1. Shooting at a target, with each shot the athlete scored only eight, nine, or ten points (all these points were scored at least once). Making more than 11 shots, in total he scored 100 points. How many 8-point shots did the athlete make?
Solution. $8 x+9 y+10 z=100, x, y, z \in \mathbb{N} \Rightarrow 8(x+y+z)<100$

$$
\begin{gathered}
\Rightarrow 11<x+y+z<\frac{100}{8}=12.5 \Rightarrow x+y+z=12 \\
\Rightarrow\left\{\begin{array} { c } 
{ x + y + z = 1 2 } \\
{ 8 x + 9 y + 1 0 z = 1 0 0 }
\end{array} \Rightarrow \left\{\begin{array} { c } 
{ x + y + z = 1 2 } \\
{ 9 6 + y + 2 z = 1 0 0 }
\end{array} \Rightarrow \left\{\begin{array}{c}
x+y+z=12 \\
y+2 z=4
\end{array} \Rightarrow y=2, z=1, x=9\right.\right.\right.
\end{gathered}
$$

Answer: 9.