2012 HMMT November General Round Problem 7

Find the number of ordered 2012-tuples of integers (x_1, x_2, \ldots, x_{2012}), with each integer between 0 and 2011 inclusive, such that the sum x_1 + 2x_2 + 3x_3 + \cdots + 2012x_{2012} is divisible by 2012.

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2012 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.