MathDB

Let S \equal{} \{x_1, x_2, \ldots, x_{k \plus{} l}\} be a (k \plus{} 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 \minus{} \frac {1}{l}\sum_{x_j\in S\setminus A} x_j\right |\le \frac {k \plus{} l}{2kl} Prove that the number of nice subsets is at least \dfrac{2}{k \plus{} l}\dbinom{k \plus{} l}{k}. Proposed by Andrey Badzyan, Russia

Problem Statement