A spherical planet has the equator of length $1$. On this planet, $N$ circular roads of length $1$ each are to be built and used for several trains each. The trains must have the same constant positive speed and never stop or collide. What is the greatest possible sum of lengths of all the trains? The trains are arcs of zero width with endpoints removed (so that if only endpoints of two arcs have coincided then it is not a collision). Solve the problem for : (a) $N=3$ (4 points) (b) $N=4$ (6 points) Alexandr Berdnikov
Problem
Source: Tournament of Towns Spring 2016
Tags: combinatorics
ythomashu
28.02.2018 20:01
3/2 for N=3 and 2 for N=4 N= 3 creates 6 intersection points with 1/4 length between. By pigeonhole, 6 is the maximum number of trains. Each train has length 1/4 because trains with greater length will occupy two or more intersection points. N=4 creates 12 intersection points with 1/6 length between. [asy][asy] unitsize(35); pair A = (1,sqrt(3)); pair B = (-1,sqrt(3)); pair C = (-2,0); pair D = (-1,-sqrt(3)); pair E = (1,-sqrt(3)); pair F = (2,0); draw(A--C,orange,MidArrow); draw(D--F,orange,MidArrow); draw(C--B,green,MidArrow); draw(F--E,green,MidArrow); draw(B--A,purple,MidArrow); draw(E--D,purple,MidArrow); draw(C--D,dashed+orange,MidArrow); draw(F--A,dashed+orange,MidArrow); draw(E--C,dashed+green,MidArrow); draw(B--F,dashed+green,MidArrow); draw(A--E,dashed+purple,MidArrow); draw(D--B,dashed+purple,MidArrow); dot(A,purple); dot(B,green); dot(C,orange); dot(D,purple); dot(E,green); dot(F,orange); [/asy][/asy] [asy][asy] unitsize(25); pair A = (1,sqrt(3)); pair B = (-1,sqrt(3)); pair C = (-2,0); pair D = (-1,-sqrt(3)); pair E = (1,-sqrt(3)); pair F = (2,0); pair G = (3,sqrt(3)); pair H = (0, 2*sqrt(3)); pair I = (-3, sqrt(3)); pair J = (-3,-sqrt(3)); pair K = (0,-2*sqrt(3)); pair L = (3, -sqrt(3)); draw(H--I,blue,MidArrow); draw(J--K,blue,MidArrow); draw(L--G,blue,MidArrow); draw(A--E,green,MidArrow); draw(K--D,green,MidArrow); draw(B--H,green,MidArrow); draw(C--A,red,MidArrow); draw(D--J,red,MidArrow); draw(G--F,red,MidArrow); draw(E--C,orange,MidArrow); draw(I--B,orange,MidArrow); draw(F--L,orange,MidArrow); draw(G--H,dashed+blue,MidArrow); draw(E--K,dashed+green,MidArrow); draw(D--B,dashed+green,MidArrow); draw(H--A,dashed+green,MidArrow); draw(A--G,dashed+red,MidArrow); draw(F--D,dashed+red,MidArrow); draw(K--L,dashed+blue,MidArrow); draw(J--C,dashed+red,MidArrow); draw(C--I,dashed+orange,MidArrow); draw(I--J,dashed+blue,MidArrow); draw(B--F,dashed+orange,MidArrow); draw(L--E,dashed+orange,MidArrow); dot(A,red); dot(B,orange); dot(C,orange); dot(D,green); dot(E,green); dot(F,red); dot(G,blue); dot(H,green); dot(I,blue); dot(J,red); dot(K,blue); dot(L,orange); [/asy][/asy]