Difference between revisions of "1979 USAMO Problems"
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==Problem 3== | ==Problem 3== | ||
<math>a_1, a_2, \ldots, a_n</math> is an arbitrary sequence of positive integers. A member of the sequence is picked at | <math>a_1, a_2, \ldots, a_n</math> is an arbitrary sequence of positive integers. A member of the sequence is picked at | ||
− | random. Its value is <math>a</math>. Another member is picked at random, independently of the first. Its value is <math>b</math>. Then a third value, <math>c</math>. Show that the probability that <math>a | + | random. Its value is <math>a</math>. Another member is picked at random, independently of the first. Its value is <math>b</math>. Then a third value, <math>c</math>. Show that the probability that <math>a + b + c</math> is divisible by <math>3</math> is at least <math>\frac14</math>. |
[[1979 USAMO Problems/Problem 3 | Solution]] | [[1979 USAMO Problems/Problem 3 | Solution]] | ||
==Problem 4== | ==Problem 4== | ||
− | <math>P</math> lies between the rays <math>OA</math> and <math>OB</math>. Find <math>Q</math> on <math>OA</math> and <math>R</math> on <math>OB</math> collinear with <math>P</math> so that <math>\frac{1}{PQ} | + | <math>P</math> lies between the rays <math>OA</math> and <math>OB</math>. Find <math>Q</math> on <math>OA</math> and <math>R</math> on <math>OB</math> collinear with <math>P</math> so that <math>\frac{1}{PQ} + \frac{1}{PR}</math> is as large as possible. |
[[1979 USAMO Problems/Problem 4 | Solution]] | [[1979 USAMO Problems/Problem 4 | Solution]] | ||
==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 =| | + | 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+1}|=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]] | ||
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== See Also == | == See Also == | ||
{{USAMO box|year=1979|before=[[1978 USAMO]]|after=[[1980 USAMO]]}} | {{USAMO box|year=1979|before=[[1978 USAMO]]|after=[[1980 USAMO]]}} | ||
+ | {{MAA Notice}} |
Latest revision as of 13:04, 24 December 2015
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
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.