XXII OBM

Suppose that at instant $0$ the traffic lights turn green and the car passes through the first light at instant $t_0\ge 0$ (time is measured in seconds). The traffic lights will stay green between the time instants $150k+90$ and $150(k+1)$, for each integer $k$. The car will pass through the lights at the instants $t_0+\frac{1500}{v}r$, for each nonnegative integer $r$.

Therefore, a necessary and sufficient condition for a non-stopping trip is that, for every integer $r$, $\frac{t_0}{150}+\frac{10}{v}$ equals an integer plus a number between $0$ and $\frac{3}{5}$. This is clearly possible if $\frac{10}{v}$ is an integer (with any $t_0$ between $0$ and $90$) or if $\frac{10}{v}$ is half of an odd integer (with any $t_0$ between $0$ and $15$).

Let's show that these are the only possible situations. If $\frac{10}{v}$ is irrational then by Kronecker theorem the fractional part of $\frac{t_0}{150}+\frac{10}{v}$ is dense in $(0,1)$ and thus admits values between $\frac{3}{5}$ and $1$. If $\frac{10}{v}$, say, $\frac{10}{v}=\frac{p}{q}$, $\gcd (p,q)=1$, then the fractional part of $\frac{10}{v}$ is $\frac{pr}{q}$ takes all values of the form $\frac{s}{q}$, $0\le s\le q-1$ (for instance, consider $r=s\cdot p^{-1}\mathrm{mod}q$). So consecutive values of the fractional part of $\frac{t_0}{150}+\frac{10}{v}$ are $\frac{1}{q}$ apart; the fractional part of $\frac{t_0}{150}+\frac{10}{v}$ takes values between $\frac{3}{5}$ and $1$ if $\frac{1}{r}\le 1-\frac{3}{5}⟺r\ge 3$. Thus $r\le 2$, that is, $\frac{10}{v}$ is an integer or half of an odd integer.

We conclude that the possible values for the speed of the car are $\frac{20}{k}$ m/s, for every positive integer $k$.