Every day I eat at least an apple. However, I take care not to eat more than 50 apples in any month.

Prove that there is a sequence of consecutive days, in which I ate exactly 125 apples.

Let ai denote the number of apples consumed by the individual on the ith day.

Let Ai denote the number of apples consumed in the first i days.

Accordingly, since the optimal rate in conformity with conditions of the problem is 50 apples per month , we must have:

[sum] ai <= 600 (50 apples per month)

Now, the individual consumes atleast one apples per day, and so:

1< = A1< A2<...< A365< = 600

Adding 125 to each of the Ai's in the above relationship, we obtain:

126< = A1+125< A2+125< ...< A365+125< = 725

Now consider the set,

S = {A1, A2,..., A365, A1+125, A2+125,..., A365 +125}

Total number of elements in S = 730.

Each of the terms takes values from 1 to 725.

Consequently, by Pigeonhole Principle, it now follows that there are atleast two terms which take the same value and hence the assertion in terms of the given problem is now proved.

*Edited on ***May 9, 2008, 4:33 pm**