Problem
ALG-B1-M01-P015 Fourth Powers with Zero Sum
#15
★★★☆☆ Level 3 of 5
Let \(a+b+c=0\). Prove that \(a^4+b^4+c^4=2(a^2b^2+b^2c^2+c^2a^2)\).
Use \(a^2+b^2+c^2=-2(ab+bc+ca)\) and square.
From \(a+b+c=0\), \(a^2+b^2+c^2=-2(ab+bc+ca)\). Square: \[ (a^2+b^2+c^2)^2=4(ab+bc+ca)^2. \]
The left side is \(a^4+b^4+c^4+2(a^2b^2+b^2c^2+c^2a^2)\). The right side is \(4(a^2b^2+b^2c^2+c^2a^2+abc(a+b+c))\), and the last term is zero. This gives the required equality.
This already requires a chain of two identities.