1986 AIME Problems/Problem 2
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Contents
Problem
Evaluate the product
Solution 1 (Algebra: Specific)
We repeatedly apply the difference of squares:
Solution 2 (Algebra: Generalized)
More generally, let so that
Note that the original expression has cyclic symmetry with respect to
and
We rewrite the original expression in terms of and
then apply the difference of squares repeatedly:
~MRENTHUSIASM
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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