Difference between revisions of "1966 IMO Problems/Problem 5"
(corrected problem statement) |
|||
| Line 7: | Line 7: | ||
==Solution== | ==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== | ==See also== | ||
{{IMO box|year=1966|num-b=4|num-a=6}} | {{IMO box|year=1966|num-b=4|num-a=6}} | ||
[[Category:Olympiad Algebra Problems]] | [[Category:Olympiad Algebra Problems]] | ||
Revision as of 16:16, 23 April 2014
Problem
Solve the system of equations
$\begin{align*}|a_1 - a_2| x_2 +|a_1 - a_3| x_3 +|a_1 - a_4| x_4 = 1\\ |a_2 - a_1| x_1 +|a_2 - a_3| x_3 +|a_2 - a_4| x_4 = 1\\ |a_3 - a_1| x_1 +|a_3 - a_2| x_2 +|a_3-a_4|x_4= 1\\ |a_4 - a_1| x_1 +|a_4 - a_2| x_2 +|a_4 - a_3| x_3 = 1\end{align*}$ (Error compiling LaTeX. Unknown error_msg)
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 | ||
