2008 IMO Shortlist C5

Let $S=\left\{x_{1}, x_{2}, \ldots, x_{k+\ell}\right\}$ be a $(k+\ell)$-element set of real numbers contained in the interval $[0,1] ; k$ and $\ell$ are positive integers. A $k$-element subset $A \subset S$ is called nice if $$\left|\frac{1}{k} \sum_{x_{i} \in A} x_{i}-\frac{1}{\ell} \sum_{x_{j} \in S \backslash A} x_{j}\right| \leq \frac{k+\ell}{2 k \ell} .$$ Prove that the number of nice subsets is at least $\frac{2}{k+\ell}\left(\begin{array}{c}k+\ell \\ k\end{array}\right)$.

设 $S=\left\{x_{1}, x_{2}, \ldots, x_{k+\ell}\right\}$ 为区间 $[0,1] 中包含的$(k+\ell)$实数元素集； k$ 和 $\ell$ 是正整数。如果 $$ \left|\frac{1}{k} \sum_{x_{i} \in A} x_{i}-\frac{1}{\ell} \sum_{x_{j} \in S \backslash A} x_{j}\right| $k$ 元素子集 $A \subset S$ 被称为良好。 \leq \frac{k+\ell}{2 k \ell} 。 $$证明好子集的数量至少为$\frac{2}{k+\ell}\left(\begin{array}{c}k+\ell \\ k\end{array}\right)$。