IMO Shortlist 1973 problem 10
Dodao/la:
arhiva2. travnja 2012. Let
![a_1, \ldots, a_n](/media/m/4/6/2/4620bab4413c05b6365d3f7e207d102a.png)
be
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
positive numbers and
![0 < q < 1.](/media/m/0/1/9/019f42479949084ea4128c62a02477ae.png)
Determine
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
positive numbers
![b_1, \ldots, b_n](/media/m/2/4/8/248d225a27545137743c65d3ba21014f.png)
so that:
a.)
![k < b_k](/media/m/9/d/7/9d7cd5170e925341318b2e7af7cf9a18.png)
for all
![k = 1, \ldots, n](/media/m/5/f/0/5f021f44d697034fd2771710035474ef.png)
,
b.)
![\displaystyle q < \frac{b_{k+1}}{b_{k}} < \frac{1}{q}](/media/m/c/9/1/c9123dccc6aab9af765071919093a202.png)
for all
![k = 1, \ldots, n-1](/media/m/0/2/d/02db7dd805a7ee58f9c34c5916b369a3.png)
,
c.)
![\displaystyle \sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k](/media/m/5/2/7/527b4f0bdee47446179dc5d4d1961975.png)
.
%V0
Let $a_1, \ldots, a_n$ be $n$ positive numbers and $0 < q < 1.$ Determine $n$ positive numbers $b_1, \ldots, b_n$ so that:
a.) $k < b_k$ for all $k = 1, \ldots, n$ ,
b.) $\displaystyle q < \frac{b_{k+1}}{b_{k}} < \frac{1}{q}$ for all $k = 1, \ldots, n-1$,
c.) $\displaystyle \sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k$.
Izvor: Međunarodna matematička olimpijada, shortlist 1973