Maths Olympiad Prep

Library / /10 of 520

Geometry Difficulty 4.7 AIME Find the answer

3-5. Find all lines in space passing through a given point MM at a given distance dd from a given line ABAB.

Solution

Solve 5. Answer: lines lying in two planes tangent to a cylinder of radius dd with axis ABA B and passing through point MM, except for the line parallel to ABA B (if point MM is located at a distance dd from line ABA B, then there will be only one tangent plane and there is no need to exclude the line). Lines at a distance dd from line ABA B are precisely the lines tangent to the specified cylinder. If we select from these lines those that pass through point MM, we obtain the set described in the answer.

## Second Round

## 7 - 8 grades

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: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.