2023 IMO Problems/Problem 5
Problem
Let be a positive integer. A Japanese triangle consists of
circles arranged in an equilateral triangular shape such that for each
,
,
,
, the
row contains exactly
circles, exactly one of which is coloured red. A ninja path in a Japanese triangle is a sequence of
circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with
, along with a ninja path in that triangle containing two red circles.
[asy]
unitsize(7mm);
path q=(3,-3sqrt(3))--(-3,-3sqrt(3));
filldraw(shift(0*dir(240)+0*dir(0))*scale(0.5)*unitcircle,lightred);
filldraw(shift(1*dir(240)+0*dir(0))*scale(0.5)*unitcircle,lightred);
draw(shift(1*dir(240)+1*dir(0))*scale(0.5)*unitcircle);
draw(shift(2*dir(240)+0*dir(0))*scale(0.5)*unitcircle);
draw(shift(2*dir(240)+1*dir(0))*scale(0.5)*unitcircle);
filldraw(shift(2*dir(240)+2*dir(0))*scale(0.5)*unitcircle,lightred);
draw(shift(3*dir(240)+0*dir(0))*scale(0.5)*unitcircle);
draw(shift(3*dir(240)+1*dir(0))*scale(0.5)*unitcircle);
filldraw(shift(3*dir(240)+2*dir(0))*scale(0.5)*unitcircle,lightred);
draw(shift(3*dir(240)+3*dir(0))*scale(0.5)*unitcircle);
draw(shift(4*dir(240)+0*dir(0))*scale(0.5)*unitcircle);
draw(shift(4*dir(240)+1*dir(0))*scale(0.5)*unitcircle);
draw(shift(4*dir(240)+2*dir(0))*scale(0.5)*unitcircle);
filldraw(shift(4*dir(240)+3*dir(0))*scale(0.5)*unitcircle,lightred);
draw(shift(4*dir(240)+4*dir(0))*scale(0.5)*unitcircle);
filldraw(shift(5*dir(240)+0*dir(0))*scale(0.5)*unitcircle,lightred);
draw(shift(5*dir(240)+1*dir(0))*scale(0.5)*unitcircle);
draw(shift(5*dir(240)+2*dir(0))*scale(0.5)*unitcircle);
draw(shift(5*dir(240)+3*dir(0))*scale(0.5)*unitcircle);
draw(shift(5*dir(240)+4*dir(0))*scale(0.5)*unitcircle);
draw(shift(5*dir(240)+5*dir(0))*scale(0.5)*unitcircle);
draw((0,0)--(1/2,-sqrt(3)/2)--(0,-sqrt(3))--(1/2,-3sqrt(3)/2)--(0,-2sqrt(3))--(-1/2, -5sqrt(3)/2),linewidth(1.5));
draw(q,Arrows(TeXHead, 1));
label("", q, S);
[/asy]
Solution
https://www.youtube.com/watch?v=jZNIpapyGJQ [Video contains solutions to all day 2 problems]