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4

Part of 2015 Federal Competition For Advanced Students, P2

Problems(1)

Problem 4 -- Three Fractions Not Enough

Source: 46th Austrian Mathematical Olympiad National Competition Part 2 Problem 4

7/14/2018
Let x,y,zx,y,z be positive real numbers with x+y+z≥3x+y+z \ge 3. Prove that
1x+y+z2+1y+z+x2+1z+x+y2≤1\frac{1}{x+y+z^2} + \frac{1}{y+z+x^2} + \frac{1}{z+x+y^2} \le 1
When does equality hold?
(Karl Czakler)
Austriaalgebrainequalities