2020 AIME I Problems/Problem 1
Contents
Problem
In with
point
lies strictly between
and
on side
and point
lies strictly between
and
on side
such that
The degree measure of
is
where
and
are relatively prime positive integers. Find
Solution 1
If we set to
, we can find all other angles through these two properties:
1. Angles in a triangle sum to
.
2. The base angles of an isosceles triangle are congruent.
Now we angle chase. ,
,
,
,
,
. Since
as given by the problem,
, so
. Therefore,
, and our desired angle is
for an answer of
.
See here for a video solution: https://youtu.be/4e8Hk04Ax_E
Solution 2
Let be
in degrees.
.
By Exterior Angle Theorem on triangle
,
.
By Exterior Angle Theorem on triangle
,
.
This tells us
and
.
Thus
and we want
to get an answer of
.
Solution 3 (Official MAA)
Let . Because
is isosceles,
. Then
Because
and
are also isosceles,
Because
is isosceles,
is also
, so
, and it follows that
. The requested sum is
.
https://artofproblemsolving.com/wiki/index.php/1961_AHSME_Problems/Problem_25 (Almost Mirrored)
See here for a video solution:
Video solution
https://youtu.be/mgRNqSDCvgM ~yofro
Solution without words
~vvsss, www.deoma-cmd.ru
See Also
2020 AIME I (Problems • Answer Key • Resources) | ||
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Followed by Problem 2 | |
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