Problem
ALG-B2-M12-P005 Set 2. Pairwise sum
#5
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Let \(a,b,c\ge0\), \(a+b+c=1\). Prove \(ab+bc+ca\le\frac13\).
Hint. Use \((a+b+c)^2\ge3(ab+bc+ca)\).
From \((a+b+c)^2\ge3(ab+bc+ca)\) and \(a+b+c=1\), we get \(1\ge3(ab+bc+ca)\).
Mock set 2: the problem is intended for independent method selection without an explicit cue in the statement.