IMO Shortlist 1990 problem 6
Kvaliteta:
Avg: 0,0Težina:
Avg: 0,0 Given an initial integer
, two players,
and
, choose integers
,
,
,
alternately according to the following rules :
I.) Knowing
,
chooses any integer
such that
II.) Knowing
,
chooses any integer
such that
is a prime raised to a positive integer power.
Player
wins the game by choosing the number 1990; player
wins by choosing the number 1. For which
does :
a.)
have a winning strategy?
b.)
have a winning strategy?
c.) Neither player have a winning strategy?
![n_0 > 1](/media/m/5/7/1/57120faf1323c3827917968ca1d47258.png)
![{\mathcal A}](/media/m/0/9/5/0954230deeebe638c5df810acbe89888.png)
![{\mathcal B}](/media/m/7/c/5/7c51bf01c59a8982457e0c94531eeeef.png)
![n_1](/media/m/7/0/e/70e80a4aa999d3759ac6c6a089dcb575.png)
![n_2](/media/m/0/c/4/0c436cf14c3c06bb64ee838aaf6ba7c6.png)
![n_3](/media/m/5/d/a/5dadd276fb8440711968dbacaf9f2ff5.png)
![\ldots](/media/m/5/8/5/58542f3cc6046ef3889f8320b7487d60.png)
I.) Knowing
![n_{2k}](/media/m/8/e/0/8e04d77e9456ef459adbdd975f373f2c.png)
![{\mathcal A}](/media/m/0/9/5/0954230deeebe638c5df810acbe89888.png)
![n_{2k + 1}](/media/m/8/5/5/855c7057ef4a7716533791179e580876.png)
![n_{2k} \leq n_{2k + 1} \leq n_{2k}^2.](/media/m/6/e/d/6ed823fbf29134ed096efd665ed8e09c.png)
II.) Knowing
![n_{2k + 1}](/media/m/8/5/5/855c7057ef4a7716533791179e580876.png)
![{\mathcal B}](/media/m/7/c/5/7c51bf01c59a8982457e0c94531eeeef.png)
![n_{2k + 2}](/media/m/4/3/6/436f6f89fcdd66e3155a81ee3cc799ee.png)
![\frac {n_{2k + 1}}{n_{2k + 2}}](/media/m/2/0/4/2047bb74932c9eefaedb50ccdd31ebaf.png)
is a prime raised to a positive integer power.
Player
![{\mathcal A}](/media/m/0/9/5/0954230deeebe638c5df810acbe89888.png)
![{\mathcal B}](/media/m/7/c/5/7c51bf01c59a8982457e0c94531eeeef.png)
![n_0](/media/m/2/2/a/22a23a21b7b74b6120b58e119a1f870c.png)
a.)
![{\mathcal A}](/media/m/0/9/5/0954230deeebe638c5df810acbe89888.png)
b.)
![{\mathcal B}](/media/m/7/c/5/7c51bf01c59a8982457e0c94531eeeef.png)
c.) Neither player have a winning strategy?
Izvor: Međunarodna matematička olimpijada, shortlist 1990