Problem 1: Color all of the points in the plane using two colors,
let’s say red and blue. Prove that no matter how the coloring was done,
there must be two points, exactly 1 mile apart, which are the same color.
Problem 2: Prove that the conclusion in problem 1 is still
true, even if you use three colors, let’s say red, white, and blue.
Problem 1 Solution: Gregory Haberl, Lansing Catholic Central High School
Proof: Construct an equilateral triangle with sides of length 1 mile anywhere in the plane. Since there are three points an equal distance apart, and each one is colored red or blue, we can use the Pigeonhole Principle to conclude that at least two of the three points must be the same color.
Also solved by: Husam Alghanem (Grand Blanc H.S.); Tabitha Miller (Bedford H.S.); Chase Schuler (Forest Hills Northern H.S.); Alex Bordyukov (Grosse Pointe South H.S.); Jaehyun Jung (Jenison H.S.); Katherine Strong (Forest Hills Central H.S.); Jacob Van Oosterhout (C.M.U. Accelerated Math Program)
Problem 2 Solution: Chase Schuler, Forest Hills Northern High School
Color all of the points in the plane using three colors. Then at least
one point in the plane must be blue; otherwise, the plane would contain
only two colors, and we’re back to problem 1. Let B be this blue point
in the plane. Now, construct a circle centered at B with a radius of 
miles. The points on this circle must be (1) all blue, (2) all red, (3)
all white, or (4) some combination of colors. If all points on the circle
are the same color, then it’s easy to see that there are two points
on the circle that are 1 mile apart. So, suppose the points on the circle
are colored with at least two colors, let’s say red and white. Let
R be a red point on the circle. So, R and B are two points that are
miles apart. Now, draw two circles of radius 1 mile, centered at R and B.
The two circlesmust intersect at two points, P and Q. Because R and B are
miles apart,
we can construct two equilateral triangles,  and
 . Now, what
are the possible colors for P and Q? Either they are both white or they
are not both white. If they’re both white, we’re done; if they’re
not both white, then one is red or blue, which means it is 1 mile away from
R or B, which has the same color. |
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Also solved by: Alex Bordyukov (Grosse Pointe South H.S.)