# 1965 IMO Problems/Problem 4

## Problem

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

## Solution

Let be the product of the four real numbers.

Then, for we have: .

Multiplying by yields:

where .

If , then we have which is a solution.

So assume that . WLOG, let at least two of equal , and OR .

Case I:

Then we have:

Which has no non-zero solutions for .

Case II: AND

Then we have:

AND

So, we have as the only non-zero solution, and thus, and all permutations are solutions.

Case III: AND

Then we have:

AND

Thus, there are no non-zero solutions for in this case.

Therefore, the solutions are: ; ; ; ; .