Andělka, Barča, Honzík, Vlád’a, and Matyáš participated in a competition throwing a shuttlecock. Each of them threw once, and the total length of their throws was 41 meters. Matyáš threw the shortest, which was 90 cm less than Andělka, and she threw 60 cm less than Vlád’a. Honzík threw the farthest and hit the tape marking whole meters. If only Matyáš, Vlád’a, and Andělka competed, the average length of their throws would be 20 cm shorter.

Determine the lengths of the throws of all the named children.

(L. Dedková)
Let's denote the lengths of the throws in meters by the initials of the competitors. (All the following results are also in $m$ and this unit will not be mentioned in the calculations.) The sum of the lengths of all their throws was 41 meters,

$$
a+b+h+v+m=41
$$

and the average length of their throws was:

$$
\frac{a+b+h+v+m}{5}=8.2
$$

If only Matyáš, Vlád’a, and Andělka were to compete, the average length of their throws would be 20 cm shorter, i.e., exactly 8 m:

$$
\frac{a+v+m}{3}=8
$$

After rearrangement, we get:

$$
a+v+m=24.
$$

From the problem statement, we know that Matyáš threw the shortest, which was 90 cm less than Andělka, and Andělka threw 60 cm less than Vlád’a:

$$
a=m+0.9, \quad v=a+0.6=m+1.5.
$$

Substituting into equation (2), we get:

$$
\begin{aligned}
m+0.9+m+1.5+m & =24 \\
m & =7.2.
\end{aligned}
$$

Substituting into equations (3), we calculate the lengths of the throws by Andělka and Vlád’a:

$$
a=7.2+0.9=8.1, \quad v=7.2+1.5=8.7.
$$

According to equation (2), we replace $v$ in equation (1) with the sum $a+v+m$ equal to 24, rearrange, and obtain the equation:

$$
b+h=17.
$$

From the problem statement, we also know that Honzík threw the farthest and hit the tape marking whole meters. By comparing with the longest throw calculated so far, we determine that Honzík threw at least 9 meters. If Honzík threw exactly 9 meters, Barča would have thrown $17-9=8$ meters. If Honzík threw 10 meters (or more), Barča would have thrown 7 meters (or less). This, however, is not possible because Matyáš threw the shortest. Therefore, the problem has a unique solution, which is

$$
a=8.1, \quad b=8, \quad h=9, \quad v=8.7, \quad m=7.2.
$$

Scoring. 2 points for determining that $a+v+m=24$; 2 points for calculating the lengths of the throws by Andělka, Vlád’a, and Matyáš; 2 points for calculating the lengths of the throws by Barča and Honzík, including the justification that it is the only solution.