S1::
x² + y² + z² = xy + yz + xz
x² - xy + y² - yz + z² - xz = 0
subtract xy, yz, and zx from both sides:
x² - 2xy + y² - 2yz + z² - 2zx = -(xy + yz + zx)
add x², y², and z² to both sides:
2x² - 2xy + 2y² - 2yz + 2z² - 2zx = (x² + y² + z²) - (xy + yz + zx)
(x - y)² + (y - z)² + (z - x)² = (x² + y² + z²) - (xy + yz + zx)
But we know from S1 that the right hand side of this equation is 0! So
(x - y)² + (y - z)² + (z - x)² = 0
and (x - y) = 0, (y - z) = 0, and (z - x) = 0.
Hence x = y = z, so the question reduces to "Is 3x³ = 0?" Since x is nonzero, the answer is NO, and S1 is SUFFICIENT.
S2::
Suppose that x + y + z = 0 and x³ + y³ + z³ = 0. Since x = -(y + z) and x³ = -(y³ + z³), we have
-(y + z)³ = -(y³ + z³), or
-3y²z - 3yz² = 0, or
0 = y²z + yz², or
0 = yz(y + z)
Since y ≠ 0 and z ≠ 0 (from the prompt), we know that (y + z) = 0, and y = -z. But then x + y + z = 0 is really x + (-z) + z = 0, or x = 0.
Since this leads to a contradiction, we cannot have both (x + y + z) = 0 and (x³ + y³ + z³ = 0), so x³ + y³ + z³ ≠0, and this statement is SUFFICIENT.
But I need to put a stern warning in bold caps:
LIKE SOME OF YOUR OTHER QUESTIONS, THIS PROBLEM IS BROKEN: S1 and S2 GIVE ENTIRELY DIFFERENT ANSWERS WITH NO OVERLAP, SOMETHING THAT IS NEVER ALLOWED ON THE GMAT. (S1 says that y = z, but S2 says that y = -z, forcing z = 0 and contradicting the prompt.) THE SOURCE OF THESE QUESTIONS SHOULD BE TREATED WITH SKEPTICISM, AS THE QUESTION WRITERS DO NOT UNDERSTAND A FUNDAMENTAL ASPECT OF THE GMAT.
