1978 USAMO Problems
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
.
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
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 |