Difference between revisions of "2020 USAMTS Round 1 Problems/Problem 3"
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− | Let the points be indicated as in the | + | Let the points be indicated as in the second image in the following link: https://artofproblemsolving.com/community/c5h2314139_usamts_problem_3_round_1_year_32_how_do_you_even_draw_the_diagram |
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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 16:18, 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 second image in the following link: https://artofproblemsolving.com/community/c5h2314139_usamts_problem_3_round_1_year_32_how_do_you_even_draw_the_diagram
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