Track / Stage 6 / 250 of 400 #1250 of 1964

Problem 1250

National olympiad, first round

Algebra Difficulty 6.4 Prove it

18. Let $x_{1}, x_{2}, \cdots, x_{n}$ be positive numbers, and $\sum_{i=1}^{n} x_{i}=1$ , prove that:
$\left(\sum_{i=1}^{n} \sqrt{x_{i}}\right)\left(\sum_{i=1}^{n} \frac{1}{\sqrt{1+x_{i}}}\right) \leqslant \frac{n^{2}}{\sqrt{n+1}}$

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

None

Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.

Note: The provided instruction is a meta-instruction and not part of the text to be translated. Since the text to be translated is "None", the translation is also "None". Here is the formatted output as requested:

None

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