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special number partition

Source: KMO 2023 P8

November 4, 2023
combinatorics

Problem Statement

For a positive integer nn, if nn is a product of two different primes and n≡2(mod3)n \equiv 2 \pmod 3, then nn is called "special number." For example, 14,26,35,3814, 26, 35, 38 is only special numbers among positive integers 11 to 5050. Prove that for any finite set SS with special numbers, there exist two sets A,BA, B such that
[*] A∩B=∅,A∪B=SA \cap B = \emptyset, A \cup B = S [*] ∣∣A∣−∣B∣∣≤1||A| - |B|| \leq 1 [*] For all primes pp, the difference between number of elements in AA which is multiple of pp and number of elements in BB which is multiple of pp is less than or equal to 11.
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