Problem

Source: JOM 2025 Mock 1 P1

Tags: algebra



Given a positive real $t$, a set $S$ of nonnegative reals is called $t$-good if for any two distinct elements $a,b$ in $S$, $\frac{a+b}2\ge\sqrt{ab}+t$. For all positive reals $N$, find the maximum number of elements a $t$-good set can have, if all elements are at most $N$. (Proposed by Ho Janson)