MathDB

1

Part of 2006 USAMO

Problems(1)

USAMO Inequality chain for getting started on the contest

Source: USAMO 2006, Problem 1, proposed by Kiran Kedlaya

4/20/2006
Let pp be a prime number and let ss be an integer with 0<s<p.0 < s < p. Prove that there exist integers mm and nn with 0<m<n<p0 < m < n < p and {smp}<{snp}<sp \left \{\frac{sm}{p} \right\} < \left \{\frac{sn}{p} \right \} < \frac{s}{p} if and only if ss is not a divisor of p−1p-1. Note: For xx a real number, let ⌊x⌋\lfloor x \rfloor denote the greatest integer less than or equal to xx, and let {x}=x−⌊x⌋\{x\} = x - \lfloor x \rfloor denote the fractional part of x.
USA(J)MOUSAMOinequalitiesfloor functionmodular arithmeticpigeonhole principleHi