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Unique solution of the system of linear equations

Source: IMO LongList 1988, South Korea 1, Problem 62 of ILL

November 3, 2005
algebrasystem of equationsalgebra unsolved

Problem Statement

Let x=p,y=q,z=r,w=sx = p, y = q, z = r, w = s be the unique solution of the system of linear equations x+ai⋅y+ai2⋅z+ai3⋅w=ai4,i=1,2,3,4. x + a_i \cdot y + a^2_i \cdot z + a^3_i \cdot w = a^4_i, i = 1,2,3,4. Express the solutions of the following system in terms of p,q,rp,q,r and s:s: x+ai2⋅y+ai4⋅z+ai6⋅w=ai8,i=1,2,3,4. x + a^2_i \cdot y + a^4_i \cdot z + a^6_i \cdot w = a^8_i, i = 1,2,3,4. Assume the uniquness of the solution.
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