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 :
$$ (Error compiling LaTeX. ! Missing $ inserted.)A(1) + zB(1) + z^2C(1) = 0$$ (Error compiling LaTeX. ! Missing $ inserted.) $$ (Error compiling LaTeX. ! Missing $ inserted.)A(1) + z^3B(1) + z^6C(1) = 0(A(1), B(1), C(1)) = (0,0,0)A(1) = 0(x-1)A(x)\blacksquare$
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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