Show circle with points coloured red and blue must have monochromatic red equilateral triangle
Clash Royale CLAN TAG#URR8PPP
up vote
4
down vote
favorite
Colour each point on a circle of radius $frac12$ 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
edited Dec 9 at 22:16
Jean Marie
28.7k41849
asked Dec 9 at 21:31
Prasiortle
1075
add a comment |
up vote
4
down vote
favorite
Colour each point on a circle of radius $frac12$ 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
edited Dec 9 at 22:16
Jean Marie
28.7k41849
asked Dec 9 at 21:31
Prasiortle
1075
1
What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
– John Hughes
Dec 9 at 21:41
Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
– Prasiortle
Dec 9 at 21:53
add a comment |
up vote
4
down vote
favorite
up vote
4
down vote
favorite
Colour each point on a circle of radius $frac12$ 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
edited Dec 9 at 22:16
Jean Marie
28.7k41849
asked Dec 9 at 21:31
Prasiortle
1075
Colour each point on a circle of radius $frac12$ 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
edited Dec 9 at 22:16
Jean Marie
28.7k41849
asked Dec 9 at 21:31
Prasiortle
1075
edited Dec 9 at 22:16
Jean Marie
28.7k41849
asked Dec 9 at 21:31
Prasiortle
1075
edited Dec 9 at 22:16
Jean Marie
28.7k41849
edited Dec 9 at 22:16
Jean Marie
28.7k41849
edited Dec 9 at 22:16
Jean Marie
28.7k41849
28.7k41849
asked Dec 9 at 21:31
Prasiortle
1075
asked Dec 9 at 21:31
Prasiortle
1075
asked Dec 9 at 21:31
Prasiortle
1075
1075
1
What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
– John Hughes
Dec 9 at 21:41
Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
– Prasiortle
Dec 9 at 21:53
add a comment |
1
What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
– John Hughes
Dec 9 at 21:41
Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
– Prasiortle
Dec 9 at 21:53
1
1
What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
– John Hughes
Dec 9 at 21:41
What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
– John Hughes
Dec 9 at 21:41
Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
– Prasiortle
Dec 9 at 21:53
Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
– Prasiortle
Dec 9 at 21:53
add a comment |
1 Answer
1
active
oldest
votes
up vote
10
down vote
Make all the red points that are a distance exactly $frac 2pi3$ 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.
edited Dec 10 at 5:22
Acccumulation
6,6512616
answered Dec 9 at 21:41
Ross Millikan
290k23196369
add a comment |
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
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
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
up vote
10
down vote
Make all the red points that are a distance exactly $frac 2pi3$ 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.
edited Dec 10 at 5:22
Acccumulation
6,6512616
answered Dec 9 at 21:41
Ross Millikan
290k23196369
add a comment |
up vote
10
down vote
Make all the red points that are a distance exactly $frac 2pi3$ 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.
edited Dec 10 at 5:22
Acccumulation
6,6512616
answered Dec 9 at 21:41
Ross Millikan
290k23196369
add a comment |
up vote
10
down vote
up vote
10
down vote
Make all the red points that are a distance exactly $frac 2pi3$ 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.
edited Dec 10 at 5:22
Acccumulation
6,6512616
answered Dec 9 at 21:41
Ross Millikan
290k23196369
Make all the red points that are a distance exactly $frac 2pi3$ 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.
edited Dec 10 at 5:22
Acccumulation
6,6512616
answered Dec 9 at 21:41
Ross Millikan
290k23196369
edited Dec 10 at 5:22
Acccumulation
6,6512616
edited Dec 10 at 5:22
Acccumulation
6,6512616
edited Dec 10 at 5:22
Acccumulation
6,6512616
6,6512616
answered Dec 9 at 21:41
Ross Millikan
290k23196369
answered Dec 9 at 21:41
Ross Millikan
290k23196369
answered Dec 9 at 21:41
Ross Millikan
290k23196369
290k23196369
add a comment |
add a comment |
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
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
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
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
1
What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
– John Hughes
Dec 9 at 21:41
Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
– Prasiortle
Dec 9 at 21:53