Team selection test for the 54th IMO

First, we shall prove the following Lemma. If $P(x)=b_0{x}^n+b_1{x}^{n-1}+b_2{x}^{n-2}+\cdots +b_{n-1}x+b_n$ has $n$ real distinct roots then $(n-1)b_1^2-2n{b}_0{b}_2>0$.

Proof. First differentiate $n-2$ times the function $f(x)$. As a result we have quadratic function having two real distinct roots and therefore its discriminant is positive.

Consider the polynomial $P(x)=(x-a_1)(x-a_3)\dots (x-a_{2n-1})+(x-a_2)(x-a_1)\dots (x-a_{2n})$. It is straightforward to verify that it satisfies the condition of the lemma and the corresponding inequality is exactly the desired inequality $$(n-1)S^2>4n(A_1+A_2).$$