Show circle with points coloured red and blue must have monochromatic red equilateral triangle












4












$begingroup$


Colour each point on a circle of radius $frac{1}{2}$ red or blue, such that the region of blue points has length $1$. Prove that we can inscribe an equilateral triangle in the circle such that all three vertices are red.



I think the Pigeonhole Principle will be involved, but don't quite see how to apply it. The length condition also seems a bit hard to work with, so any hints or suggestions would be much appreciated.






combinatorics discrete-mathematics







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
    $endgroup$
    – John Hughes
    Dec 9 '18 at 21:41










  • $begingroup$
    Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
    $endgroup$
    – Prasiortle
    Dec 9 '18 at 21:53
















4












$begingroup$


Colour each point on a circle of radius $frac{1}{2}$ red or blue, such that the region of blue points has length $1$. Prove that we can inscribe an equilateral triangle in the circle such that all three vertices are red.



I think the Pigeonhole Principle will be involved, but don't quite see how to apply it. The length condition also seems a bit hard to work with, so any hints or suggestions would be much appreciated.






combinatorics discrete-mathematics







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
    $endgroup$
    – John Hughes
    Dec 9 '18 at 21:41










  • $begingroup$
    Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
    $endgroup$
    – Prasiortle
    Dec 9 '18 at 21:53














4












4








4


1



$begingroup$


Colour each point on a circle of radius $frac{1}{2}$ red or blue, such that the region of blue points has length $1$. Prove that we can inscribe an equilateral triangle in the circle such that all three vertices are red.



I think the Pigeonhole Principle will be involved, but don't quite see how to apply it. The length condition also seems a bit hard to work with, so any hints or suggestions would be much appreciated.






combinatorics discrete-mathematics







share|cite|improve this question











$endgroup$




Colour each point on a circle of radius $frac{1}{2}$ red or blue, such that the region of blue points has length $1$. Prove that we can inscribe an equilateral triangle in the circle such that all three vertices are red.



I think the Pigeonhole Principle will be involved, but don't quite see how to apply it. The length condition also seems a bit hard to work with, so any hints or suggestions would be much appreciated.






combinatorics discrete-mathematics




combinatorics discrete-mathematics






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 9 '18 at 22:16









Jean Marie

29.6k42050




29.6k42050










asked Dec 9 '18 at 21:31









PrasiortlePrasiortle

1525




1525








  • 1




    $begingroup$
    What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
    $endgroup$
    – John Hughes
    Dec 9 '18 at 21:41










  • $begingroup$
    Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
    $endgroup$
    – Prasiortle
    Dec 9 '18 at 21:53














  • 1




    $begingroup$
    What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
    $endgroup$
    – John Hughes
    Dec 9 '18 at 21:41










  • $begingroup$
    Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
    $endgroup$
    – Prasiortle
    Dec 9 '18 at 21:53








1




1




$begingroup$
What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
$endgroup$
– John Hughes
Dec 9 '18 at 21:41




$begingroup$
What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
$endgroup$
– John Hughes
Dec 9 '18 at 21:41












$begingroup$
Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
$endgroup$
– Prasiortle
Dec 9 '18 at 21:53




$begingroup$
Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
$endgroup$
– Prasiortle
Dec 9 '18 at 21:53










1 Answer
1






active

oldest

votes


















10












$begingroup$

Make all the red points that are a distance exactly $frac {2pi}3$ away from a blue point blue. The measure of the blue points is now no more than $3$, but the circumference of the circle is $pi$. There is at least $pi-3$ of the circle still colored red and any of the red points is on an all red equilateral triangle.






share|cite|improve this answer











$endgroup$













    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "69"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3033047%2fshow-circle-with-points-coloured-red-and-blue-must-have-monochromatic-red-equila%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    10












    $begingroup$

    Make all the red points that are a distance exactly $frac {2pi}3$ away from a blue point blue. The measure of the blue points is now no more than $3$, but the circumference of the circle is $pi$. There is at least $pi-3$ of the circle still colored red and any of the red points is on an all red equilateral triangle.






    share|cite|improve this answer











    $endgroup$


















      10












      $begingroup$

      Make all the red points that are a distance exactly $frac {2pi}3$ away from a blue point blue. The measure of the blue points is now no more than $3$, but the circumference of the circle is $pi$. There is at least $pi-3$ of the circle still colored red and any of the red points is on an all red equilateral triangle.






      share|cite|improve this answer











      $endgroup$
















        10












        10








        10





        $begingroup$

        Make all the red points that are a distance exactly $frac {2pi}3$ away from a blue point blue. The measure of the blue points is now no more than $3$, but the circumference of the circle is $pi$. There is at least $pi-3$ of the circle still colored red and any of the red points is on an all red equilateral triangle.






        share|cite|improve this answer











        $endgroup$



        Make all the red points that are a distance exactly $frac {2pi}3$ away from a blue point blue. The measure of the blue points is now no more than $3$, but the circumference of the circle is $pi$. There is at least $pi-3$ of the circle still colored red and any of the red points is on an all red equilateral triangle.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 10 '18 at 5:22









        Acccumulation

        6,9042618




        6,9042618










        answered Dec 9 '18 at 21:41









        Ross MillikanRoss Millikan

        295k23198371




        295k23198371






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3033047%2fshow-circle-with-points-coloured-red-and-blue-must-have-monochromatic-red-equila%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Probability when a professor distributes a quiz and homework assignment to a class of n students.

            Image
            0 0 $begingroup$ Need help with this problem. Suppose our lazy professor collects a quiz and a homework assignment from a class of n students one day, then distributes both the quizzes and the homework assignments back to the class in a random fashion for grading. Each student receives one quiz and one homework assignment to grade. (a) What is the probability that every student receives someone else's quiz to grade, and someone else's homework to grade? (b) What is the probability that no student receives both their own quiz and their own homework assignment to grade? In this case, some students may receive their own quiz, and others may receive their own homework assignment. (c) Compute the limiting probability as n approaches infinity in each case. ...
            Read more

            Aardman Animations

            Image
            Aardman Animations, Ltd. Aardman Animations Tipo privada Atividade Animação em stop-motion , Animação em CGI Fundação 1972 Fundador(es) Peter Lord David Sproxton Sede Bristol, Inglaterra  Reino Unido Proprietário(s) Dreamworks Pessoas-chave Peter Lord David Sproxton Nick Park Divisões Aardman Features Aardman Digital Aardman Commercials Aardman Broadcast Aardman International Aardman Rights Aardman Effects Aardman 3-D Systems Aardman Nathan Love AardBoiled Website oficial aardman.com ...
            Read more

            Are they similar matrix

            Image
            0 $begingroup$ Do $begin{bmatrix} 0&i&0\0&0&1\0&0&0 end{bmatrix} $ and $begin{bmatrix} 0&0&0\-i&0&0\0&1&0 end{bmatrix} $ are similar.Is this True/false Clearly both are nilpotent and one is conjucate transpose of other but how to know if they are similar.i'm stuck. Please help me linear-algebra vector-spaces eigenvalues-eigenvectors generalizedeigenvector share | cite | improve this question edited Dec 10 '18 at 6:38 Vasanth Kris asked Dec 10 '18 at 6:19 ...
            Read more