2023 IMO 第 5 题

Let $n$ be a positive integer. A Japanese triangle consists of $1 + 2 + \dots +n$ circles arranged in an equilateral triangular shape such that for each $1\leq i\leq n$ , the $i^{\mathrm{th}}$ row contains exactly $i$ circles, exactly one of which is colored red. A ninja path in a Japanese triangle is a sequence of $n$ circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with $n = 6$ , along with a ninja path in that triangle containing two red circles.

In terms of $n$ , find the greatest $k$ such that in each Japanese triangle there is a ninja path containing at least $k$ red circles.

令 $n$ 为正整数。日本三角形由以等边三角形排列的 $1 + 2 + \dots +n$ 个圆组成，因此对于每个 $1\leq i\leq n$ ， $i^{\mathrm{th}}$ 行恰好包含 $i$ 个圆，其中一个圆为红色。日本三角形中的忍者路径是一系列 $n$ 个圆圈，从顶行开始，然后重复地从一个圆圈移动到紧邻其下方的两个圆圈之一，最后在底行结束。这是一个 $n = 6$ 的日本三角形示例，以及该三角形中包含两个红色圆圈的忍者路径。

根据 $n$ ，找到最大的 $k$ ，使得每个日本三角形中都有一条包含至少 $k$ 个红色圆圈的忍者路径。