2022 SSMO Speed Round Problems/Problem 10
Problem
In a circle centered at with radius
we have non-intersecting chords
and
is outisde of quadrilateral
and
Let
and
Suppose that
. If
and
, then
for
and squareless
Find
Solution
Let .
Then, by power of the point we have that
and subtracting gives that
.
Since we know that
, dividing gives
that
so
and
.
Then, by law of cosines, it follows that
which implies that
.
Then,
which implies that
so the
answer is then
.