1978 USAMO Problems/Problem 2
Problem
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] 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("$"+lbl[i]+"'$", P, Q); label("$"+lbl[i]+"$",D2+rotate(theta)*(r*P), rotate(theta)*Q); }[/asy]](http://latex.artofproblemsolving.com/6/f/a/6fa08871047fa32076357c574b19dfeb124210b6.png)
Solution
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See Also
| 1978 USAMO (Problems • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
