1978 USAMO #2
Source:
August 16, 2011
AMCUSA(J)MOUSAMOgeometrygeometric transformationhomothetyratio
Problem Statement
and are square maps of the same region, drawn to different scales and superimposed as shown in the figure. Prove that there is only one point on the small map that lies directly over point of the large map such that and each represent the same place of the country. Also, give a Euclidean construction (straight edge and compass) for .[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
real theta = -100, r = 0.3; pair D2 = (0.3,0.76);
string[] lbl = {'A', 'B', 'C', 'D'}; draw(unitsquare); draw(shift(D2)*rotate(theta)*scale(r)*unitsquare);
for(int i = 0; i < lbl.length; ++i) {
pair Q = dir(135-90*i), P = (.5,.5)+Q/2^.5;
label("", P, Q);
label("",D2+rotate(theta)*(r*P), rotate(theta)*Q);
}[/asy]