Library / Italian Mathematical Olympiad /59 of 70

Italian Mathematical Olympiad

Geometry Difficulty 6.0 National Olympiad Find the answer Italy

Problem:

From a point $L$ two straight roads depart, forming an acute angle $\alpha$ . Along one of the two roads there are two lampposts, positioned at $P$ and $Q$ , such that $L P=40~\mathrm{m}$ and $L Q=90~\mathrm{m}$ . Eva is at $E$ on the other road, and she sees the two lampposts under an angle $P \widehat{E} Q$ . At what distance from $L$ is Eva, if $P \widehat{E} Q$ has the maximum possible amplitude?

Pick one

Solution

Solution:

The answer is (B). Let $f$ be the road on which Eva is located and $s$ the one on which the lampposts are located. Consider the circle passing through $P, Q, E$ , which exists since $P, Q$ belong to $s$ while $E$ does not belong to it, otherwise the angle $P \hat{E} Q$ would be 0, and suppose by contradiction that it is not tangent to $f$ . Then there exists the further intersection point $E'$ of this circle with $f$ , and let $E''$ be a point on the arc $E E'$ not containing $P$ and $Q$ ; we have $P \hat{E} Q = P \hat{E}'' Q$ since they subtend the same arc $P Q$ ; let $D$ be the intersection of $f$ and $P E''$ , then $P \hat{D} Q > P \hat{E}'' Q = P \hat{E} Q$ since it is an exterior angle of the triangle $P D E''$ not adjacent to the angle at $E''$ . Then the point $E$ could not give the maximum angle, and therefore the circle through $P, Q, E$ must necessarily be tangent to $f$ at $E$ . But then, by the tangent-secant theorem, $L E^2 = L P \cdot L Q$ , hence $L E = \sqrt{40 \cdot 90} = 60$ .

(Kuzmin)

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.