MathDB

Sum of x is 1

Source: CMO-1996-5

January 3, 2006

inequalitiestrigonometrycalculusderivativeintegration

Problem Statement

Let

n

be a natural number. Suppose that

x_0=0

and that

x_i>0

for all

i\in\{1,2,\ldots ,n\}

. If

\sum_{i=1}^nx_i=1

, prove that

1\leq\sum_{i=1}^{n} \frac{x_i}{\sqrt{1+x_0+x_1+\ldots +x_{i-1}}\sqrt{x_i+\ldots+x_n}} < \frac{\pi}{2}