2004 AIME II Problem 14

Consider a string of n 7's, 7777\cdots77, into which + signs are inserted to produce an arithmetic expression. For example, 7+77+777+7+7=875 could be obtained from eight 7's in this way. For how many values of n is it possible to insert + signs so that the resulting expression has value 7000?

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.