Problem
ALG-B2-M12-P009 Set 3. Division by a sum
#9
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Prove for \(a,b,c>0\): \[\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\ge\frac{a+b+c}{2}.\]
Hint. Apply Cauchy and find the sum of denominators.
By Cauchy, \[\sum\frac{a^2}{b+c}\ge\frac{(a+b+c)^2}{2(a+b+c)}=\frac{a+b+c}{2}.\]
Mock set 3: the problem is intended for independent method selection without an explicit cue in the statement.