Difference between revisions of "1982 USAMO Problems"
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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]] |
Revision as of 17:23, 28 December 2015
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
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.