1976 USAMO Problems/Problem 5
Problem
If , , , and are all polynomials such that prove that is a factor of .
Solutions
Solution 1
In general we will show that if is an integer less than and and are polynomials satisfying then , for all integers . For the problem, we may set , , and then note that since , is a factor of .
Indeed, let be the th roots of unity other than 1. Then for all integers , for all integers . This means that the th degree polynomial has distinct roots. Therefore all its coefficients must be zero, so for all integers , as desired.
Solution 2
Let be three distinct primitive fifth roots of unity. Setting , we have These equations imply that or But by symmetry, Since , it follows that . Then, as noted above, so is a factor of , as desired.
Solution 3
Let be three of the 5th roots of unity not equal to one that satisfy as a result. Plugging them into the equation gives the linear system of equations in :
Direct observation gives as a solution, and there are no others because the system of equations is linear. Hence, , and so is a factor of , as desired.
Note: We can generalize this approach to prove the claim in Solution 1 in an equally fast, concise, and readily understandable fashion.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
1976 USAMO (Problems • Resources) | ||
Preceded by Problem 4 |
Followed by Last Question | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
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