Problem: Find the smallest integer n such that (x2+y2+z2)2⩽n(x4+y4+z4) for all real numbers x,y and z.
Answer Choices:
A. 2
B. 3
C. 4
D. 6
E. There is no such integer n
Solution:
Let a=x2,b=y2 and c=z2. Then
0⩽(a−b)2+(b−c)2+(c−a)2a2+b2+c2ab+bc+ca⩽1a2+b2+c2a2+b2+c2+2(ab+bc+ca)⩽3;(a+b+c)2⩽3(a2+b2+c2)
Therefore n⩽3. Choosing a=b=c>0 shows n is not less than three.