Suppose a1, a2, . . . , an is a list of n numbers with the following properties:

Question

The sum of those n numbers is 500.
The sum of the smallest three of those numbers is 48.
The sum of the largest two of those numbers is 35.
(Note: There might be some repetitions among the numbers ak in that list.)
(a) What are the possible values of n ? Explain your reasoning.
(b) Can each of those values of n occur if we require all the numbers ak to be integers?

Answer 1

find two consecutive positive integer sum of the whose squares is 365