2008 AIME I Problems/Problem 8
Problem
Find the positive integer such that
Contents
Solution
Solution 1
Since we are dealing with acute angles, .
Note that , by tangent addition. Thus,
.
Applying this to the first two terms, we get .
Now, .
We now have . Thus,
; and simplifying,
.
Solution 2 (generalization)
From the expansion of , we can see that
and
If we divide both of these by
, then we have
which makes for more direct, less error-prone computations. Substitution gives the desired answer.
Solution 3: Complex Numbers
Adding a series of angles is the same as multiplying the complex numbers whose arguments they are. In general, , is the argument of
. The sum of these angles is then just the argument of the product
and expansion give us . Since the argument of this complex number is
, its real and imaginary parts must be equal.
So we set them equal and expand the product to get:
Therefore,
equals
.
See also
2008 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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