1965 IMO Problems/Problem 4
Contents
[hide]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: ; ; ; ; .
Solution 2
We have to solve the system of equations
Subtract (2) from (1) and factor. We get
,
which implies or .
Similarly, subtracting (3) and then (4) from (1) and factoring, we get
They imply or , and or .
We will consider four possibilities:
1.
2. and
3. and
4.
Note that in fact, there are four more possibilities, but they just correspond to permutations in of cases 2. and 3., so there is no harm in not dealing with them explicitly.
Case 1. Plug in equation (1). We get . This is an equation of degree whose only real root is . We get the solution .
Case 2. Plug into , and get . We get or . The first solution, , yields , and using (4), . The second solution, , yields , and using (4), . We get the solution Because of the permutations of , we also get the solutions .
Case 3. Plug in in (4) and (3), and get and . Now plug in (1), and get . This equation becomes . yields which we already know, and yields (or some other values, depending on where we plug ), which do not give a possible solution.
Case 4. plugging into (4), (3), (2), we get . Plugging into (1), we get . The solutions of this equation are . Note that and yield solutions we already know, and is impossible.
Thus, the five solutions to the problem are .
[Solution by pf02, November 2024]
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 |