UA-UMO 2021

For arbitrary positive reals $a\ge b \ge c$ prove the inequality: $$\frac{a^2+b^2}{a+b}+\frac{a^2+c^2}{a+c}+\frac{c^2+b^2}{c+b}\ge (a+b+c)+ \frac{(a-c)^2}{a+b+c}$$ (Anton Trygub)

The sequence $a_1,a_2, ..., a_{2n}$ of integers is such that each number occurs in no more than $n$ times. Prove that there are two strictly increasing sequences of indices $b_1,b_2, ..., b_{n}$ and $c_1,c_2, ..., c_{n}$ are such that every positive integer from the set $\{1,2,...,2n\}$ occurs exactly in one of these two sequences, and for each $1\le i \le n$ is true the condition $a_{b_i} \ne a_{c_i}$ . (Anton Trygub)