\section*{Problem 1 - 251021}

Give all triples \((a, b, c)\) of integers \(a, b, c\) with \(a \leq b \leq c\) and \(a \cdot b \cdot c=1985\)!
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The prime factorization of 1985 is \(5 \cdot 397 = 1985\).

If we include the factor 1, there are exactly the following representations for natural numbers \(a, b, c\):
(1) \(1 \cdot 5 \cdot 397 = 1985\)
(2) \(1 \cdot 1 \cdot 1985 = 1985\)

Since all integer triples are sought, the cases where exactly two of the factors have a negative sign must also be considered:

From (1), this results in exactly the representations \((-1) \cdot (-5) \cdot 397\), \((-1) \cdot 5 \cdot (-397)\), and \(1 \cdot (-5) \cdot (-397)\).

From (2), the representations \((-1) \cdot (-1) \cdot 1985\) and \(1 \cdot (-1) \cdot (-1985)\) are derived.

In total, there are exactly the seven mentioned triples that satisfy the problem statement.