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An apple (or more) a day... (Posted on 2007-05-28) Difficulty: 3 of 5
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.

No Solution Yet Submitted by Federico Kereki    
Rating: 4.0000 (1 votes)

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Solution Solution (Sppoiler Ahead) Comment 1 of 1

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
  Posted by K Sengupta on 2007-05-28 15:07:28

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