# 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, 9 months 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).

- 2 years, 9 months ago

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 $$24+38=62$$.

- 2 years, 9 months ago