2006 USAMO Problems
Revision as of 20:17, 4 July 2006 by Ragnarok23 (talk | contribs)
Day 1
Problem 1
Let be a prime number and let
be an integer with
. Prove that there exists integers
and
with
and
![$\frac{sm}{p}$](http://latex.artofproblemsolving.com/3/e/d/3ed232b94ed9ff8da7fb5d9f8b5593d9c1e6d6f1.png)
![$\frac{sn}{p}$](http://latex.artofproblemsolving.com/9/6/8/968de9fced2c629642619fd43e91865a80b56c6f.png)
![${\frac{s}{p}}$](http://latex.artofproblemsolving.com/8/b/0/8b019b237f57073f7b250eaede1699dfcfe93eff.png)
if and only if is not a divisor of
.
Note: For a real number, let
denote the greatest integer less than or equal to
, and let
denote the fractional part of x