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Math Help - Hard question.

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
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    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.

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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    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?
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  3. #3
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    Quote Originally Posted by Also sprach Zarathustra View Post
    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?
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  4. #4
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    Point 1. Hint: can you find three equidistant points on the plane?

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

    p_{11} p_{12}

    p_{21} p_{22}

    p_{31} p_{33}

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

    Now, can you find 6 points with this property?
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