Difference between revisions of "1982 USAMO Problems"
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<math>(*) </math> <math>\frac{S_{m+n}}{m+n}=\frac{S_m}{m}\frac{S_n}{n}</math> | <math>(*) </math> <math>\frac{S_{m+n}}{m+n}=\frac{S_m}{m}\frac{S_n}{n}</math> | ||
| − | for <math>(m,n)=(2,3),(3,2),(2,5)</math>, or <math>(5,2)</math>. Determine | + | 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>x+y+z=0</math>. |
[[1982 USAMO Problems/Problem 2 | Solution]] | [[1982 USAMO Problems/Problem 2 | Solution]] | ||
Revision as of 10:33, 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
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
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 | ||
