Difference between revisions of "1966 IMO Problems/Problem 5"
Aditya1135 (talk | contribs) (→Problem) |
|||
| Line 2: | Line 2: | ||
Solve the system of equations | Solve the system of equations | ||
| − | <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 11:47, 1 September 2017
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.
Subtracting the equation for i=4 from that for i=3 and dividing by (a3 - a4) we get:
- x1 - x2 - x3 + x4 = 0.
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.
Hence x2 = x3 = 0, and x1 = x4 = 1/(a1 - a4).
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 | ||
