Problem:
Let ,
,
be complex numbers such that
Prove that .
Proof:
Method I:
Substituting in
gives
, so
.
Likewise, and
. Then,
, that is,
yielding
.
Method II:
Using the relations between the roots and the coefficients, it follows that ,
,
are the roots of polynomial
, where
. Hence,
implying
, and the conclusion follows.
More complex fun later,
Nalin Pithwa