2008 AIME I Problems/Problem 8
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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.
See also
| 2008 AIME I (Problems • Resources) | ||
| Preceded by Problem 7 | Followed by Problem 9 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
