Difference between revisions of "1966 IMO Problems/Problem 5"
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Solve the system of equations | Solve the system of equations | ||
| − | <math>|a_1 - a_2| x_2 +|a_1 - a_3| x_3 +|a_1 - a_4| x_4 = 1\\ |a_2 - a_1| x_1 +|a_2 - a_3| x_3 +|a_2 - a_4| x_4 = 1\\ |a_3 - a_1| x_1 +|a_3 - a_2| x_2 +|a_3-a_4|x_4= 1\\ |a_4 - a_1| x_1 +|a_4 - a_2| x_2 +|a_4 - a_3| x_3 = 1</math> | + | <math>\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ |a_1 - a_2| x_2 + |a_1 - a_3| x_3 + |a_1 - a_4| x_4 = 1 \\ |
| + | |a_2 - a_1| x_1 + \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ |a_2 - a_3| x_3 + |a_2 - a_4| x_4 = 1 \\ | ||
| + | |a_3 - a_1| x_1 + |a_3 - a_2| x_2 + \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ |a_3-a_4| x_4 = 1 \\ | ||
| + | |a_4 - a_1| x_1 + |a_4 - a_2| x_2 + |a_4 - a_3| x_3 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = 1</math> | ||
where <math>a_1, a_2, a_3, a_4</math> are four different real numbers. | where <math>a_1, a_2, a_3, a_4</math> are four different real numbers. | ||
Revision as of 19:55, 22 September 2024
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 | ||
