IMO Shortlist 2003 problem A1
Dodao/la:
arhiva2. travnja 2012. Let
![a_{ij}](/media/m/b/6/c/b6cda4f0bdc70b1c45ec2b74d35bcdbd.png)
(with the indices
![i](/media/m/3/2/d/32d270270062c6863fe475c6a99da9fc.png)
and
![j](/media/m/7/9/e/79ebb10f98eb80d16b0c785d9d682a72.png)
from the set
![\left\{1,\ 2,\ 3\right\}](/media/m/d/4/5/d45e68024378eaa14a5116304eaab3c5.png)
) be real numbers such that
![a_{ij}>0](/media/m/c/b/e/cbed05529a022a8abdb5f9b334a4be40.png)
for
![i = j](/media/m/6/b/0/6b0305f3cd73a22cbcf05385646a5b4d.png)
;
![a_{ij}<0](/media/m/e/7/3/e732c08e6334e0f41a256aeca1603ed7.png)
for
![i\neq j](/media/m/d/0/f/d0f78061e1617137185624c9f5992813.png)
.
Prove the existence of positive real numbers
![c_{1}](/media/m/c/3/b/c3b499e0dedfead7a0ba6b740fdb803e.png)
,
![c_{2}](/media/m/2/3/1/2313dacfe602a87702e9aebeed055786.png)
,
![c_{3}](/media/m/7/3/7/73786f1a9ff86048a34b022fa8d89d61.png)
such that the numbers
![a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3}](/media/m/d/2/4/d241b0a67866053ce89c84cf2a370a3a.png)
,
![a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3}](/media/m/4/5/d/45d260cf5eea8c48eecc88c64b32e982.png)
,
are either all negative, or all zero, or all positive.
%V0
Let $a_{ij}$ (with the indices $i$ and $j$ from the set $\left\{1,\ 2,\ 3\right\}$) be real numbers such that
$a_{ij}>0$ for $i = j$;
$a_{ij}<0$ for $i\neq j$.
Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers
$a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3}$,
$a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3}$,
$a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}$
are either all negative, or all zero, or all positive.
Izvor: Međunarodna matematička olimpijada, shortlist 2003