Difference between revisions of "1982 USAMO Problems"
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==Problem 1== | ==Problem 1== | ||
| − | In a party with <math>1982</math> persons, among any group of four there is at least one person who knows each of the other three. What is the minimum number of people in the party who know everyone else | + | In a party with <math>1982</math> persons, among any group of four there is at least one person who knows each of the other three. What is the minimum number of people in the party who know everyone else? |
[[1982 USAMO Problems/Problem 1 | Solution]] | [[1982 USAMO Problems/Problem 1 | Solution]] | ||
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for <math>(m,n)=(2,3),(3,2),(2,5)</math>, or <math>(5,2)</math>. Determine ''all'' other pairs of integers <math>(m,n)</math> if any, so that <math>(*)</math> holds for all real numbers <math>x,y,z</math> such that <math>S_1=0</math>. | for <math>(m,n)=(2,3),(3,2),(2,5)</math>, or <math>(5,2)</math>. Determine ''all'' other pairs of integers <math>(m,n)</math> if any, so that <math>(*)</math> holds for all real numbers <math>x,y,z</math> such that <math>S_1=0</math>. | ||
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[[1982 USAMO Problems/Problem 2 | Solution]] | [[1982 USAMO Problems/Problem 2 | Solution]] | ||
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==Problem 4== | ==Problem 4== | ||
| − | Prove that there exists a positive integer <math>k</math> such that <math>k\cdot2^n+1</math> is composite for every integer <math>n</math>. | + | Prove that there exists a positive integer <math>k</math> such that <math>k\cdot2^n+1</math> is composite for every positive integer <math>n</math>. |
[[1982 USAMO Problems/Problem 4 | Solution]] | [[1982 USAMO Problems/Problem 4 | Solution]] | ||
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== See Also == | == See Also == | ||
{{USAMO box|year=1982|before=[[1981 USAMO]]|after=[[1983 USAMO]]}} | {{USAMO box|year=1982|before=[[1981 USAMO]]|after=[[1983 USAMO]]}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 21:36, 19 March 2020
Problems from the 1982 USAMO.
Problem 1
In a party with 
persons, among any group of four there is at least one person who knows each of the other three. What is the minimum number of people in the party who know everyone else?
Problem 2
Let 
with 
real. It is known that if 
,
for 
, or 
. Determine all other pairs of integers 
if any, so that holds for all real numbers such that .
Problem 3
If a point 
is in the interior of an equilateral triangle 
and point 
is in the interior of 
, prove that
,
where the isoperimetric quotient of a figure 
is defined by
![$I.Q.(F) = \frac{\text{Area (F)}}{\text{[Perimeter (F)]}^2}$](http://latex.artofproblemsolving.com/5/9/6/59605bef5fa499b868bb9d25da48807658139b0a.png)
Problem 4
Prove that there exists a positive integer 
such that 
is composite for every positive integer 
.
Problem 5
, and 
are three interior points of a sphere 
such that 
and 
are perpendicular to the diameter of through 
, and so that two spheres can be constructed through , 
, and which are both tangent to . Prove that the sum of their radii is equal to the radius of .
See Also
| 1982 USAMO (Problems • Resources) | ||
| Preceded by 1981 USAMO |
Followed by 1983 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. 
