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 1
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.
Solution 2
Let and where is the common root. From Vieta's Formulas, we have: and . We see that $s,t,u \notequal 0$ (Error compiling LaTeX. ! Undefined control sequence.). Dividing by , we have: Also, we have: Substituting into the above, we have:
Thus either or . We check to see that is indeed a possible value to satisfy the requirements. If , then from , we have , and from , we have , which also satisfies the requirements.
Thus, the only possible a values are: .
1971 Canadian MO (Problems) | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • | Followed by Problem 5 |