MathDB

2

Part of 2020 China Second Round Olympiad

Problems(1)

Minimum of the sum of some nonnegative reals

Source: 2020 China Second Round (A) P2

9/13/2020
Let n≥3n\geq3 be a given integer, and let a1,a2,⋯ ,a2n,b1,b2,⋯ ,b2na_1,a_2,\cdots,a_{2n},b_1,b_2,\cdots,b_{2n} be 4n4n nonnegative reals, such that a1+a2+⋯+a2n=b1+b2+⋯+b2n>0,a_1+a_2+\cdots+a_{2n}=b_1+b_2+\cdots+b_{2n}>0, and for any i=1,2,⋯ ,2n,i=1,2,\cdots,2n, aiai+2≥bi+bi+1,a_ia_{i+2}\geq b_i+b_{i+1}, where a2n+1=a1,a_{2n+1}=a_1, a2n+2=a2,a_{2n+2}=a_2, b2n+1=b1.b_{2n+1}=b_1. Detemine the minimum of a1+a2+⋯+a2n.a_1+a_2+\cdots+a_{2n}.
algebrainequalitiesn-variable inequality