IMO 2023 | PROBLEM 5 - 9 July 2023

Problem 5.

Let $n$ be a positive integer. A Japanese triangle consists of $1 + 2 + · · · + n$ circles arranged in an equilateral triangular shape such that for each $i = 1, 2, . . . , n$ , the $i^{th}$ row contains exactly $i$ circles, exactly one of which is coloured 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.

proposed by Merlijn Staps, Netherlands