Problem
ALG-B2-M12-P001 Set 1. Sum of three fractions
#1
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Prove for \(a,b,c>0\): \[\frac{a^2}{ab+ac+bc}+\frac{b^2}{ab+ac+bc}+\frac{c^2}{ab+ac+bc}\ge1.\]
Hint. Use \((a+b+c)^2\ge3(ab+bc+ca)\).
The left side is \(\frac{a^2+b^2+c^2}{ab+bc+ca}\). Since \(a^2+b^2+c^2\ge ab+bc+ca\), it is at least \(1\).
Mock set 1: the problem is intended for independent method selection without an explicit cue in the statement.