Japan 2007

Question

Find one of the polynomials f(x, y, z) whose degree is 3, with real coefficients, that satisfy the following conditions. • f(x, y, z) + x is divisible by y + z • f(x, y, z) + y is divisible by z + x • f(x, y, z) + z is divisible by x + y A polynomial P(x, y, z) is divisible by a polynomial Q(x, y, z) means that there exists a polynomial R(x, y, z) that satisfies P(x, y, z) = Q(x, y, z)R(x, y, z).

Accepted Answer

f(x, y, z) = k(x + y)(y + z)(z + x) - x - y - z, (k 0) satisfies the conditions (in fact, only this form is a solution). Put g(x, y, z) = f(x, y, z) + x + y + z. Then the conditions are equivalent to the condition that g(x, y, z) is divisible by x + y, y + z, z + x. And the condition that the degree of f(x, y, z) is 3 and f is with real coefficients are equivalent to the condition that the degree of g(x, y, z) is 3 and g is with real coefficients. Now g(x, y, z) = k(x + y)(y + z)(z + x), (k 0) satisfies all the conditions. Then f(x, y, z) = k(x + y)(y + z)(z + x) - x - y - z, (k 0) satisfies all the conditions too.