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One of USAMO's hardest NT problems

Source: USAMO 1999 Problem 3

October 3, 2005

modular arithmeticgeometryalgebrapolynomialfractional partUSAMO

Problem Statement

Let

p > 2

be a prime and let

a,b,c,d

be integers not divisible by

p

, such that

\left\{ \dfrac{ra}{p} \right\} + \left\{ \dfrac{rb}{p} \right\} + \left\{ \dfrac{rc}{p} \right\} + \left\{ \dfrac{rd}{p} \right\} = 2

for any integer

r

not divisible by

p

. Prove that at least two of the numbers

a+b

,

a+c

,

a+d

,

b+c

,

b+d

,

c+d

are divisible by

p

. (Note:

\{x\} = x - \lfloor x \rfloor

denotes the fractional part of

x

.)

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