Maths Olympiad Prep

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Algebra Difficulty 6.1 National olympiad Prove it

Example 21 (2003 National Training Team Selection Test) Let x,y,z∈R+x, y, z \in \mathbf{R}^{+}, prove that
(∑x)2⋅∑1y2+z2>10\left(\sum x\right)^{2} \cdot \sum \frac{1}{y^{2}+z^{2}}>10

Solution

Given that we have already proved (∑x)4⩾16∑y2z2+11xyz∑x\left(\sum x\right)^{4} \geqslant 16 \sum y^{2} z^{2}+11 x y z \sum x (Equation (14)), we have
(∑x)2⩾4∑y2z2\left(\sum x\right)^{2} \geqslant 4 \sqrt{\sum y^{2} z^{2}}

Therefore, to prove Equation (24), it suffices to prove
∑y2z2⋅∑1y2+z2⩾52\sqrt{\sum y^{2} z^{2}} \cdot \sum \frac{1}{y^{2}+z^{2}} \geqslant \frac{5}{2}

This inequality can be derived from the above Equation (24).

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