Cube and Square

Do there exist 1985 distinct positive integers such that the sum of their squares is a cube and the sum of their cubes is a square?

 

Solution

Answer: yes.

Take any n integers a_i. Suppose that ∑ a_i² = b and ∑ a_i³ = c. Now multiply each a_i by b⁴c³. The sum of their squares becomes b⁹c⁶ which is a cube and the sum of their cubes becomes b¹²c¹⁰ which is a square.