Maths Olympiad Prep

Library / /201 of 520

Algebra Difficulty 6.4 National olympiad Prove it

Example 3 Let x1,x2,⋯ ,xn∈R+x_{1}, x_{2}, \cdots, x_{n} \in \mathbf{R}^{+}, and ∑i=1nx1=1\sum_{i=1}^{n} x_{1}=1, prove: x121−x1+x221−x2+xn21−xn⩾1n−1\frac{x_{1}^{2}}{1-x_{1}}+\frac{x_{2}^{2}}{1-x_{2}}+\frac{x_{n}^{2}}{1-x_{n}} \geqslant \frac{1}{n-1}.

Solution

[(1−x1)+(1−x2)+⋯+(1−xn)](x121−x1+x221−x2+⋯+xn21−xn)⩾(x1+x2+⋯+xn)2 Given ∑i=1nx1=1, we get x121−x1+x221−x2+⋯+xn21−xn⩾1n−1.\begin{aligned} & {\left[\left(1-x_{1}\right)+\left(1-x_{2}\right)+\cdots+\left(1-x_{n}\right)\right]\left(\frac{x_{1}^{2}}{1-x_{1}}\right.} \\ + & \left.\frac{x_{2}^{2}}{1-x_{2}}+\cdots+\frac{x_{n}^{2}}{1-x_{n}}\right) \geqslant\left(x_{1}+x_{2}+\cdots+x_{n}\right)^{2} \\ & \text { Given } \sum_{i=1}^{n} x_{1}=1, \text { we get } \frac{x_{1}^{2}}{1-x_{1}}+\frac{x_{2}^{2}}{1-x_{2}}+\cdots+\frac{x_{n}^{2}}{1-x_{n}} \\ \geqslant & \frac{1}{n-1} . \end{aligned}

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.