Task 2 - 330942
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Determine all triples \((a, b, c)\) of positive integers \(a, b, c\) none of which is greater than 100 and for which the inequalities \(a+b \geq 101, a+c \leq 101, a+b+c \geq 201\) hold!
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If \(a+c<101\) or \(b<100\), then due to \(a+c \leq 101\) and \(b \leq 100\) also \(a+c+b<101+100=201\), in contradiction to the last given inequality. Therefore, \(a+c=101\) and \(b=100\) must hold, so that the last two inequalities are satisfied in any case.

From \(a+c=101\) it follows that for every positive integer \(1 \leq a \leq 100\), \(c=101-a\) is also a positive integer not greater than 100. Finally, for all these \(a\), since \(b=100\), \(a+b \geq 101\) also holds, so all conditions are satisfied. There cannot be any further solutions. Thus, exactly the elements of the set

\[
\{(a, 100,101-a) \mid a \in \mathbb{Z} \wedge 1 \leq a \leq 100\}
\]

satisfy the conditions of the problem statement.

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