Let n be a positive integer with n>1, and let a1, a2, ..., an be positive integers such that a1<a2<...<an and 1a1+1a2+...+1an≤1. Prove that (1a21+x2+1a22+x2+...+1a2n+x2)2≤12⋅1a1(a1−1)+x2 for all real numbers x.
Source: MOP 2005 Homework - Blue Group #7
Tags: inequalities, inequalities unsolved
Let n be a positive integer with n>1, and let a1, a2, ..., an be positive integers such that a1<a2<...<an and 1a1+1a2+...+1an≤1. Prove that (1a21+x2+1a22+x2+...+1a2n+x2)2≤12⋅1a1(a1−1)+x2 for all real numbers x.