Belorusija 2012

Question

Non-zero real numbers a, b, c, d satisfy the equalities a + b + c + d = 0, 1a + 1b + 1c + 1d + 1abcd = 0. Find all possible values of the product (ab - cd)(c + d).

Accepted Answer

Answer: -1. By condition, 1a + 1b + 1c + 1d = -1abcd bcd + cda + dab + abc = -1. So aligned -1 = bcd + cda + dab + abc &= (bcd + cda) + (dab + abc) = cd(b + a) + ab(c + d) = &= [a + b + c + d = 0 a + b = -(c + d)] = ab(c + d) - cd(c + d) = (ab - cd)(c + d). aligned Therefore, (ab - cd)(c + d) = -1 for all admissible values of a, b, c, d.