## Task 3 - 330723

Participants report on their four-day cycling tour:

Michael: "On the second day, we covered exactly $7 \mathrm{~km}$ more than on the third day."

Martin: "On the second and third day, we drove a total of $105 \mathrm{~km}$."

Matthias: "On the first day, exactly 516 and on the fourth day exactly 14 of the total distance of all four days were achieved."

Show that these statements uniquely determine how many kilometers were covered on each of the four days, and give these four distances!
On the second and third day together, according to Michael's statement, $7 \mathrm{~km}$ and twice the distance of the third day were driven. Since this was $105 \mathrm{~km}$ according to Martin's statement, the double distance of the third day was $(105-7) \mathrm{km}=98 \mathrm{~km}$. Therefore, on the third day, $98 \mathrm{~km} : 2 = 49 \mathrm{~km}$, and on the second day, $(49+7) \mathrm{km} = 56 \mathrm{~km}$ were driven.

On the first and fourth day together, according to Matthias' statement, $\frac{5}{16} + \frac{1}{4} = \frac{9}{16}$ of the total distance was driven. The $105 \mathrm{~km}$ of the second and third days were consequently the remaining $\frac{7}{16}$ of the total distance; this therefore amounted to $\frac{7}{16} \cdot 105 \mathrm{~km} = 240 \mathrm{~km}$.

Thus, on the first day, $\frac{5}{16} \cdot 240 \mathrm{~km} = 75 \mathrm{~km}$, and on the fourth day, $\frac{1}{4} \cdot 240 \mathrm{~km} = 60 \mathrm{~km}$ were driven. This completes the required proof, and the four distances for each day are given.