2005 AIME II Problems/Problem 9
For how many positive integers less than or equal to is true for all real ?
Recall the trigonometric identities and hold for all real . If our original equation holds for all , it must certainly hold for . Thus, the question is equivalent to asking for how many positive integers we have that holds for all real .
if and only if either or for some integer . So from the equality of the real parts we need either , in which case , or we need , in which case will depend on and so the equation will not hold for all real values of . Checking in the equation for the imaginary parts, we see that it works there as well, so exactly those values of congruent to work. There are of them in the given range.
This problem begs us to use the familiar identity . Notice, since . Using this, is recast as . Hence we must have . Thus since is a multiple of exactly one quarter of the residues are congruent to hence we have .
De Moivre's Theorem states that However, our given expression is . We use the identity to substitute things in, getting . We now use De Moivre's, getting . We want this to be equal to . This is only true when the two angles are the same, or when their difference is a multiple of . . is only a multiple of when is a multiple of , or when . There are therefore possible values of .
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