1984 AIME Problems/Problem 13
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Problem
Find the value of
Contents
Solution
Solution 1
We know that so we can repeatedly apply the addition formula,
. Let
,
,
, and
. We have
,
So
and
,
so
.
Thus our answer is .
Solution 2
Apply the formula repeatedly. Using it twice on the inside, the desired sum becomes
. This sum can then be tackled by taking the cotangent of both sides of the inverse cotangent addition formula shown at the beginning.
Solution 3
On the coordinate plane, let ,
,
,
,
,
,
,
,
, and
. We see that
,
,
, and
. The sum of these four angles forms the angle of triangle
, which has a cotangent of
, which must mean that
. So the answer is
.
Solution 4
Recall that and that
. Then letting
and
, we are left with
Expanding , we are left with
See also
1984 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |