IMO Shortlist 2008 problem C5


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April 2, 2012
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Let S = \{x_1, x_2, \ldots, x_{k + l}\} be a (k + l)-element set of real numbers contained in the interval [0, 1]; k and l are positive integers. A k-element subset A\subset S is called nice if
\left |\frac {1}{k}\sum_{x_i\in A} x_i - \frac {1}{l}\sum_{x_j\in S\setminus A} x_j\right |\le \frac {k + l}{2kl}
Prove that the number of nice subsets is at least \dfrac{2}{k + l}\dbinom{k + l}{k}.

Proposed by Andrey Badzyan, Russia
Source: Međunarodna matematička olimpijada, shortlist 2008