IMO 1965/B1 solution

IMO 1965

Problem B1

Find all sets of four real numbers such that the sum of any one and the product of the other three is 2.

Answer

1,1,1,1 or 3,-1,-1,-1.

Solution

Let the numbers be x₁, ... , x₄. Let t = x₁x₂x₃x₄. Then x₁ + t/x₁ = 2. So all the x_i are roots of the quadratic x² - 2x + t = 0. This has two roots, whose product is t.

If all x_i are equal to x, then x³ + x = 2, and we must have x = 1. If not, then if x₁ and x₂ are unequal roots, we have x₁x₂ = t and x₁x₂x₃x₄ = t, so x₃x₄ = 1. But x₃ and x₄ are still roots of x² - 2x + t = 0. They cannot be unequal, otherwise x₃x₄ = t, which gives t = 1 and hence all x_i = 1. Hence they are equal, and hence both 1 or both -1. Both 1 gives t = 1 and all x_i = 1. Both -1 gives t = -3 and hence x_i = 3, -1, -1, -1 (in some order).

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.

7th IMO 1965