Competition problem — Geometry — Maths Olympiad Prep

Library / /286 of 841

Geometry Difficulty 5.0 AIME, harder Find the answer

Let $A B C D$ be a rectangle such that $A B=20$ and $A D=24$ . Point $P$ lies inside $A B C D$ such that triangles $P A C$ and $P B D$ have areas 20 and 24, respectively. Compute all possible areas of triangle $P A B$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

There are four possible locations of $P$ as shown in the diagram. Let $O$ be the center. Then, $[P A O]=10$ and $[P B O]=12$ . Thus, $[P A B]=[A O B] \pm[P A O] \pm[P B O]=120 \pm 10 \pm 12$ , giving the four values $98,118,122$ , and 142.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.