Let
![S = \{x_1, x_2, \ldots, x_{k + l}\}](/media/m/4/9/2/492a43054e05905878a112529ffdd20c.png)
be a
![(k + l)](/media/m/7/8/0/78086c25e7847afb830fbf1c636106e1.png)
-element set of real numbers contained in the interval
![[0, 1]](/media/m/7/f/0/7f04c41d91b0555ac792351e836cbd4e.png)
;
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
and
![l](/media/m/e/e/9/ee975101080f37986f56baaf4c3cdcd2.png)
are positive integers. A
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
-element subset
![A\subset S](/media/m/9/9/e/99e9ed029bd9aa26a53e5391592cd11b.png)
is called nice if
Prove that the number of nice subsets is at least
![\dfrac{2}{k + l}\dbinom{k + l}{k}](/media/m/d/3/9/d3924e48cc2e729e3479cdc2fb8e2f45.png)
.
Proposed by Andrey Badzyan, Russia
%V0
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