MathDB

6

Part of 2002 Iran Team Selection Test

Problems(1)

x_1,\dots,x_n

Source: Iran TST 2002

9/27/2006
Assume x1,x2,…,xn∈R+x_{1},x_{2},\dots,x_{n}\in\mathbb R^{+}, ∑i=1nxi2=n\sum_{i=1}^{n}x_{i}^{2}=n, ∑i=1nxi≥s>0\sum_{i=1}^{n}x_{i}\geq s>0 and 0≤λ≤10\leq\lambda\leq1. Prove that at least ⌈s2(1−λ)2n⌉\left\lceil\frac{s^{2}(1-\lambda)^{2}}n\right\rceil of these numbers are larger than λsn\frac{\lambda s}{n}.
ceiling functioninequalities proposedinequalities