1966 IMO Problems/Problem 5

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