Difference between revisions of "1979 USAMO Problems"
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Revision as of 19:07, 3 July 2013
Problems from the 1979 USAMO.
Problem 1
Determine all non-negative integral solutions if any, apart from permutations, of the Diophantine Equation
.
Problem 2
is the north pole.
and
are points on a great circle through
equidistant from
.
is a point on the equator. Show that the great circle through
and
bisects the angle
in the spherical triangle
(a spherical triangle has great circle arcs as sides).
Problem 3
is an arbitrary sequence of positive integers. A member of the sequence is picked at
random. Its value is
. Another member is picked at random, independently of the first. Its value is
. Then a third value,
. Show that the probability that
is divisible by
is at least
.
Problem 4
lies between the rays
and
. Find
on
and
on
collinear with
so that $\frac{1}{PQ}\plus{} \frac{1}{PR}$ (Error compiling LaTeX. Unknown error_msg) is as large as possible.
Problem 5
Let be distinct subsets of
with
. Prove that
for some pair
.
See Also
1979 USAMO (Problems • Resources) | ||
Preceded by 1978 USAMO |
Followed by 1980 USAMO | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.