MathDB
Inequality with some sums

Source: IMOC 2021 A6

August 11, 2021
inequalitiesAM-GMalgebra

Problem Statement

Let nn be some positive integer and a1,a2,…,ana_1 , a_2 , \dots , a_n be real numbers. Denote S0=∑i=1nai2,S1=∑i=1naiai+1,S2=∑i=1naiai+2,S_0 = \sum_{i=1}^{n} a_i^2 , \hspace{1cm} S_1 = \sum_{i=1}^{n} a_ia_{i+1} , \hspace{1cm} S_2 = \sum_{i=1}^{n} a_ia_{i+2}, where an+1=a1a_{n+1} = a_1 and an+2=a2.a_{n+2} = a_2.
1. Show that S0−S1≥0S_0 - S_1 \geq 0. 2. Show that 33 is the minimum value of CC such that for any nn and a1,a2,…,an,a_1 , a_2 , \dots , a_n, there holds C(S0−S1)≥S1−S2C(S_0 - S_1) \geq S_1 - S_2.
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