IMO Shortlist 1998 problem C2
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Avg: 6,0 Let
be an integer greater than 2. A positive integer is said to be attainable if it is 1 or can be obtained from 1 by a sequence of operations with the following properties:
1.) The first operation is either addition or multiplication.
2.) Thereafter, additions and multiplications are used alternately.
3.) In each addition, one can choose independently whether to add 2 or
4.) In each multiplication, one can choose independently whether to multiply by 2 or by
.
A positive integer which cannot be so obtained is said to be unattainable.
a.) Prove that if
, there are infinitely many unattainable positive integers.
b.) Prove that if
, all positive integers except 7 are attainable.
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
1.) The first operation is either addition or multiplication.
2.) Thereafter, additions and multiplications are used alternately.
3.) In each addition, one can choose independently whether to add 2 or
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
4.) In each multiplication, one can choose independently whether to multiply by 2 or by
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
A positive integer which cannot be so obtained is said to be unattainable.
a.) Prove that if
![n\geq 9](/media/m/6/0/c/60c562f2c742670b9285a9107fdc5f62.png)
b.) Prove that if
![n=3](/media/m/6/e/8/6e8cc663572ec564892ed13a28debcb1.png)
Izvor: Međunarodna matematička olimpijada, shortlist 1998