MathDB

x_1,\dots,x_n

Source: Iran TST 2002

September 27, 2006

ceiling functioninequalities proposedinequalities

Problem Statement

Assume

x_{1},x_{2},\dots,x_{n}\in\mathbb R^{+}

\sum_{i=1}^{n}x_{i}^{2}=n

\sum_{i=1}^{n}x_{i}\geq s>0

and

0\leq\lambda\leq1

. Prove that at least

\left\lceil\frac{s^{2}(1-\lambda)^{2}}n\right\rceil

of these numbers are larger than

\frac{\lambda s}{n}