Task 3A - 161043A
In a sporting triathlon, a clear ranking of the athletes (marked by position numbers 1, 2, 3, ...) was determined in each of the three sports.

In each of the three sports, points were awarded for the first five places such that the point value (a natural number \(>0\)) decreased with increasing position number and from the 2nd place onwards, the point difference between adjacent places was always constant.

This point system was the same for each of the three sports.

After two competitions, it turned out that the first three places in each of these two sports were always taken by athletes A, B, and C (not necessarily in that order).

Each of the athletes A and B had 17 points after two competitions, and athlete C had 16 points after two competitions.

In the overall ranking of the triathlon (sum of the three achieved point values), athlete D won. Athlete C came in second.

Determine the position numbers for athletes C and D in each of the three sports that satisfy these conditions!
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After 2 competitions, athletes A, B, C have achieved a total of 50 points (17/17/16) with the places won ranging from 1 to 3.

Under these conditions, the distribution of placements can only look like this: A and B the 1st/3rd and 3rd/1st places, and for \( \mathrm{C} \) two 2nd places. Let \( x \in \mathbb{N} \) be the point value for 1st place and with \( \mathrm{C}=16:(x-k)=8 \).

\[
\begin{aligned}
& 2 x+2(x-k)+2(x-(k+l))=50 ; \quad 2 x+16+16-2 l=50 \\
& \Longrightarrow x-l=9 \quad \Longrightarrow l=k-1 \quad \text { (1) }
\end{aligned}
\]

The sum \( P \) forms the total points of the three competitions, consisting of 75 points for the first three places plus the points for the 4th and 5th places:

\[
P=75+3(x-(k+2 l))+3(x-(k+3 l) \quad \Longrightarrow P=123-15 l
\]

The points per competition: \( P / 3=41-5 l \) and for the 4th and 5th places \( P_{4 ; 5} / 3=41-25-5 l=16-5 l \). Now it can be easily estimated that \( l<3 \) must be true and can only take the values 1 or 2.

Athlete D must achieve more than 18 points to win, so that C can achieve at least 18 points for 2nd place.

The points for D: \( D_{p}=2(x-(k+2 l))+x \geq 19 \) With \( x=9+l \) and \( k=l+1 \)

\[
D_{p}=2(9+l-(l+1+2 l))+9+l=25-3 l
\]

For \( l=1 \), C gets: \(\{2 ; 2 ; 5\}\) and for D: \(\{4 ; 4 ; 1\}\)

For \( l=2 \), C also gets: \(\{2 ; 2 ; 5\}\) and for D: \(\{4 ; 4 ; 1\}\)

Point distribution for \( l=1: 10,8,7,6,5 \) and for \( l=2: 11,8,6,4,2 \)