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 the original expression.
Let be the measure of the angle 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 | |
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