Combinatorics · Parity · Pair counting · Block designs
A class has \(30\) students, and every day exactly three of them are on duty in the school kitchen. Prove that the duty roster cannot be arranged so that every two students of the class are on duty together exactly once.
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Serbian Municipal Competition 2001, high school grade I, category A, problem 4. Organized by the Mathematical Society of Serbia (DMS). Source