Paolo Perfetti, Universitá Degli Studi di Roma “Tor Vergata”, Rome, Italy. Find a homogeneous polynomial p of degree 2 in a, b, c, and d such that for 0 < −d < a < b < c,

∫_{0}^{a} \frac{\sqrt{x(a − x)(x − b)(x − c)}}{x + d} dx = ∫_{b}^{c} \frac{\sqrt{x(a − x)(x − b)(x − c)}}{x + d} dx

if and only if \sqrt{d(a + d)(b + d)(c + d)} = p(a, b, c, d).

P.S. Actually d > 0 otherwise the first integral diverges.