Difference between revisions of "1982 USAMO Problems"
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==Problem 2== | ==Problem 2== | ||
| − | Show that if <math>m, n</math> are positive integers such that <math>\frac{\left(x^{m+n} + y^{m+n} + z^{m+n}\right)}{(m+n)} =\frac{ (x^m + y^m + z^m)}{\frac{m \left(x^n + y^n + z^n\right)}{n}}</math> for all real <math>x, y, z</math> with sum <math>0</math>, then <math>(m, n) = (2, 3) </math> or <math>(2, 5)</math>. | + | Show that if <math>m, n</math> are positive integers such that <math>\frac{\left(x^{m+n} + y^{m+n} + z^{m+n}\right)}{(m+n)} =\frac{ (x^m + y^m + z^m)}{\frac{m} \left(x^n + y^n + z^n\right)}{n}}</math> for all real <math>x, y, z</math> with sum <math>0</math>, then <math>(m, n) = (2, 3) </math> or <math>(2, 5)</math>. |
[[1982 USAMO Problems/Problem 2 | Solution]] | [[1982 USAMO Problems/Problem 2 | Solution]] | ||
Revision as of 11:35, 30 September 2012
Problem 1
A graph has 
points. Given any four points, there is at least one joined to the other three. What is the smallest number of points which are joined to 
points?
Problem 2
Show that if 
are positive integers such that $\frac{\left(x^{m+n} + y^{m+n} + z^{m+n}\right)}{(m+n)} =\frac{ (x^m + y^m + z^m)}{\frac{m} \left(x^n + y^n + z^n\right)}{n}}$ (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 | ||
