2010 IMO Shortlist C6

Given a positive integer $k$ and other two integers $b>w>1$. There are two strings of pearls, a string of $b$ black pearls and a string of $w$ white pearls. The length of a string is the number of pearls on it. One cuts these strings in some steps by the following rules. In each step: (i) The strings are ordered by their lengths in a non-increasing order. If there are some strings of equal lengths, then the white ones precede the black ones. Then $k$ first ones (if they consist of more than one pearl) are chosen; if there are less than $k$ strings longer than 1 , then one chooses all of them. (ii) Next, one cuts each chosen string into two parts differing in length by at most one. (For instance, if there are strings of $5,4,4,2$ black pearls, strings of $8,4,3$ white pearls and $k=4$, then the strings of 8 white, 5 black, 4 white and 4 black pearls are cut into the parts $(4,4),(3,2),(2,2)$ and $(2,2)$, respectively.) The process stops immediately after the step when a first isolated white pearl appears. Prove that at this stage, there will still exist a string of at least two black pearls. (Canada)

给定一个正整数 $k$ 和另外两个整数 $b>w>1$。有两串珍珠，一串$b$黑珍珠和一串$w$白珍珠。绳子的长度就是绳子上珍珠的数量。人们按照以下规则分步骤切割这些字符串。在每个步骤中： (i) 字符串按其长度以非递增顺序排序。如果有一些长度相等的字符串，则白色字符串在黑色字符串之前。然后选择第 $k$ 个(如果它们由多于一颗珍珠组成)；如果长度大于 1 的字符串少于 $k$，则选择所有字符串。 (ii) 接下来，将每个选定的字符串切成长度最多相差一的两部分。 (例如，如果有 $5,4,4,2$ 黑珍珠串，$8,4,3$ 白珍珠串，$k=4$，则将 8 颗白珍珠、5 颗黑珍珠、4 颗白珍珠和 4 颗黑珍珠串分别切割成 $(4,4)、(3,2)、(2,2)$ 和 $(2,2)$ 部分。)当第一个孤立的白珍珠出现时，该过程立即停止。证明在这个阶段，仍然存在一串至少两颗黑珍珠。 (加拿大)