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IMO ShortList 1999, combinatorics problem 7

Source: IMO ShortList 1999, combinatorics problem 7

November 14, 2004

countingSubsetsSet systemsDivisibilitycombinatoricsIMO Shortlistcombinatorial inequality

Problem Statement

Let

p >3

be a prime number. For each nonempty subset

T

of

\{0,1,2,3, \ldots , p-1\}

, let

E(T)

be the set of all

(p-1)

-tuples

(x_1, \ldots ,x_{p-1} )

, where each

x_i \in T

and

x_1+2x_2+ \ldots + (p-1)x_{p-1}

is divisible by

p

and let

|E(T)|

denote the number of elements in

E(T)

. Prove that

|E(\{0,1,3\})| \geq |E(\{0,1,2\})|

with equality if and only if

p = 5

.

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