IrMO 2019 Problem 6

The number $2019$ has the following nice properties: (a) It is the sum of the fourth powers of fuve distinct positive integers. (b) It is the sum of six consecutive positive integers. In fact, $2019 = 1^4 + 2^4 + 3^4 + 5^4 + 6^4$ (1) $2019 = 334 + 335 + 336 + 337 + 338 + 339$ (2) Prove that $2019$ is the smallest number that satises both (a) and (b). (You may assume that (1) and (2) are correct!)