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Arithmetic progression in equation system

Source: IMO Longlist 1967, Mongolia 4

October 14, 2005
linear algebramatrixalgebrasystem of equationsarithmetic sequenceIMO ShortlistIMO Longlist

Problem Statement

In what case does the system of equations x+y+mz=ax+my+z=bmx+y+z=c\begin{matrix} x + y + mz = a \\ x + my + z = b \\ mx + y + z = c \end{matrix} have a solution? Find conditions under which the unique solution of the above system is an arithmetic progression.
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