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An arm wrestler is the champion for a period of 75 hours. (Here, by an hour, we mean a period starting from an exact hour, such as 1p.m., until the next hour.)

The arm wrestler had at least one match an hour, but no more than 125 total matches. Show that there is a period of consecutive hours during which the arm wrestler had exactly 24 matches.

I am completely baffled by this problem. What does this problem actually mean?? I have tried my level best to understand it but no luck.

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    I think it means that an arm wrestler is winning the matches for 75 hours. In the 75 hours, no more than 125 matches were played by him, so I think it might be 125 matches that he played, otherwise why would it say? Then you are supposed to show that in one hour, he played 24 matches. Does that help? – Stevo Oct 24 '20 at 10:38
  • Yes I understand now. But still not sure how to proceed. –  Oct 24 '20 at 10:42
  • You could look at this:https://math.stackexchange.com/questions/97397/pigeonhole-principle-at-least-1-match-hr-for-75-hrs-at-most-125-matches-then?rq=1 That might help – Stevo Oct 24 '20 at 10:56
  • I think the Pigeon Hole principle will be important. The arm wrestler has at least 1 match an hour, so if he only had 75 matches, then any contiguous twenty four period will suffice. Then show that there is a way to add 51 matches such that there is no contiguous period of time that has exactly 24 matches and when you delete any 1 match then you can find a contiguous period of time with 24 matches. – Laars Helenius Oct 24 '20 at 11:00
  • @102152111 that may not be the right interpretation. Laars Helenius's interpretation makes sense. – Math Lover Oct 24 '20 at 11:23
  • @MathLover ok. I suppose that is correct. Thank you for noting – Stevo Oct 24 '20 at 11:29
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    Bringing in arm-wrestling like this is ridiculous, IMHO. And who's going to arm-wrestle for $75$ hours straight? The problem is this: Let $(a_i,\ldots,a_{75})$ be a sequence of strictly positive integers whose sum is $\le 125$. Show that there exist $24$ consecutive elemnts whose sum is $24$. – TonyK Oct 24 '20 at 15:28

2 Answers2

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Let $m_i$ be the number of matches fought in hour $j$, so we have $m_i \ge 1$ for $1 \le i \le 75$. Define $$s_n = \sum_{i=1}^n m_i$$ for $1 \le n \le 75$. If we consider the values $s_n$ modulo $24$, there are $24$ possible slots and $75$ numbers, so there must be some slot that contains at least $4$ numbers, by the pigeonhole principle. Let's say the $4$ numbers are $s_a, s_b, s_c$ and $s_d$, with $a<b<c<d$, so $s_a=s_b=s_c=s_d \pmod{24}$. Then $s_b-s_a = s_c-s_b=s_d-s_c = 0 \pmod{24}$, so $$\sum_{i=a+1}^b m_i = \sum_{i=b+1}^c m_i= \sum_{c+1}^d m_i = 0 \pmod{24} \tag{*}$$ Therefore each one of the three sums above must be one of the values $0, 24, 48, 72 \dots$ etc.

Zero is ruled out as a sum because we know $m_i \ge 1$ for all $i$. Can all three sums be $48$ or greater? No, because then the total of the three sums would be at least $144$, and we know the total number of matches was no more than $125$. So at least one of the sums listed in $(*)$ is equal to $24$, i.e. exactly $24$ matches were fought in one of the intervals $a+1$ to $b$, $b+1$ to $c$, or $c+1$ to $d$.

awkward
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  • could you please explain why modulo 24 –  Oct 24 '20 at 15:46
  • @PrajwalJ Well, we could choose a different modulus, but the goal is to show there is an interval in which exactly $24$ matches were fought, so we choose a modulus of $24$. – awkward Oct 24 '20 at 22:53
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The problem seeks a partition of 125 into 75 parts with 2 parts that total to 24 (we can take these to be consecutive parts).

Without loss of generality, let us take the first two parts to be 12, 12 each with 2 hours done and 73 hours remaining to be accounted.

This leaves us with 125 - 12 = 113 matches to be played in 73 hours.

113 can be partitioned into 73 parts (eg: take 1 each for 72 parts and the rest in the last part).

Therefore, we have proved that there is at least a period of 2 consecutive hours where the total matches were exactly 24. There are also multiple 24 consecutive hour periods where the total matches would equal 24.

vvg
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