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S(1)S(3)...S(2n-1) >=S(2)S(4)...S(2n) , sum of digits inequality

Source: 2021 3nd Final Mathematical Cup Junior Division P3 FMC

October 13, 2021
inequalitiesnumber theorysum of digitsalgebra

Problem Statement

For every positive integer nn, s(n)s(n) denotes the sum of the digits in the decimal representation of nn. Prove that for every integer n≥5n \ge 5, we have S(1)S(3)...S(2n−1)≥S(2)S(4)...S(2n)S(1)S(3)...S(2n-1) \ge S(2)S(4)...S(2n)
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