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Found problems: 1

2024 Australian Mathematical Olympiad, P3
Tags: rearrangement , absolute difference , permutation , algebra
Let $a_1, a_2, \ldots, a_n$ be positive reals for $n \geq 2$. For a permutation $(b_1, b_2, \ldots, b_n)$ of $(a_1, a_2, \ldots, a_n)$, define its $\textit{score}$ to be $$\sum_{i=1}^{n-1}\frac{b_i^2}{b_{i+1}}.$$ Show that some two permutations of $(a_1, a_2, \ldots, a_n)$ have scores that differ by at most $3|a_1-a_n|$.