Slovenija 2008

Question

If a(b + c) + b(c + a) + c(a + b) = ab + bc + ca, then a^2(b+c) + b^2(a+c) + c^2(a+b)abc is an integer.

Accepted Answer

The given equation implies ab + bc + ca = 0, so we can write a^2(b+c) + b^2(a+c) + c^2(a+b)abc = a(ab+ac) + b(ba+bc) + c(ca+cb)abc Since ab + ac = -bc, ba + bc = -ca and ca + cb = -ab we have a(ab + ac) + b(ba + bc) + c(ca + cb)abc = a(-bc) + b(-ca) + c(-ab)abc = -3abcabc = -3.