Difference between revisions of "1978 USAMO Problems"
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==Problem 2== | ==Problem 2== | ||
| − | + | <math>ABCD</math> and <math>A'B'C'D'</math> 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 <math>O</math> on the small map that lies directly over point <math>O'</math> of the large map such that <math>O</math> and <math>O'</math> each represent the same place of the country. Also, give a Euclidean construction (straight edge and compass) for <math>O</math>. | |
| + | |||
| + | <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> | ||
[[1978 USAMO Problems/Problem 2 | Solution]] | [[1978 USAMO Problems/Problem 2 | Solution]] | ||
Revision as of 14:25, 17 September 2012
Problems from the 1978 USAMO.
Problem 1
Given that 
are real numbers such that
,
.
Determine the maximum value of 
.
Problem 2
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)
Problem 3
You are told that all integers from 
to 
inclusive can be expressed as a sum of positive integers whose reciprocals sum to 1. Show that the same is true for all integers greater than .
Problem 4
Show that if the angle between each pair of faces of a tetrahedron is equal, then the tetrahedron is regular. Does a tetrahedron have to be regular if five of the angles are equal?
Problem 5
There are 9 delegates at a conference, each speaking at most three languages. Given any three delegates, at least 2 speak a common language. Show that there are three delegates with a common language.
See Also
| 1978 USAMO (Problems • Resources) | ||
| Preceded by 1977 USAMO |
Followed by 1979 USAMO | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
