\section*{Task 2 - 321022}

In a strength sports competition, Robert, Stefan, and Tilo are participating. Robert manages to do 20 pull-ups. Stefan aims to achieve at least \(80 \%\) of the combined performance of Robert and Tilo; Tilo wants to achieve at least \(60 \%\) of the combined performance of Robert and Stefan.

Is there a smallest possible number of pull-ups for Stefan and Tilo such that both goals are met?

If so, determine this number!
}

Let \(s\) and \(t\) be the number of pull-ups performed by Stefan and Tilo, respectively. Then, \(s \geq \frac{4}{5} \cdot (20 + t) = 16 + \frac{4}{5} t\) and \(t \geq \frac{3}{5} \cdot (20 + s) = 12 + \frac{3}{5} s\) should hold.

Since neither can perform a negative number of pull-ups, it follows that \(s \geq 16\). Substituting this, we get \(t \geq 12 + \frac{3}{5} \cdot 16 = 12 + \frac{48}{5}\), so, since \(t\) is an integer, \(t \geq 22\).

Substituting this value into the first inequality gives \(s \geq 16 + \frac{4}{5} \cdot 22 = 16 + \frac{88}{5}\), so \(s \geq 34\). Substituting this value into the second inequality gives \(s \geq 12 + \frac{3}{5} \cdot 34\), so \(s \geq 33\).

Substituting this better estimate into the first inequality gives \(s \geq 16 + \frac{4}{5} \cdot 33\), so \(s \geq 43\), and substituting this value into the second inequality gives \(t \geq 12 + \frac{3}{5} \cdot 43\), so \(t \geq 38\).

Substituting this into the first inequality gives \(s \geq 16 + \frac{4}{5} \cdot 38\), so \(s \geq 47\), and substituting this into the second inequality gives \(t \geq 12 + \frac{3}{5} \cdot 47\), so \(t \geq 41\).

Substituting this into the first inequality gives \(s \geq 16 + \frac{4}{5} \cdot 41\), so \(s \geq 49\). Substituting this into the second inequality gives \(t \geq 12 + \frac{3}{5} \cdot 49\), so \(t \geq 42\).

Substituting this into the first inequality gives \(s \geq 16 + \frac{4}{5} \cdot 42\), so \(s \geq 50\). Substituting this into the second inequality gives \(t \geq 12 + \frac{3}{5} \cdot 50 = 42\).

Thus, \(s = 50\) and \(t = 42\) are the smallest values that satisfy both inequalities, so the smallest value of \(s + t = 92\) is obtained, which is the smallest such value (since \(s \geq 50\) and \(t \geq 42\)).