sussy baka stop intersecting in my lattice points
Source: 2022 AMC 10A #25
November 11, 2022
AMCAMC 102022 AMC 10a2022 AMClattice pointscoordinate geometrysquare
Problem Statement
Let , , and be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the x-axis. The left edge of and the right edge of are on the -axis, and contains as many lattice points as does . The top two vertices of are in , and contains of the lattice points contained in . See the figure (not drawn to scale).[asy]
//kaaaaaaaaaante314
size(8cm);
import olympiad;
label(scale(.8)*"", (0,60), N);
label(scale(.8)*"", (60,0), E);
filldraw((0,0)--(55,0)--(55,55)--(0,55)--cycle, yellow+orange+white+white);
label(scale(1.3)*"", (55/2,55/2));
filldraw((0,0)--(0,28)--(-28,28)--(-28,0)--cycle, green+white+white);
label(scale(1.3)*"",(-14,14));
filldraw((-10,0)--(15,0)--(15,25)--(-10,25)--cycle, red+white+white);
label(scale(1.3)*"",(3.5,25/2));
draw((0,-10)--(0,60),EndArrow(TeXHead));
draw((-34,0)--(60,0),EndArrow(TeXHead));[/asy]The fraction of lattice points in that are in is 27 times the fraction of lattice points in that are in . What is the minimum possible value of the edge length of plus the edge length of plus the edge length of ?