2017 AIME I Problems/Problem 15
Problem 15
The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths and
as shown, is
where
and
are positive integers,
and
are relatively prime, and
is not divisible by the square of any prime. Find
Solution
Lemma. If satisfy
, then the minimal value of
is
.
Proof. Recall that the distance between the point and the line
is given by
. In particular, the distance between the origin and any point
on the line
is at least
.
---
Let the vertices of the right triangle be and let
be two of the vertices of the equilateral triangle. Then, the third vertex of the equilateral triangle is
. This point must lie on the hypotenuse
, i.e.
must satisfy
which can be simplified to
By the lemma, the minimal value of is
so the minimal area of the equilateral triangle is
and hence the answer is
.
See Also
2017 AIME I (Problems • Answer Key • Resources) | ||
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