Thread: Hard question.

We coloring the plane R^2 in two different colors.

1. Prove that there is 2 points that in 1 unit distance from each other.

2. Prove that exist a rectangle that his edge points colored with the same color.

Let be a group wit 10 naturals numbers.

1. Is there a non-empty subset that the sum of element divides by 10?

2. Is there a non-empty subset that the sum of elements divides by 11?

Originally Posted by Also sprach Zarathustra

We coloring the plane R^2 in two different colors.

1. Prove that there is 2 points that in 1 unit distance from each other.

Does the problem statement require that these two points have the same color?

Point 1. Hint: can you find three equidistant points on the plane?

Point 2. Hint: Suppose we found 6 points located like this:







Let be the color of and assume that the triple equals the triple (pointwise). Can you find the required rectangle?

Now, can you find 6 points with this property?