Difference between revisions of "2018 AIME I Problems/Problem 15"
(→Problem 15) |
Happyman25 (talk | contribs) m |
||
Line 17: | Line 17: | ||
By S.B. | By S.B. | ||
LaTeX by willwin4sure | LaTeX by willwin4sure | ||
+ | |||
+ | ==See Also== | ||
+ | {{AIME box|year=2018|n=I|num-b=12|num-a=14}} | ||
+ | {{MAA Notice}} |
Revision as of 23:32, 9 November 2018
Problem 15
David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, , which can each be inscribed in a circle with radius
. Let
denote the measure of the acute angle made by the diagonals of quadrilateral
, and define
and
similarly. Suppose that
,
, and
. All three quadrilaterals have the same area
, which can be written in the form
, where
and
are relatively prime positive integers. Find
.
Solution
Suppose our four sides lengths cut out arc lengths of ,
,
, and
, where
. Then, we only have to consider which arc is opposite
. These are our three cases, so
Our first case involves quadrilateral
with
,
,
, and
.
Then, by Law of Sines, and
. Therefore,
so our answer is
.
By S.B. LaTeX by willwin4sure
See Also
2018 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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.