Combinatorics · Colourings · Periodicity · Divisibility · Residue classes
Every positive integer is painted in one of two colours, one of which is called red. The painting is periodic with period \(d\): the numbers \(x\) and \(x + d\) always receive the same colour.
Suppose there exist positive integers \(a\), \(b\), \(c\) such that for every positive integer \(x\) exactly one of the three numbers
\[ x + a, \qquad x + b, \qquad x + c \]
is red. Prove that \(d\) is divisible by \(3\).
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Serbian Regional Competition (Okruzno takmicenje) 2020, high school grade I, category A, problem 2. Organized by the Mathematical Society of Serbia (DMS). Source