IMO Shortlist 1970 problem 2
Dodao/la:
arhiva2. travnja 2012. We have
![0\le x_i<b](/media/m/f/1/c/f1c99a042f639e83abbfc23724e91ec2.png)
for
![i=0,1,\ldots,n](/media/m/1/2/8/1289938bf4c86a5e46fffad358c9b79f.png)
and
![x_n>0,x_{n-1}>0](/media/m/e/4/2/e4224f5344ff40d1bf44ec7aeffd9929.png)
. If
![a>b](/media/m/9/3/e/93ec187b4eea9a155615a5025c8701f3.png)
, and
![x_nx_{n-1}\ldots x_0](/media/m/c/5/0/c50bdfecfa0c7f7311ebceefe58287fd.png)
represents the number
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
base
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
and
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
base
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
, whilst
![x_{n-1}x_{n-2}\ldots x_0](/media/m/7/b/1/7b192e39904bb480a27609917c625348.png)
represents the number
![A'](/media/m/9/2/6/9267b8bcabe1ad2df0d51dab3364714b.png)
base
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
and
![B'](/media/m/a/1/a/a1a88eb7f35fee4f41c66bfb0c902f51.png)
base
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
, prove that
![A'B<AB'](/media/m/2/4/8/24878d10ed170da94cd6d5368df77d39.png)
.
%V0
We have $0\le x_i<b$ for $i=0,1,\ldots,n$ and $x_n>0,x_{n-1}>0$. If $a>b$, and $x_nx_{n-1}\ldots x_0$ represents the number $A$ base $a$ and $B$ base $b$, whilst $x_{n-1}x_{n-2}\ldots x_0$ represents the number $A'$ base $a$ and $B'$ base $b$, prove that $A'B<AB'$.
Izvor: Međunarodna matematička olimpijada, shortlist 1970