MathDB

Set and Function

Source: China TST 2003

June 29, 2006

functioncombinatorics unsolvedcombinatorics

Problem Statement

Let

A= \{a_1,a_2, \cdots, a_n \}

and

B=\{b_1,b_2 \cdots, b_n \}

be two positive integer sets and

|A \cap B|=1

.

C= \{ \text{all the 2-element subsets of A} \} \cup \{ \text{all the 2-element subsets of B} \}

. Function

f: A \cup B \to \{ 0, 1, 2, \cdots 2 C_n^2 \}

is injective. For any

\{x,y\} \in C

, denote

|f(x)-f(y)|

as the \textsl{mark} of

\{x,y\}

. If

n \geq 6

, prove that at least two elements in

C

have the same \textsl{mark}.

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