Let $n$ be a positive integer. Morgane has coloured the integers $1,2,\ldots,n$. Each of them is coloured in exactly one colour. It turned out that for all positive integers $a$ and $b$ such that $a<b$ and $a+b \leqslant n$, at least two of the integers among $a$, $b$ and $a+b$ are of the same colour. Prove that there exists a colour that has been used for at least $2n/5$ integers. (Vincent Jugé)