Difference between revisions of "2020 USAMTS Round 1 Problems/Problem 3"
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=Solution 1= | =Solution 1= | ||
+ | Let the points be indicated as in the figure below. | ||
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We claim the answer is <math>2+\sqrt3.</math> | We claim the answer is <math>2+\sqrt3.</math> | ||
Revision as of 16:16, 22 October 2020
The bisectors of the internal angles of parallelogram with
determine a quadrilateral with the same area as
. Determine, with proof, the value of
.
Solution 1
Let the points be indicated as in the figure below.
Image.PNG
We claim the answer is
Lemma :
is a rectangle.
is a parallelogram.
as
bisects
and
bisects
By the same logic,
is a parallelogram.
2.
and
and
By
and
we can conclude that
is a rectangle.
Let
and
Thus,
and
By the same logic,
and
Because
we have
Solution by Sp3nc3r