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 .
We can rewrite as and as . This means that , by De Moivre's Theorem. This theorem also tells us that , so . By the same line of reasoning, we have .
For the statement in the question to be true, we must have . The left hand side simplifies to . We cancel the denominators and find that the only thing that needs to be true is that . This is true if , and there are such numbers between and . Solution by Zeroman
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