Difference between revisions of "1993 AIME Problems/Problem 15"
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[[Category:Intermediate Geometry Problems]] | [[Category:Intermediate Geometry Problems]] | ||
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Revision as of 18:26, 4 July 2013
Problem
Let be an altitude of
. Let
and
be the points where the circles inscribed in the triangles
and
are tangent to
. If
,
, and
, then
can be expressed as
, where
and
are relatively prime integers. Find
.
Solution
From the Pythagorean Theorem, , and
. Subtracting those two equations yields
. After simplification, we see that
, or
. Note that
. Therefore we have that
. Therefore
.
Now note that ,
, and
. Therefore we have
.
Plugging in and simplifying, we have
.
See also
1993 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last question | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.