1995 Putnam A4

Suppose we have a necklace of $n$ beads. Each bead is

labeled with an integer and the sum of all these labels is $n-1$.

Prove that we can cut the necklace to form a string whose

consecutive labels $x_{1},x_{2},\dots,x_{n}$ satisfy

$$

\sum_{i=1}^{k} x_{i} \leq k-1 \qquad \text{for} \quad k=1,2,\dots,n.

$$

假设我们有一条由 $n$ 颗珠子组成的项链。每颗珠子都是

用一个整数标记，所有这些标签的总和是 $n-1$。

证明我们可以把项链剪成一根绳子，

连续标签 $x_{1},x_{2},\dots,x_{n}$ 满足

$$

\sum_{i=1}^{k} x_{i} \leq k-1 \qquad \text{for} \quad k=1,2,\dots,n。

$$