Difference between revisions of "1979 USAMO Problems"
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==Problem 5== | ==Problem 5== | ||
− | Let <math>A_1,A_2,...,A_{n+1}</math> be distinct subsets of <math>[n]</math> with <math>|A_1|=|A_2|=\cdots =|A_n|=3</math>. Prove that <math>|A_i\cap A_j|=1</math> for some pair <math>\{i,j\}</math>. | + | Let <math>A_1,A_2,...,A_{n+1}</math> be distinct subsets of <math>[n]</math> with <math>|A_1|=|A_2|=\cdots =|A_n|=3</math>. Prove that <math>|A_i\cap A_j|=1</math> for some pair <math>\{i,j\}</math>. Note that <math>[n] = {1, 2, 3, ..., n}</math>, or, alternatively, <math>{x: 1 \le x \le n}</math>. |
[[1979 USAMO Problems/Problem 5 | Solution]] | [[1979 USAMO Problems/Problem 5 | Solution]] |
Revision as of 00:22, 19 April 2014
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
. Note that
, or, alternatively,
.
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.