SAUDI ARABIAN IMO Booklet 2023

a) Denote $x_i,y_i$ as the roots of the $i$-th polynomial, then by Vieta's theorem, $a_i=-(x_i+y_i)$ and $b_i=x_i{y}_i$. Thus $a_i{b}_i=-x_i{y}_i(x_i+y_i)$ which is always even, implying that at most 1 number among $a_i,b_i$ is odd. Hence, there are at most 10 odd values among their coefficients.

The equality case occurs when $(1,2),(3,4),\dots ,(9,10),(-1,-2),\dots ,(-9,-10)$ are roots of the given polynomials.

b) By Vieta's theorem, we need to find the minimum and maximum value of $$T=x_1{y}_1+x_2{y}_2+\cdots +x_{10}{y}_{10}.$$ Note that for all $x,y\in \mathbb{R}$, $xy\ge -\frac{x^2+y^2}{2}$, thus $$T\ge -\frac{1}{2}(x_1^2+y_1^2+x_2^2+y_2^2+\cdots +x_{10}^2+y_{10}^2)=-(1^2+2^2+\cdots +10^2)=-385.$$ On the other hand, for $x,y\in \mathbb{Z}$ and $x\ne y$ then $(x-y{)}^2\ge 1$ so $xy\le \frac{x^2+y^2-1}{2}$, thus $$T\le \frac{1}{2}(x_1^2+y_1^2+x_2^2+y_2^2+\cdots +x_{10}^2+y_{10}^2)-5=380.$$ Hence, we can conclude that:

$\max T=380$, attained when $(1,2),(3,4),\dots ,(9,10),(-1,-2),\dots ,(-9,-10)$ are roots of 10 polynomials. $\min T=-385$, attained when $(1,-1),(2,-2),\dots ,(10,-10)$ are roots of 10 polynomials.