1971 Canadian MO Problems/Problem 4
Revision as of 23:49, 27 July 2006 by Boy Soprano II (talk | contribs) (Added solution and category tag)
Problem
Determine all real numbers such that the two polynomials
and
have at least one root in common.
Solution
Let this root be . Then we have
Now, if , then we're done, since this satisfies the problem's conditions. If
, then we can divide both sides by
to obtain
. Substituting this value into the first polynomial gives
It is easy to see that this value works for the second polynomial as well.
Therefore the only possible values of are
and
. Q.E.D.