14) Salvo and Maria (S and M in the figure) are separated by a long wall, inclined at an angle $\alpha$ to the straight line connecting them (where $00$; knowing that Salvo and Maria are 10 meters and 8 meters, respectively, from the nearest door, which of the two doors should Salvo cross to reach Maria by the shortest possible route?

$\begin{array}{lll}\text { (A) the door } P_{1} & \text { (B) the door } P_{2} & \text { (C) it is indifferent }\end{array}$

(D) it depends on the distance $d$ between the doors

(E) it depends on the inclination $\alpha$ of the wall
14. The answer is $\mathbf{( B )}$.

Using the Pythagorean theorem, we calculate the distances $M P_{1}=\sqrt{d^{2}+64}$ and $S P_{2}=\sqrt{d^{2}+100}$. The shortest path obtained by passing through the door $P_{1}$ then has a length of $L_{1}=10+\sqrt{d^{2}+64}$, while the shortest path passing through the door $P_{2}$ has a length of $L_{2}=8+\sqrt{d^{2}+100}$. The inequality $L_{1}>L_{2}$ can be written as

$$
\sqrt{d^{2}+100}-\sqrt{d^{2}+64}d^{2}+80$, and squaring it again we get $4 d^{2}>80^{2}-100 \cdot 64=0$, which is satisfied for every $d \neq 0$.

[Problem proposed by S. Di Trani]