13.381 Coal mined at point A is sold at q rubles per ton, while coal mined at point B is sold at $p\%$ more expensive. Points A and B are connected by a road $s$ km long. In which zone of this road AB are the consumers of coal located, for whom the purchase and delivery of coal from B is cheaper than from A, if the transportation of 1 ton of coal for a distance of 1 km costs $r$ rubles? At what point on the road AB is the enterprise located, the expenses for which the consumption of coal do not depend on the choice of point $A$ or $B$?
Solution. Let the coal consumer be located $x$ km from V. Then the costs for purchasing and delivering 1 ton of coal from A are $r(s-x)+q$, and from V $-r x+\frac{q(100+p)}{100}$.

Therefore, the inequality $r(s-x)+q \leq \frac{100 r x+q(100+p)}{100}$ is equivalent to $x \geq \frac{s}{2}-\frac{q p}{200 r}$. From the last inequality, it follows that

Answer: if $\frac{s}{2}-\frac{q p}{200 r} < x$, the costs of delivering coal from A will be cheaper than from V, and at $x=\frac{s}{2}-\frac{q p}{200 r}$, the costs will not depend on the choice of point A or V.