2024 AIME I 第 6 题

Question

考虑在一个

8 \times 8

方格中，从左下角沿网格线走到右上角、长度为

16

的路径。求这样的路径中，恰好改变方向四次的路径数，如下图中的例子所示。

Consider the paths of length

16

that follow the lines from the lower left corner to the upper right corner on an

8 \times 8

grid. Find the number of such paths that change direction exactly four times, as in the examples shown below.

答案

答案：294

知识点：格路分拆与有序分拆

难度评级：2340

小提示：

恰好转向四次的路径由五段交替向右和向上的直线段组成

A path with exactly four turns consists of five straight runs that alternate between rightward and upward

大提示：

若路径从向右开始，则三段向右和两段向上的长度都是正整数，并且各自总和为

8

；先数这些组成，再将结果乘二

If the path starts rightward, the three rightward runs and two upward runs are positive integers summing to

8

each — count the compositions, then double

解答：

恰好改变方向四次的路径由五段最长的直线段组成，方向在向右与向上之间交替。若第一段向右，模式为

R, U, R, U, R

：三段向右的正长度总和为

8

，两段向上的正长度总和为

8

。这样的组成数分别为

\binom{7}{2} = 21

和

\binom{7}{1} = 7

，给出

21 \cdot 7 = 147

条路径。

从向上开始的路径由对称性也有

147

条。总数为

147 + 147 = 294

。

A path that changes direction exactly four times consists of five maximal straight runs, alternating between rightward and upward moves. If the first run is rightward, the pattern is

R, U, R, U, R:

three rightward runs with positive lengths summing to

8,

and two upward runs with positive lengths summing to

8.

The counts of such compositions are

\binom{7}{2} = 21

and

\binom{7}{1} = 7,

giving

21 \cdot 7 = 147

paths.

Paths starting upward are counted symmetrically, another

147.

The total is

147 + 147 = 294.

2024 AIME I 第 6 题

答案：294

难度评级：2340

小提示：

大提示：

解答：

其他年份的第 6 题