1966 IMO Problems/Problem 5
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Problem
Solve the system of equations
where 
are four different real numbers.
Solution
Take a1 > a2 > a3 > a4. Subtracting the equation for i=2 from that for i=1 and dividing by (a1 - a2) we get:
![\[- x1 + x2 + x3 + x4 = 0.\]](http://latex.artofproblemsolving.com/f/4/0/f4090428c51248fedca0dad530bb6bcc9c43ac7a.png)
Subtracting the equation for i=4 from that for i=3 and dividing by (a3 - a4) we get:
![\[- x1 - x2 - x3 + x4 = 0.\]](http://latex.artofproblemsolving.com/8/9/4/894aa6afa34cc8d124ed7818f5bb3767749ad0b7.png)
Hence x1 = x4. Subtracting the equation for i=3 from that for i=2 and dividing by (a2 - a3) we get:
![\[- x1 - x2 + x3 + x4 = 0.\]](http://latex.artofproblemsolving.com/7/1/e/71e9abe49a9d910150c4163438842584a6ae8529.png)
Hence 
, and 
.
See also
| 1966 IMO (Problems) • Resources | ||
| Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
| All IMO Problems and Solutions | ||
