MathDB
Positive sequence

Source: USAMO 1994

October 23, 2005
functioninequalities unsolvedinequalities

Problem Statement

Let  a1,a2,a3,… \, a_1, a_2, a_3, \ldots \, be a sequence of positive real numbers satisfying  ∑j=1naj≥n \, \sum_{j=1}^n a_j \geq \sqrt{n} \, for all  n≥1\, n \geq 1. Prove that, for all  n≥1, \, n \geq 1, \, ∑j=1naj2>14(1+12+⋯+1n). \sum_{j=1}^n a_j^2 > \frac{1}{4} \left( 1 + \frac{1}{2} + \cdots + \frac{1}{n} \right).
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