Algebra · Maximum function · Inequalities · Optimisation · Equality case
Determine the smallest possible value of the expression
\[ F = \max\{x,\, 1 - y\} + \max\{y,\, 2 - z\} + \max\{z,\, 3 - x\}, \]
where \(x\), \(y\), \(z\) are real numbers, and find all triples \((x, y, z)\) at which this smallest value is attained.
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Serbian Regional Competition (Okruzno takmicenje) 2020, high school grade I, category A, problem 5. Organized by the Mathematical Society of Serbia (DMS). Source