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Swiss tst

Source: Swiss IMO Team Selection Test 1997

May 7, 2017

algebraSequence

Problem Statement

1. A finite sequence of integers

a_0,a_1,...,a_n

is called quadratic if

|a_k -a_{k-1}| = k^2

for

n\geq k\geq1

. (a) Prove that for any two integers

b

and

c

, there exist a natural number

n

and a quadratic sequence with

a_0 = b

and

a_n =c

. (b) Find the smallest natural number

n

for which there exists a quadratic sequence with

a_0 = 0

and

a_n = 1997

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