In a non-decreasing list of N real numbers, the average is 599. If the first element of the list is replaced with four times its original value, the new average increases to 800. How many different values can the first element in the list take?

Let the sum of N numbers be S. S/N = 599, so S = 599N. Let the first element be x. The new sum is S - x + 4x = S + 3x. The new average is (S + 3x)/N = 800. Substituting S, 599N + 3x = 800N, so 3x = 201N, or x = 67N. Since the list is non-decreasing, x must be <= 599. Thus 67N <= 599, meaning N can be 1, 2, 3, 4, 5, 6, 7, 8. There are 8 possible values for N, but the problem implies specific constraints on N.