IMO Shortlist 2003 problem A6
Dodao/la:
arhiva2. travnja 2012. Let
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
be a positive integer and let
![(x_1,\ldots,x_n)](/media/m/e/a/2/ea2eafe236bc0db78d30be351b6eec1a.png)
,
![(y_1,\ldots,y_n)](/media/m/8/f/c/8fc74e34cf14c6cac0095e32c7ddc9d1.png)
be two sequences of positive real numbers. Suppose
![(z_2,\ldots,z_{2n})](/media/m/2/8/f/28fae5bb39ce11db058132620022610d.png)
is a sequence of positive real numbers such that
![z_{i+j}^2 \geq x_iy_j \qquad](/media/m/a/4/2/a42ccb2a4231350d2afb47a0f553ea85.png)
for all
![1\le i,j \leq n](/media/m/3/5/3/353b390b10004857a7caf46f33a0211f.png)
.
Let
![M=\max\{z_2,\ldots,z_{2n}\}](/media/m/d/e/c/dec43bbf6d35c536516777ed120ee7c5.png)
. Prove that
comment
Edited by Orl.
%V0
Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that $z_{i+j}^2 \geq x_iy_j \qquad$ for all $1\le i,j \leq n$.
Let $M=\max\{z_2,\ldots,z_{2n}\}$. Prove that
$$\biggl(\frac{M+z_2+\cdots+z_{2n}}{2n}\biggr)^2\ge \biggl(\frac{x_1+\cdots+x_n}{n}\biggr)\biggl(\frac{y_1+\cdots+y_n}{n}\biggr...$$
comment
Edited by Orl.
Izvor: Međunarodna matematička olimpijada, shortlist 2003