AMC 12B #22

[stupidityismygam]

Let [unparseable or potentially dangerous latex formula] be a cyclic quadrilateral. The side lengths of [unparseable or potentially dangerous latex formula] are distinct integers less than 15 such that [unparseable or potentially dangerous latex formula]. What is the largest possible value of [unparseable or potentially dangerous latex formula]?

[sinkokuyou]

btw, it should say "cyclic quadrilateral"

[sinkokuyou]

well first, if the question wants to know something about the diagonal, and gives you something about the sides of triangles, we'll directly think of LawofCos:
set [unparseable or potentially dangerous latex formula] and due to cyclic, [unparseable or potentially dangerous latex formula]
In [unparseable or potentially dangerous latex formula]
In [unparseable or potentially dangerous latex formula]
as we all know, cosx=-cos(180-x) and it says: (AB)(AD)=(BC)(CD)
so, after sorting the eqation, we have [unparseable or potentially dangerous latex formula]
since it says distinct integers less than 15, we can take a guess that one is 14, and the other side is 12, which gives us the other two sides are 7 and 6.
so the the diagonal is [unparseable or potentially dangerous latex formula]