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exist m,n such a_m = a_n and a_{m+1} = b_{n+1} in an English alphabet

Source: Singapore Open Math Olympiad 2003 2nd Round p1 SMO

April 2, 2020

Wordsalphabetcombinatorics

Problem Statement

A sequence

(a_1,a_2,...,a_{675})

is given so that each term is an alphabet in the English language (no distinction is made between lower and upper case letters). It is known that in the sequence

a

is never followed by

b

and

c

is never followed by

d

. Show that there are integers

m

and

n

with

1 \le m < n \le 674

such that

a_m = a_n

and

a_{m+1} = a_{n+1}

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