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Maximum number of balanced sets with nonempty intersections

Source: Romania TST 1 2012, Problem 5

May 3, 2012
arithmetic sequencealgebra proposedalgebra

Problem Statement

Let pp and qq be two given positive integers. A set of p+qp+q real numbers a1<a2<⋯<ap+qa_1<a_2<\cdots <a_{p+q} is said to be balanced iff a1,…,apa_1,\ldots,a_p were an arithmetic progression with common difference qq and ap,…,ap+qa_p,\ldots,a_{p+q} where an arithmetic progression with common difference pp. Find the maximum possible number of balanced sets, so that any two of them have nonempty intersection.
Comment: The intended problem also had "pp and qq are coprime" in the hypothesis. A typo when the problems where written made it appear like that in the exam (as if it were the only typo in the olympiad). Fortunately, the problem can be solved even if we didn't suppose that and it can be further generalized: we may suppose that a balanced set has m+nm+n reals a1<⋯<am+n−1a_1<\cdots <a_{m+n-1} so that a1,…,ama_1,\ldots,a_m is an arithmetic progression with common difference pp and am,…,am+n−1a_m,\ldots,a_{m+n-1} is an arithmetic progression with common difference qq.
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