\section*{Problem 1 - 131011}

Determine all sets \(\{a, b, c\}\) of rational numbers \(a, b, c\) with the property that \(\left\{\frac{10}{3} ;-\frac{5}{12} ; \frac{9}{4}\right\}\) is the set of sums of any two numbers from \(\{a, b, c\}\)!
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To solve are the systems of equations
I. \(a+b=10 / 3 \quad ; \quad a+c=-5 / 12 \quad ; \quad b+c=9 / 4\)

II. \(a+b=10 / 3 \quad ; \quad a+c=9 / 4 \quad ; \quad b+c=-5 / 12\)

III. \(a+b=-5 / 12 \quad ; \quad a+c=10 / 3 \quad ; \quad b+c=9 / 4\)

IV. \(a+b=-5 / 12 \quad ; \quad a+c=9 / 4 \quad ; \quad b+c=10 / 3\)
V. \(a+b=9 / 4 \quad ; \quad a+c=10 / 3 \quad ; \quad b+c=-5 / 12\)

VI. \(a+b=9 / 4 \quad ; \quad a+c=-5 / 12 \quad ; \quad b+c=10 / 3\)

Each of these systems of equations has the solutions \(3, -3 / 4\), and \(1 / 3\). The set sought is thus \(\{-3 / 4, 1 / 3, 3\}\).