Problem

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 8.4

Tags: algebra, quadratics



The board contains $20$ non-constant linear functions, not necessarily distinct. For each pair $(f, g)$ of these functions ($190$ pairs in total), Victor writes on the board a quadratic function $f(x)\cdot g(x) - 2$, and Solomiya writes on the board a quadratic function $f(x)g(x)-1$. Victor calculated that exactly $V$ of his quadratic functions have a root, and Solomiya calculated that exactly $S$ of her quadratic functions have a root. Find the largest possible value of $S-V$. Remarks. A linear function $y = kx+b$ is called non-constant if $k\neq 0$. Proposed by Oleksiy Masalitin