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usamo 99/4

Source:

January 19, 2004
USAMOinequalitiesfunctioninequalities proposed

Problem Statement

Let a1,a2,…,ana_{1}, a_{2}, \dots, a_{n} (n>3n > 3) be real numbers such that a_{1} + a_{2} + \cdots + a_{n} \geq n \qquad \mbox{and} \qquad a_{1}^{2} + a_{2}^{2} + \cdots + a_{n}^{2} \geq n^{2}. Prove that max⁡(a1,a2,…,an)≥2\max(a_{1}, a_{2}, \dots, a_{n}) \geq 2.
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