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Triangles with non rational sides and integral area

Problem 390

Published on Saturday, 23rd June 2012, 08:00 pm; Solved by 390; Difficulty rating: 60%

Consider the triangle with sides √5, √65 and √68. It can be shown that this triangle has area 9.

S(n) is the sum of the areas of all triangles with sides √(1+b2), √(1+c2) and √(b2+c2) (for positive integers b and c ) that have an integral area not exceeding n.

The example triangle has b=2 and c=8.


Find S(1010).