2011 IMO Shortlist C1

Let $n>0$ be an integer. We are given a balance and $n$ weights of weight $2^{0}, 2^{1}, \ldots, 2^{n-1}$. In a sequence of $n$ moves we place all weights on the balance. In the first move we choose a weight and put it on the left pan. In each of the following moves we choose one of the remaining weights and we add it either to the left or to the right pan. Compute the number of ways in which we can perform these $n$ moves in such a way that the right pan is never heavier than the left pan. ## $\mathrm{C} 2$ Suppose that 1000 students are standing in a circle. Prove that there exists an integer $k$ with $100 \leq k \leq 300$ such that in this circle there exists a contiguous group of $2 k$ students, for which the first half contains the same number of girls as the second half.

令 $n>0$ 为整数。我们得到一个余额和 $n$ 个权重，权重为 $2^{0}, 2^{1}, \ldots, 2^{n-1}$。在一系列 $n$ 移动中，我们将所有重量放在天平上。在第一步中，我们选择一个重物并将其放在左盘上。在以下每个动作中，我们选择剩余的重量之一，并将其添加到左侧或右侧的平底锅中。计算我们可以执行这些 $n$ 移动的方式数量，使得右平底锅永远不会比左平底锅重。 ## $\mathrm{C} 2$ 假设 1000 名学生站成一圈。证明存在一个整数 $k$，$100 \leq k \leq 300$，使得在这个圆圈中存在一个连续的 $2 k$ 学生组，其中前半部分包含与后半部分相同数量的女孩。