1978 USAMO Problems
Problems from the 1978 USAMO.
Given that are real numbers such that
Determine the maximum value of .
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 .
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.
(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?
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.
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