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. | |
[[1982 USAMO Problems/Problem 1 | Solution]] | [[1982 USAMO Problems/Problem 1 | Solution]] | ||
Revision as of 11:31, 6 March 2013
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
Show that if 
are positive integers such that $\frac{\left(x^{m+n} + y^{m+n} + z^{m+n}\right)}{m+n}=\left(\frac{x^m + y^m + z^m}{m}\right) \left(\dfrac{x^n + y^n + z^n}{n}}\right)$ (Error compiling LaTeX. Unknown error_msg) for all real 
with sum 
, then 
or 
.
Problem 3
is a point inside the equilateral triangle 
. 
is a point inside 
. Show that 
Problem 4
Show that there is a positive integer 
such that, for every positive integer 
, 
is composite.
Problem 5
is the center of a sphere 
. Points 
are inside , 
is perpendicular to 
and 
, and there are two spheres through 
, and 
which touch . Show that the sum of their radii equals 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 | ||
