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A combinatorics problem from pre RMO 2015

A subset \(B\) of the set of first 100 positive integer has the property that no two elements of \(B\) sum upto 125. What is the maximum number of elements in \(B\)?

Note by Nihar Mahajan
2 years ago

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Maximum number of elements in R is 62. From 1 to 24 and either one of (62, 63), (61, 64), (60, 65), . . . . ., (26, 99), (25, 100).

Rajen Kapur - 2 years ago

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Yes , correct. The basic idea is complementation. We find \(a,b\) such that \(a+b=125\) and we have 38 pairs of it. So excluding such pairs , the remaining numbers that 1 to 24 must at least be included in set \(B\). To make number of elements maximum , we include either all \(a\) or \(b\) making the max. number of elements as \(24+38=62\).

Nihar Mahajan - 2 years ago

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