IMO 2011 Shortlist A7

Let a, b, and c be positive real numbers satisfying min(a+b,b+c,c+a) > √ 2 and a2 +b2 +c2 = 3. Prove that a (b + c − a)2...

imomathematicsolympiadshortlistalgebra

IMO 2011 Shortlist A7

Category: Algebra

Problem

Let a, b, and c be positive real numbers satisfying min(a+b,b+c,c+a) > √ 2 and a2 +b2 +c2 = 3. Prove that a (b + c − a)2 + b (c + a − b)2 + c (a + b − c)2 ≥ (abc)2 .