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: ; ; ; ; .
See Also
1965 IMO (Problems) • Resources | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
All IMO Problems and Solutions |