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Integral median

Problem 513 Published on Sunday, 26th April 2015, 01:00 am; Solved by 234;
Difficulty rating: 60%

ABC is an integral sided triangle with sides a≤b≤c.
mc is the median connecting C and the midpoint of AB.
F(n) is the number of such triangles with c≤n for which mc has integral length as well.
F(10)=3 and F(50)=165.

Find F(100000).