Problem 311
Biclinic Integral Quadrilaterals
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+AD2 ≤ N.
We can verify that B(10 000) = 49 and B(1 000 000) = 38239.
Find B(10 000 000 000).
双斜整数四边形
ABCD是一个各边长为整数的凸四边形，满足1 ≤ AB < BC < CD < AD。
BD的长为整数，O是BD的中点，且AO的长也是整数。
如果AO = CO ≤ BO = DO，我们称ABCD为双斜整数四边形。
例如，如下的四边形是双斜整数四边形：
AB = 19，BC = 29，CD = 37，AD = 43，BD = 48，以及AO = CO = 23。
记B(N)是所有满足AB2+BC2+CD2+AD2 ≤ N的双斜整数四边形ABCD的数目。
我们可以验证B(10 000) = 49以及B(1 000 000) = 38239。
求B(10 000 000 000)。