2005 USAMO 第 4 题

Legs $L_1, L_2, L_3, L_4$ of a square table each have length $n$, where $n$ is a positive integer. For how many ordered 4-tuples $(k_1, k_2, k_3, k_4)$ of nonnegative integers can we cut a piece of length $k_i$ from the end of leg $L_i \; (i = 1,2,3,4)$ and still have a stable table?

(The table is stable if it can be placed so that all four of the leg ends touch the floor. Note that a cut leg of length 0 is permitted.)

方桌的桌腿 $L_1、L_2、L_3、L_4$ 的长度均为 $n$，其中 $n$ 为正整数。对于多少个非负整数的有序 4 元组 $(k_1, k_2, k_3, k_4)$，我们可以从边 $L_i \; 的末尾切出一段长度为$k_i$的片段； (i = 1,2,3,4)$ 并且仍然有稳定的表？

(如果可以将桌子放置成所有四个腿端都接触地板，则桌子是稳定的。请注意，允许长度为 0 的切割腿。)