Building a new track – 2

Number Theory Level 3

Now, back at our home stadium, we want to run 5 sprints:

40 m , 45 m , 50 m , 55 m 40\si{m}, 45\si{m}, 50\si{m}, 55\si{m} and 60 m 60\si{m}

The track has a finish line and starting lines for the following distances, all with respect to the finish line,

50 m , 60 m , 75 m , 80 m , 100 m , 110 m 50\si{m}, 60\si{m}, 75\si{m}, 80\si{m}, 100\si{m}, 110\si{m}

We now realize, that it's impossible to run 45m and 55m only with these marks, so we have to guess them.

How many lines would have to be added, only between the 110m and the finish line, to make it possible to run all of the following distances?

{ 5 m , 10 m , 15 m , 20 m , … , 95 m , 100 m , 105 m , 110 m } \{5\si{m}, 10\si{m}, 15\si{m}, 20\si{m}, \ldots, 95\si{m}, 100\si{m}, 105\si{m}, 110\si{m} \}


The answer is 3.

Henry U by Henry U , 17, Germany

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1 solution

Henry U
Dec 28, 2018

With the given lines, we can run all of the following distances

possible? distance start end yes 5 80 75 yes 10 60 50 yes 15 75 60 yes 20 80 60 yes 25 75 50 yes 30 80 50 yes 35 110 80 yes 40 100 60 no 45 yes 50 50 0 no 55 yes 60 60 0 no 65 no 70 yes 75 75 0 yes 80 80 0 no 85 no 90 no 95 yes 100 100 0 no 105 yes 110 110 0 \begin{array}{cccc} \text{possible?} & \text{distance} & \text{start} & \text{end} \\ \text{yes}&5&80&75\\ \text{yes}&10&60&50\\ \text{yes}&15&75&60\\ \text{yes}&20&80&60\\ \text{yes}&25&75&50\\ \text{yes}&30&80&50\\ \text{yes}&35&110&80\\ \text{yes}&40&100&60\\ \text{no}&45\\ \text{yes}&50&50&0\\ \text{no}&55\\ \text{yes}&60&60&0\\ \text{no}&65\\ \text{no}&70\\ \text{yes}&75&75&0\\ \text{yes}&80&80&0\\ \text{no}&85\\ \text{no}&90\\ \text{no}&95\\ \text{yes}&100&100&0\\ \text{no}&105\\ \text{yes}&110&110&0 \end{array}

First, consider 105. The only (convenient) ways are to introduce a 105 m 105\si{m} -line and run till the finish, or draw a 5 m 5\si{m} line and run from 110 m 110\si{m} to this. The later option is by far better because it makes many more distances possible. Our table now looks like this

possible? distance start end yes 5 80 75 yes 10 60 50 yes 15 75 60 yes 20 80 60 yes 25 75 50 yes 30 80 50 yes 35 110 80 yes 40 100 60 now 45 50 5 yes 50 50 0 now 55 60 5 yes 60 60 0 no 65 now 70 75 5 yes 75 75 0 yes 80 80 0 no 85 no 90 now 95 100 5 yes 100 100 0 now 105 110 5 yes 110 110 0 \begin{array}{cccc} \text{possible?} & \text{distance} & \text{start} & \text{end} \\ \text{yes}&5&80&75\\ \text{yes}&10&60&50\\ \text{yes}&15&75&60\\ \text{yes}&20&80&60\\ \text{yes}&25&75&50\\ \text{yes}&30&80&50\\ \text{yes}&35&110&80\\ \text{yes}&40&100&60\\ \text{now}&45&50&5\\ \text{yes}&50&50&0\\ \text{now}&55&60&5\\ \text{yes}&60&60&0\\ \text{no}&65\\ \text{now}&70&75&5\\ \text{yes}&75&75&0\\ \text{yes}&80&80&0\\ \text{no}&85\\ \text{no}&90\\ \text{now}&95&100&5\\ \text{yes}&100&100&0\\ \text{now}&105&110&5\\ \text{yes}&110&110&0 \end{array}

If it was possible to draw one more line that makes running the 3 missing distances possible, then 85m and 90m, which are 5 away from each other would both have to start at this new line and end at two lines that are also 5m away, the only possibility for this is 75m and 80m, but then the new line wouldn't fit on the track. This already proves that we need at least two more lines.

For these, we can choose pretty freely, for example one at 10m and one at 15m.

possible? distance start end yes 5 80 75 yes 10 60 50 yes 15 75 60 yes 20 80 60 yes 25 75 50 yes 30 80 50 yes 35 110 80 yes 40 100 60 now 45 50 5 yes 50 50 0 yes 55 60 5 yes 60 60 0 now 65 75 10 yes 70 75 5 yes 75 75 0 yes 80 80 0 now 85 100 15 now 90 100 10 yes 95 100 5 yes 100 100 0 yes 105 110 5 yes 110 110 0 \begin{array}{cccc} \text{possible?} & \text{distance} & \text{start} & \text{end} \\ \text{yes}&5&80&75\\ \text{yes}&10&60&50\\ \text{yes}&15&75&60\\ \text{yes}&20&80&60\\ \text{yes}&25&75&50\\ \text{yes}&30&80&50\\ \text{yes}&35&110&80\\ \text{yes}&40&100&60\\ \text{now}&45&50&5\\ \text{yes}&50&50&0\\ \text{yes}&55&60&5\\ \text{yes}&60&60&0\\ \text{now}&65&75&10\\ \text{yes}&70&75&5\\ \text{yes}&75&75&0\\ \text{yes}&80&80&0\\ \text{now}&85&100&15\\ \text{now}&90&100&10\\ \text{yes}&95&100&5\\ \text{yes}&100&100&0\\ \text{yes}&105&110&5\\ \text{yes}&110&110&0 \end{array}

This means that we need a total of 3 \boxed{\boxed{3}} new lines.

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