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A set of inequalities on a sequence [Serbia TST 2017, D2, P2]

Source: Serbia TST 2017, Day 2, Problem 2

May 22, 2017
algebrainequalitiesSequence

Problem Statement

Let n≥2n \geq 2 be a positive integer and {xi}i=0n\{x_i\}_{i=0}^n a sequence such that not all of its elements are zero and there is a positive constant CnC_n for which: (i) x1+⋯+xn=0x_1+ \dots +x_n=0, and (ii) for each ii either xi≤xi+1x_i\leq x_{i+1} or xi≤xi+1+Cnxi+2x_i\leq x_{i+1} + C_n x_{i+2} (all indexes are assumed modulo nn). Prove that a) Cn≥2C_n\geq 2, and b) Cn=2C_n=2 if and only 2∣n2 \mid n.
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