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Another easy inequality with square roots

Source: ARO Regional stage 2023 10.10

February 16, 2023
algebraRussiainequalities proposed

Problem Statement

Prove that for all positive reals x,y,zx, y, z, the inequality (x−y)3x2+y2+(y−z)3y2+z2+(z−x)3z2+x2≥0(x-y)\sqrt{3x^2+y^2}+(y-z)\sqrt{3y^2+z^2}+(z-x)\sqrt{3z^2+x^2} \geq 0 is satisfied.
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