1971 Canadian MO Problems/Problem 4
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.
1971 Canadian MO (Problems) | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • | Followed by Problem 5 |