IMO Shortlist 1993 problem C1
Kvaliteta:
Avg: 0,0Težina:
Avg: 6,0 a) Show that the set
of all positive rationals can be partitioned into three disjoint subsets.
satisfying the following conditions:
where
stands for the set
for any two subsets
of
and
stands for
b) Show that all positive rational cubes are in
for such a partition of
c) Find such a partition
with the property that for no positive integer
both
and
are in
that is,
![\mathbb{Q}^{ + }](/media/m/6/8/1/68175b3a50ef693f217095e3c7186a3f.png)
![A,B,C](/media/m/6/0/1/6012c28979f41c54e9b40b9fc855aa34.png)
![BA = B; B^2 = C; BC = A;](/media/m/3/5/5/355847a3ad78909acd845584b1efce04.png)
![HK](/media/m/1/6/c/16c38f8d1bdbcdca5b5573926d5999bf.png)
![\{hk: h \in H, k \in K\}](/media/m/8/d/1/8d1df8ead6964d929ccdd9f85b74eea5.png)
![H, K](/media/m/4/a/4/4a43e626cbdbdbccd58d33c1d5f84928.png)
![\mathbb{Q}^{ + }](/media/m/6/8/1/68175b3a50ef693f217095e3c7186a3f.png)
![H^2](/media/m/f/d/e/fde2b7d72887d56a408ea40db9561878.png)
![HH.](/media/m/f/c/e/fcefa339c83df3f2cc6265c043139bc5.png)
b) Show that all positive rational cubes are in
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![\mathbb{Q}^{ + }.](/media/m/e/4/d/e4db378b26d26a54cfe2ebcaf03242b0.png)
c) Find such a partition
![\mathbb{Q}^{ + } = A \cup B \cup C](/media/m/0/c/3/0c3413559c3a245f55d8506ee9a0e97b.png)
![n \leq 34,](/media/m/e/2/7/e27b0bc416013351c7ab32b24a2fd5d5.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![n + 1](/media/m/3/6/d/36dc98984132471cc8b030d766fd893a.png)
![A,](/media/m/8/6/5/865743fba196abcc2b01372b2f0205c1.png)
![\text{min} \{n \in \mathbb{N}: n \in A, n + 1 \in A \} > 34.](/media/m/7/5/4/754e7f6c2289b99245ff90e25f2b88a5.png)
Izvor: Međunarodna matematička olimpijada, shortlist 1993