In this note we present a construction of an infinite family of diagonal quintic threefolds defined over Q \mathbb {Q} each containing infinitely many rational points. As an application, we prove that there are infinitely many quadruples B = ( B 0 , B 1 , B 2 , B 3 ) B=(B_{0}, B_{1}, B_{2}, B_{3}) of co-prime integers such that for a suitable chosen integer b b (depending on B B ), the equation B 0 X 0 5 + B 1 X 1 5 + B 2 X 2 5 + B 3 X 3 5 = b B_{0}X_{0}^5+B_{1}X_{1}^5+B_{2}X_{2}^5+B_{3}X_{3}^{5}=b has infinitely many positive integer solutions.