Problem

Source: China TST 2002 Quiz

Tags: inequalities, inequalities unsolved



Given $ n \geq 3$, $ n$ is a integer. Prove that: \[ (2^n - 2) \cdot \sqrt{2i-1} \geq \left( \sum_{j=0}^{i-1}C_n^j + C_{n-1}^{i-1} \right) \cdot \sqrt{n}\] where if $ n$ is even, then $ \displaystyle 1 \leq i \leq \frac{n}{2}$; if $ n$ is odd, then $ \displaystyle 1 \leq i \leq \frac{n-1}{2}$.