MathDB
Hard problem

Source: 2023 Japan MO Finals 5

February 11, 2023
combinatorics

Problem Statement

Let S={1,2,…,3000}S=\{1,2,\dots,3000\}. Determine the maximum possible integer XX that satisfies the condition: For all bijective function f:S→Sf:S\rightarrow S, there exists bijective function g:S→Sg:S\rightarrow S such that ∑k=13000(max⁡{f(f(k)),f(g(k)),g(f(k)),g(g(k))}−min⁡{f(f(k)),f(g(k)),g(f(k)),g(g(k))})≥X\displaystyle\sum_{k=1}^{3000}\left(\max\{f(f(k)),f(g(k)),g(f(k)),g(g(k))\}-\min\{f(f(k)),f(g(k)),g(f(k)),g(g(k))\}\right)\geq X
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