Maths Olympiad Prep

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Algebra Difficulty 6.3 National olympiad Prove it

Question 1: Given x>0,y>0,z>0x>0, y>0, z>0, prove:
x2−xy+y2+y2−yz+z2⩾x2+xz+z2\begin{array}{l} \sqrt{x^{2}-x y+y^{2}}+\sqrt{y^{2}-y z+z^{2}} \\ \geqslant \sqrt{x^{2}+x z+z^{2}} \end{array}

The equality holds if 1y=1x+1z\frac{1}{y}=\frac{1}{x}+\frac{1}{z}.

Solution

None

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None

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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.