Competition problem — Geometry — Olympiad Maths Prep

Library / /3 of 30

Geometry Difficulty 5.1 AIME, harder Prove it Belarus

Is it possible to mark 8 distinct points (1 red, 3 blue, and 4 green points) on a plane so that the sum of the distances between the red point and the blue points is 6, the sum of the distances between the red point and the green points is 16, and the sum of the distances between the blue points and the green points is 56?

Solution

For example, the points can be marked on the straight line in the following way: the blue points have the coordinates $-3$ , $-1$ , $2$ , the red point has the coordinate $0$ , and the green points have the coordinates $3$ , $3.5$ , $4.5$ , $5$ .

Looking for a route rather than an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.