Find all solutions x1, ... , x5 to the five equations xi + xi+2 = y xi+1 for i = 1, ... , 5, where subscripts are reduced by 5 if necessary.
Solution
Successively eliminate variables to get x1(y - 2)(y2 + y - 1)2 = 0. We have the trivial solution xi = 0 for any y. For y = 2, we find xi = s for all i (where s is arbitrary). Care is needed for the case y2 + y - 1 = 0, because after eliminating three variables the two remaining equations have a factor y2 + y - 1, and so they are automatically satisfied. In this case, we can take any two xi arbitrary and still get a solution. For example, x1 = s, x2 = t, x3 = - s + yt, x4 = - ys - yt, x5 = ys - t.
Solutions are also available in: Samuel L Greitzer, International Mathematical Olympiads 1959-1977, MAA 1978, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48-X.
© John Scholes
jscholes@kalva.demon.co.uk
21 Sep 1998
Last corrected/updated 24 Sep 2003