1986 AIME Problems/Problem 2
Contents
Problem
Evaluate the product
Solution 1 (Algebra: Generalized)
More generally, let so that
We rewrite the original expression in terms of and
then apply the difference of squares repeatedly:
Remark
From this solution, note that the original expression has cyclic symmetry with respect to and
~MRENTHUSIASM
Solution 2 (Algebra: Specific)
We group the first and last factors as well as the two middle factors, then apply the difference of squares repeatedly:
~Azjps (Solution)
~MRENTHUSIASM (Revision)
Solution 3 (Geometry)
Notice that in a triangle with side lengths and
, by Heron's formula, the area is the square root of what we are looking for.
Let angle
be opposite the
side. By the Law of Cosines,
So
. The area of the triangle is then
So our answer is
See also
1986 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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