Saudi Arabia Mathematical Competitions

Question

Find positive integers a_1 < a_2 < < a_2010 such that a_1(1!)^2010 + a_2(2!)^2010 + + a_2010(2010!)^2010 = (2011!)^2010.

Accepted Answer

Consider a_1 = 2^2010, a_2 = 3^2010 - 1, a_3 = 4^2010 - 1, , a_2010 = 2011^2010 - 1 and get gathered a_1(1!)^2010 + a_2(2!)^2010 + + a_2010(2010!)^2010 = (2!)^2010 + (3!)^2010 - (2!)^2010 + (4!)^2010 - (3!)^2010 + + + (2011!)^2010 - (2010!)^2010 = (2011!)^2010 gathered Moreover, it is clear that a_1 < a_2 < a_3 < < a_2010.