2023 IMO Shortlist C3

Let $n$ be a positive integer. We arrange $1+2+\cdots+n$ circles in a triangle with $n$ rows, such that the $i^{\text {th }}$ row contains exactly $i$ circles. The following figure shows the case $n=6$. In this triangle, a ninja-path is a sequence of circles obtained by repeatedly going from a circle to one of the two circles directly below it. In terms of $n$, find the largest value of $k$ such that if one circle from every row is coloured red, we can always find a ninja-path in which at least $k$ of the circles are red. (Netherlands)

令 $n$ 为正整数。我们将 $1+2+\cdots+n$ 个圆排列在具有 $n$ 行的三角形中，使得 $i^{\text {th }}$ 行恰好包含 $i$ 个圆。下图显示了$n=6$的情况。在这个三角形中，忍者路径是通过重复从一个圆圈到其正下方的两个圆圈之一而获得的一系列圆圈。就 $n$ 而言，找到 $k$ 的最大值，这样如果每一行中的一个圆圈被涂成红色，我们总能找到一条忍者路径，其中至少 $k$ 的圆圈是红色的。 (荷兰)