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Biclinic Integral Quadrilaterals

Problem 311

Published on Saturday, 20th November 2010, 10:00 pm; Solved by 320

ABCD is a convex, integer sided quadrilateral with 1 ≤ AB < BC < CD < AD.
BD has integer length. O is the midpoint of BD. AO has integer length.
We'll call ABCD a biclinic integral quadrilateral if AO = CO ≤ BO = DO.

For example, the following quadrilateral is a biclinic integral quadrilateral:
AB = 19, BC = 29, CD = 37, AD = 43, BD = 48 and AO = CO = 23.

Let B(N) be the number of distinct biclinic integral quadrilaterals ABCD that satisfy AB2+BC2+CD2+AD2N.
We can verify that B(10 000) = 49 and B(1 000 000) = 38239.

Find B(10 000 000 000).