IMO Shortlist 1996 problem A4
Dodao/la:
arhiva2. travnja 2012. Let
![a_{1}, a_{2}...a_{n}](/media/m/9/0/d/90db0c7d5cbffcdf4dac3044ddee71df.png)
be non-negative reals, not all zero. Show that that
(a) The polynomial
![p(x) = x^{n} - a_{1}x^{n - 1} + ... - a_{n - 1}x - a_{n}](/media/m/c/8/2/c82ffc65af318ef2f2497844c57d94f3.png)
has preceisely 1 positive real root
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
.
(b) let
![A = \sum_{i = 1}^n a_{i}](/media/m/2/f/2/2f257bc5f85baaa8618bba393cd72af3.png)
and
![B = \sum_{i = 1}^n ia_{i}](/media/m/f/5/1/f51deb84afa0653313d6103646f29683.png)
. Show that
![A^{A} \leq R^{B}](/media/m/6/8/5/685f3843bc38245ec4085dd89e73a841.png)
.
%V0
Let $a_{1}, a_{2}...a_{n}$ be non-negative reals, not all zero. Show that that
(a) The polynomial $p(x) = x^{n} - a_{1}x^{n - 1} + ... - a_{n - 1}x - a_{n}$ has preceisely 1 positive real root $R$.
(b) let $A = \sum_{i = 1}^n a_{i}$ and $B = \sum_{i = 1}^n ia_{i}$. Show that $A^{A} \leq R^{B}$.
Izvor: Međunarodna matematička olimpijada, shortlist 1996