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Algebra Difficulty 6.3 National olympiad Prove it

Example 3 Let x,y,zx, y, z be non-negative real numbers, and the sum of any two of them is not zero. Prove:
∑xcc2x2+yzy+z⩾9(x2+y2+z2)2(x+y+z)\sum_{x \mathrm{cc}} \frac{2 x^{2}+y z}{y+z} \geqslant \frac{9\left(x^{2}+y^{2}+z^{2}\right)}{2(x+y+z)}

Solution

 Prove ∑cyc2x2+yzy+z−9(x2+y2+z2)2(x+y+z)=∑cyc(y−z)2(2(x−y−z)2+yz)2(x+y)(x+z)(x+y+z)⩾0,\text { Prove } \begin{aligned} & \sum_{\mathrm{cyc}} \frac{2 x^{2}+y z}{y+z}-\frac{9\left(x^{2}+y^{2}+z^{2}\right)}{2(x+y+z)} \\ = & \sum_{\mathrm{cyc}} \frac{(y-z)^{2}\left(2(x-y-z)^{2}+y z\right)}{2(x+y)(x+z)(x+y+z)} \\ \geqslant & 0, \end{aligned}

so inequality (5) holds.

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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.