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Induction screaming sequence

Source: 2024 Turkey EGMO TST P4

February 12, 2024
algebraSequenceInequalityinduction

Problem Statement

Let (an)n=1∞(a_n)_{n=1}^{\infty} be a strictly increasing sequence such that inequality an(an−2an−1)+an−1(an−1−2an−2)≥0a_n(a_n-2a_{n-1})+a_{n-1}(a_{n-1}-2a_{n-2})\geq 0 holds for all n≥3n \geq 3. Prove that for all n≥2n\geq2 the inequality an≥an−1+an−2+⋯+a1a_n \geq a_{n-1}+a_{n-2}+\dots+a_1 holds as well.
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