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Source: 2023 IMO Shortlist N3

July 17, 2024
IMO Shortlistnumber theoryAZE BMO TSTAZE EGMO TSTTST

Problem Statement

For positive integers nn and k≥2k \geq 2, define Ek(n)E_k(n) as the greatest exponent rr such that krk^r divides n!n!. Prove that there are infinitely many nn such that E10(n)>E9(n)E_{10}(n) > E_9(n) and infinitely many mm such that E10(m)<E9(m)E_{10}(m) < E_9(m).
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