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Bosnia and Herzegovina JBMO TST 2014 Problem 4

Source: Bosnia and Herzegovina Junior Balkan Mathematical Olympiad TST 2014

September 16, 2018

combinatoricsiterationsInvariants

Problem Statement

It is given

5

numbers

1

3

5

7

9

. We get the new

5

numbers such that we take arbitrary

4

numbers(out of current

5

numbers)

a

b

c

and

d

and replace them with

\frac{a+b+c-d}{2}

\frac{a+b-c+d}{2}

\frac{a-b+c+d}{2}

and

\frac{-a+b+c+d}{2}

. Can we, with repeated iterations, get numbers:

a)

0

2

4

6

and

8

b)

3

4

5

6

and

7