1. ## 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.

2. ## 10 naturals

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?

3. 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?

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

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

$\displaystyle p_{11}$ $\displaystyle p_{12}$

$\displaystyle p_{21}$ $\displaystyle p_{22}$

$\displaystyle p_{31}$ $\displaystyle p_{33}$

Let $\displaystyle c_{ij}$ be the color of $\displaystyle p_{ij}$ and assume that the triple $\displaystyle (c_{11}, c_{21}, c_{31})$ equals the triple $\displaystyle (c_{12}, c_{22}, c_{32})$ (pointwise). Can you find the required rectangle?

Now, can you find 6 points with this property?