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For any positive reals x, y, z it holds-[IMO LongList 1971]

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January 1, 2011
inequalitiesinequalities proposed

Problem Statement

Let a,b,ca, b, c be positive real numbers, 0<a≤b≤c0 < a \leq b \leq c. Prove that for any positive real numbers x,y,zx, y, z the following inequality holds: (ax+by+cz)(xa+yb+zc)≤(x+y+z)2⋅(a+c)24ac.(ax+by+cz) \left( \frac xa + \frac yb+\frac zc \right) \leq (x+y+z)^2 \cdot \frac{(a+c)^2}{4ac}.
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