1959 IMO Problems/Problem 3
Problem
Let be real numbers. Consider the quadratic equation in :
Using the numbers , form a quadratic equation in , whose roots are the same as those of the original equation. Compare the equations in and for .
Solution
Let the original equation be satisfied only for . Then we wish to construct a quadratic with roots .
Clearly, the sum of the roots of this quadratic must be
and the product of its roots must be
Thus the following quadratic fulfils the conditions:
Now, when we let , our equations are
and
This simplifies to previous equation. The first root of the first equation corresponds to and the second root of the first equation corresponds to .
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
1959 IMO (Problems) • Resources) | ||
Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 4 |
All IMO Problems and Solutions |