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100 numbers in a circle, min number of good pairs

Source: 2022 South Russian Girls MO - Assara Seniors p2

March 1, 2024
combinatorics

Problem Statement

Numbers 1,2,3,...,1001, 2, 3, . . . , 100 are arranged in a circle in some order. A good pair is a pair of numbers of the same parity, between which there are exactly 33 numbers. What is the smallest possible number good pairs?
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