Running track

To the Akčesú races there came 25 runners. The running track was however narrow, and therefore always only five runners could run at once. What surprised Sára and Arthur the most however was the fact that the Te-TiVá people do not have stopwatches, nor other devices with which they could measure the runners' performance precisely. They could compare the performance of the runners only within one race according to who came in at what position. Fortunately, Te-TiVá runners are experienced sportsmen and everyone has balanced performances, so one runner runs each of his races equally fast. And in fact there is never a tie, since coming in at the same position as someone else is, according to the Te-TiVá people, very boring. What is the smallest number of races with 5 competitors that the competitive runners had to run, so that the Te-TiVá people could from among the 25 runners determine the three fastest?

Final Answer:

n =  13

Step-by-step explanation:

$$\begin{gathered} n_1=25 \\ {n}_2=n_1/5\cdot 3=25/5\cdot 3=15 \\ {n}_3=n_2/5\cdot 3=15/5\cdot 3=9 \\ {n}_4=\lceil n_3/5\cdot 3\rceil =\lceil 9/5\cdot 3\rceil =6 \\ {n}_5=\lceil n_4/5\cdot 3\rceil =\lceil 6/5\cdot 3\rceil =4 \\ n=5+3+2+2+1=13 \end{gathered}$$

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