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
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 |