Balkan Mathematical Olympiad

Let $x_A^{(n)}$ (resp. $x_B^{(n)}$) be the $x$-coordinates of the first (resp. second) ant's position after $n$ minutes. Then $x_A^{(n)}-x_B^{(n)}\in \{q^n,-q^n,0\}$, and so $x_A^{(n)},x_B^{(n)}$ are given by polynomials in $q$ with coefficients in $\{-1,1,0\}$. So if the ants meet after $n$ minutes, then $$0=x_A^{(n)}-x_B^{(n)}=P(q),$$ where $P$ is a polynomial with degree at most $n$ and coefficients in $\{-2,-1,0,1,2\}$. Thus if $q=\frac{a}{b}$ ($a,b\in \mathbb{N}$), we have $a\mid 2$ and $b\mid 2$, i.e. $q\in \{\frac{1}{2},1,2\}$.

It is clearly possible when $q=1$.

We argue that $q=\frac{1}{2}$ is not possible. Assume that the ants diverge for the first time after the $k$ minute, for $k\ge 0$. Then $$\left|x_B^{(k+1)}-x_A^{(k+1)}\right|+\left|y_B^{(k+1)}-y_A^{(k+1)}\right|=2q^k.(1)$$ But also $\left|x_A^{(l+1)}-x_A^{(l)}\right|+\left|y_A^{(l+1)}-y_A^{(l)}\right|=q^l$ for each $l\ge k+1$, and so $$\left|x_A^{(kn)}-x_A^{(k+1)}\right|+\left|y_B^{(n)}-y_A^{(k+1)}\right|\le q^{k+1}+q^{k+2}+\cdots +q^{n-1}(2)$$ $$|x_B^{(n)}-x_A^{(n)}|+|y_B^{(n)}-y_A^{(n)}|\ge 2q^k-2(q^{k+1}+q^{k+2}+\cdots +q^{n-1})$$ which is strictly positive for $q=\frac{1}{2}$. So for any $n\ge k+1$, the ants cannot meet after $n$ minutes. Thus $q\ne \frac{1}{2}$.

Finally, we show that $q=2$ is also not possible. Suppose to the contrary that there is a pair of routes for $q=2$, meeting after $n$ minutes. Now consider rescaling the plane by a factor $2^{-n}$, and looking at the routes in the opposite direction. This would then be an example for $q=\frac{1}{2}$ and we have just shown that this is not possible.

Therefore, the only possible value is $q=1$.