Problem:
Given and a point on one of its sides, call line the splitting line of through if passes through and divides into two polygons of equal perimeter. Let be a triangle where and and are positive integers. Let and be the midpoints of and , respectively, and suppose that the splitting lines of through and intersect at . Find the perimeter of .
Solution:
Look at the splitting line through point M. After extending BC, create point P such that
Then, observe how this splitting line splits into 2 congruent segments, yielding that the splitting line is a midline of , which yields that it's parallel to
Notice so is isosceles, and after angle chasing we see that
After using law of cosines, we get the equation,
Notice that the RHS is divisible by 3, and so we can write and have
From here, we can rewrite the above expression as
and this directly yields
After this, notice that we can bound by using AM GM.
Substitute this into the original expression
And rearranging and solving yields
Now, we just take the expression using multiple mods to see that and which is nice because so we only have to try using our bounding in previous steps.
Thus, we see that yields valid solution yielding our answer of
The problems on this page are the property of the MAA's American Mathematics Competitions