MathDB
a_1,...a_n

Source: Iran TST 2005

April 20, 2005
inequalitiesinequalities proposed

Problem Statement

Suppose that a1 a_1, a2 a_2, ..., an a_n are positive real numbers such that a1≤a2≤⋯≤an a_1 \leq a_2 \leq \dots \leq a_n. Let {{a_1 \plus{} a_2 \plus{} \dots \plus{} a_n} \over n} \equal{} m; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{a_1^2 \plus{} a_2^2 \plus{} \dots \plus{} a_n^2} \over n} \equal{} 1. Suppose that, for some i i, we know ai≤m a_i \leq m. Prove that: n \minus{} i \geq n \left(m \minus{} a_i\right)^2
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