MathDB

Sum (m_i+1/m_i)^2>=k(k/S+S/k) where Sum m_i=S.

Source: Vietnam MO 1980 P2

March 17, 2011

functioninequalitiesinequalities proposed

Problem Statement

Let

m_1, m_2, \cdots ,m_k

be positive numbers with the sum

S

. Prove that \displaystyle\sum_{i=1}^k\left(m_i +\frac{1}{m_i}\right)^2 \ge k\left(\frac{k}{S}+\frac{S}{k}\right)^2