Question

MTRP class 10 :

A plane is coloured with 3 colours red , green and blue . Prove that there exists 2 points with the same colour in the plane with a distance d for all d

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Answer 1

THOUGH OUR ASSUMPTION:-


✔️✔️✔️In the attachment above...

➖Let o = red colour

➖Let the radius be √3d.

✔️✔️Now;

If we notice! there are 2 possibilities:-

⭕️Case 1

➖All the points on circle are red.


✴️➖➖or ➖➖✴️

⭕️Case 2


➖There's a point that is either green or blue.

Now

✔️✔️In case1 :-

⭕️The endpoints of chord of length d will be of the colour red.....

⭕️Let there be green point R.

⭕️we have to now find two points A and B that form two equil. ΔLes OAB and PAB of side d.

⭕️ points are intersections of the two circle O(d) and R(d) where they are centered at O and R and both with radius d.

⭕️One of A and B may be either red or green.

⭕️So .....Together with O or R we would have a monochromatic pair*

✔️✔️monochromatic pair* :- connecting v to each vertex of V. 


⭕️Else they both are blue and at distance d and so form a monochromatic pair!!!!!!!!!!!!!!



$\huge\bf{\boxed {\red{\mathfrak{thank \: you :)}}}}$

Answer 2

check the attachment
Their was a mistake in last answer