\section*{Task 1 - 091011}

In an internationally composed cycling race, the following race situation arose.

The field of participants was split into exactly three groups (leading group, main field, last group). Each rider was in one of these groups. Exactly 14 riders were in the last group, none of whom were from the GDR. Exactly 90 percent of the remaining riders formed the main field. Some, but not all, of the GDR riders were in the main field. The leading group consisted of exactly one twelfth of the total field of participants. Of the teams represented there, the Polish were the weakest and the Soviet the strongest.

a) How many riders participated in total?

b) How many GDR riders were in the leading group?

c) How many teams were represented in the leading group?
}

Notations:

\(x\) is the total number of drivers, \(s\) the number of drivers in the leading group, \(h\) the number of drivers in the main field, and \(l\) the number of drivers in the last group.

The indices stand for the respective countries: \(s\) (Soviet), \(d\) (German), \(p\) (Polish).

Then the following relationships apply:

\[
\begin{array}{r}
x=s+m+l \\
l_{d}=0 \\
h_{d} \neq 0 \\
s=\frac{1}{12} x
\end{array}
\]

\[
\begin{array}{r}
l=14 \\
h=0.9 \cdot(x-l) \\
s_{d} \neq 0 \\
s_{s}>s_{d}>s_{p}
\end{array}
\]

From (1), (2), (4), and (7), it follows: \(x=\frac{1}{12} x+0.9 \cdot(x-14)+14 \Rightarrow \frac{11}{12} x=0.9 x+1.4 \Rightarrow x=84\). This directly yields \(s=7\).

From (8), it follows that \(s_{s} \geq 3\) and \(s_{p} \geq 1\) (since \(s_{d} \geq 2\)). However, \(s=7 \geq s_{s}+s_{d}+s_{p} \geq 3+2+1=6\). If there were a fourth team in the leading group, it would be represented by at most one driver, which contradicts the fact that the Polish team has the fewest drivers (at least one). Therefore, the leading group consists of drivers from exactly 3 teams. The number 7 can be divided into 3 summands such that the smallest and largest summands are not equal to the middle one, only through the variant: \(7=1+2+4\). Consequently, \(s_{d}=2\), so there were 2 German drivers in the leading group.

a) A total of 84 drivers participated in the cycling race.

b) Of these, 2 drivers were in the leading group.

c) Drivers from three teams were in the leading group.