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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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.
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 $d$.