Math contest problem?

Question

I'm a judge in a high school math contest. Over the years, we've had really hard problems. One of my favoriates is this one.

Santa Claus and his Elves paint a geometric plane in two colors. Prove there are two points, exactly 1-mile apart, that are the same color.

Hints: Every single spot, no matter how small, is painted in one of the two colors. It doesn't matter which two colors you use (as long as they are different colors). Your proof should be only a few small sentences in length. It doesn't matter what pattern the plane is painted in, so your proof shouldn't assume a specific pattern (don't assume a checkerboard pattern). I'll post the answer over the weekend if anyone wants to check back.

Answer 1

Best Answer - Chosen by Asker

Take an equilateral triangle ABC, with side length 1 mile.

Assume point A is color one. Then point B is either color one or color two.

If point B is color one then proven, but if point B is color two then look at point C.

If point C is color one it is one mile from A and then proven, if it is color two then it is one mile from point B and one again proven.

So there must be two points one mile apart that are the same color.

Nice Question, you just had to think a little...

Asker's Rating:

Asker's Comment:

Good job - you got the same answer that the contest creator (Dr. Alexander Soifer) got.

Answer 2

Take some equilateral triangle of side length 1 mile in the plane. By the Pigeonhole Principle at least two of its vertices must be the same color. Thus there exists two points of the same color exactly 1 mile apart.

Answer 3

consider an equilateral triangle with one mle sides. The vertices (since of only two colors) must contain at least one duplicate color. qed

Answer 4

Let point A be red. Then if one of the points on the circle of radius 1 mile around A are blue, otherwise we'd have our two points. Now take one of those points on the radius that's blue and draw a circle of radius one mile around it. It'll intersect the original circle twice, and those points will be blue too.

Answer 5

No clue!

Answer 6

If the points are painted in rings so that exactly 1 mile straight out of the center of the circle is a different color, the 1 mile long line can be "drawn" as a chord on the circle. with a radius of 2 miles, a 1 mile chord is feasible.

Answer 7

naa no one is going to the ? is to difficult to understand no offense