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 11:02, 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.