Suppose there are $3$ runners $R1, R2$ and $R3$. The betting odds paid on each runner are $r1:1$, $r2:1$ and $r3:1$. This means that if, for example, you bet $\$2$ for runner 1 winning, you will make $2((r1+1)/1)-2$ if runner 1 ends up winning.
The actual probabilities of each runner winning are unspecified. How much should you bet on each runner so that you always end up with a profit?
Is it simply a matter of solving the following system of inequalities (let $a,b$ and $c$ be the amount you bet for each of the runners) for $a,b$ and $c$:
$$a(r1+1)-a-b-c >0 \\ b(r2+1)-b-a-c >0 \\ c(r3+1)-c-b-a >0$$