Question

Assume that 2(r+b) points, with rbe N, are drawn on a circle, equally spaced (thus, the rays from the center of the circle to any two adjacent points are at an angle of (r + b) radians). Assume also that 2r of the points are colored red and that the other 2b are colored blue. (You do not get to choose which points get which color.) Must there be at least one rotation of the circle through its center that puts r red points and blue points on each side of the vertical line through the center? Provide either a proof that there is such a rotation, or a counterexample having no such rotation.

Answer 1

Assume that 2(r+b) points, with rbe N, are drawn on a circle, equally spaced (thus, the rays from the center of the circle

Sol) Given To assume 2 (rab) points with r, BEN are drawn an a circle Name 2 (rtb) Porints as R,, Rz, Rze..... Rar B,,B2,B2b with Ri are red in colour * Bi is blue in colour * we have to arrange there on the circle R., R2, Rz are in anti-clock wire divection B.; B2, B3 are in clock wise direction. R, is on ix-axis. Let us Consider lined, at an angle (2x-1) 2 (4ab) with positive ta ares

* line bisects angle Ro Roth 2 (orth) 28th] in 0 : (2011) 2(r+b) To = -(2021) th 2 crtb) ..da (2b+1) 2 ertb) Here fae, The angle submitted by Bro with N-axis bith auti-clock wise * The angle bisector of augle la, Bb will make an angle of thg 7 1 1rty) with no anis clock wise. (261) is angle a made byd, with granis clock wise These are exactly half red pants on each side of the bisecting angle RroRral Exactly half blue points on each side bisecting angle Blobbal

* he makes angle o with x-axis in anti-clock wise and clock wise. *d makes with xnaxis in same direction ..so litida are on same lines. which are exactly half red and half blue.