Difference between revisions of "2006 USAMO Problems/Problem 1"
Ragnarok23 (talk | contribs) |
Ragnarok23 (talk | contribs) |
||
| Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
| + | Let '''<math>p</math>''' be a prime number and let '''<math>s</math>''' be an integer with '''<math>0 < s < p</math>.''' Prove that there exists integers '''<math>m</math>''' and '''<math>n</math>''' with '''<math>0 < m < n < p</math>''' and | ||
| + | |||
| + | '''<center>{<math>\frac{sm}{p}</math>} < {<math>\frac{sn}{p}</math>}< <math>{\frac{s}{p}}</math></center>''' | ||
| + | |||
| + | if and only if '''<math>s</math>''' is not a divisor of '''<math>p-1</math>.''' | ||
| + | |||
| + | Note: For <math>x</math> a real number, let <math>\lfloor x \rfloor</math> denote the greatest integer less than or equal to <math>x</math>, and let <math>\{x\} = x - \lfloor x \rfloor</math> denote the fractional part of x. | ||
== Solution == | == Solution == | ||
== See Also == | == See Also == | ||
*[[2006 USAMO Problems]] | *[[2006 USAMO Problems]] | ||
Revision as of 12:02, 12 July 2006
Problem
Let 
be a prime number and let 
be an integer with 
. Prove that there exists integers 
and 
with 
and
} < {
}< 
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.
