1
$\begingroup$

The problem I'm working on says:

A basketball player has been training for 112 hours during 12 days. He has trained an integer number of hours every day. Prove that there was two consecutive days where he has trained for at least 19 hours.

I'm following this rationale to try prove it:

As we are interested in two consecutive days we split the 12 days into 6 pairs: {1,2}, {3,4}, {5,6}, {7,8}, {9,10}, {11, 12}

Using a corollary of the Pigeonhole principle we know that the sum of one the pairs is greater than 18, but not 19.

Does this mean that I can't prove it this way?

  • 0
    I agree about LaTeX in general although for this post it is not really necessary. You shouldn't group 1 and 12 together, they are not consecutive days.2011-12-28

2 Answers 2

5

Your grouping works just fine. The average over the $6$ groups is $\dfrac{112}{6}$, which is greater than $18$. So at least one of the group sums is greater than $18$. Since all group sums are integers, at least one of these must be $\ge 19$.

  • 0
    @Javi: There is no need to assume *positive* integer number of hours, $\ge 0$ is fine. Actually, negative integer also allows the proof to go through. If you have a sequence $a_1,a_2, \dots, a_{12}$ of $12$ integers, with sum $112$, then some two consecutive elements of the sequence add up to $19$ or more. But negative integers would not make much sense for basketball pracice.2011-12-28
2

This can be proved easily. Suppose the statement is NOT true. Then any given pair of consecutive days he trains for at most 18 hours. So he trains a total of at most 18 * 6 = 108 hours. This is a contradiction since we are told that he trains a total of 112 hours.

I hope this helps! :)

  • 0
    Javi, cuando la matemática es intuitiva es mejor no complicarse al divino cohete. Saludos y suerte con tus estudios!2011-12-28