Difference between revisions of "2016 AIME II Problems/Problem 5"
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Solution modified/fixed from Shaddoll's solution. | Solution modified/fixed from Shaddoll's solution. | ||
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+ | {{AIME box|year=2016|n=II|num-b=4|num-a=6}} |
Revision as of 22:29, 16 May 2016
Triangle has a right angle at
. Its side lengths are pariwise relatively prime positive integers, and its perimeter is
. Let
be the foot of the altitude to
, and for
, let
be the foot of the altitude to
in
. The sum
. Find
.
Solution
Note that by counting the area in 2 ways, the first altitude is . By similar triangles, the common ratio is
for reach height, so by the geometric series formula, we have
. Multiplying by the denominator and expanding, the equation becomes
. Cancelling
and multiplying by
yields
, so
and
. Checking for Pythagorean triples gives
and
, so
Solution modified/fixed from Shaddoll's solution.
See also
2016 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |