Any solution?

OMO Fall 2015 Problem #20 was an extended version of this problem. Problems pdf Solutions pdf

This is also 2023 PKU Autumn Camp T5(first problem of that day), many people failed solving it. Say Alice is first and Bob is second, then Alice has a winning strategy in the game. Alice starts by writing $5.$ If Bob writes $y,$ Alice then writes $4y-5$. Suppose Bob now has a winning strategy. Let $x$ be the number Bob writes. Then by mcnugget theorem $x < 4y - 5$ and $x$ cannot be expressed linearly using 5 and $y,$ $4y - 5 - x$ can be expressed linearly using $5$ and $y$. That is, there exist $u, v$ such that $4y - 5 - x = 5u + yv$. This implies that $4y - 5$ can be expressed linearly using $x$, $5,$ and $y$. However, instead of writing $4y - 5,$ Alice can write $x$ in second round. This means now Alice have the winning, contradiction! Therefore, Alice has a winning strategy.$\Box$