c+d=0, a^2+b^2+c^2+d^2=12, abcd=-3. \end{cases} \end{aligned}$$

A positive integer $n\geq 2$ and a set $S$ consisting of $2n$ disting positive integers smaller or equal to $n^2$ are given. Prove that there exists a positive integer $r\in \{1, 2, ..., n\}$ that can be written in the form $r=a-b$ , for $a, b\in \mathbb{S}$ in at least $3$ different ways.

A strictly decreasing function $f:(0, \infty)\Rightarrow (0, \infty)$ attaining all positive values and positive numbers $a_1\ne b_1$ are given. Numbers $a_2, b_2, a_3, b_3, ...$ satisfy $$a_{n+1}=a_n+f(b_n),\;\;\;\;\;\; b_{n+1}=b_n+f(a_n)$$ for every $n\geq 1$ . Prove that there exists a positive integer $n$ satisfying $|a_n-b_n| >2025$ .