1978 USAMO Problems: Difference between revisions
No edit summary |
|||
| Line 54: | Line 54: | ||
== See Also == | == See Also == | ||
{{USAMO box|year=1978|before=[[1977 USAMO]]|after=[[1979 USAMO]]}} | {{USAMO box|year=1978|before=[[1977 USAMO]]|after=[[1979 USAMO]]}} | ||
{{MAA Notice}} | |||
Latest revision as of 18:06, 3 July 2013
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
An integer 
will be called good if we can write
,
where 
are positive integers (not necessarily distinct) satisfying
.
Given the information that the integers 33 through 73 are good, prove that every integer 
is good.
Problem 4
(a) Prove that if the six dihedral (i.e. angles between pairs of faces) of a given tetrahedron are congruent, then the tetrahedron is regular.
(b) Is a tetrahedron necessarily regular if five dihedral angles are congruent?
Problem 5
Nine mathematicians meet at an international conference and discover that among any three of them, at least two speak a common language. If each of the mathematicians speak at most three languages, prove that there are at least three of the mathematicians who can speak the same 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 | ||
These problems are copyrighted © by the Mathematical Association of America. 
