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
The roots of the first equation are , which implies that
is one of two certain multiples of
. The roots of the second equation are
. It is straightforward to verify that they result in the same values of
.
Solution 2.
Note that
After two lines of rearrangement, the solution,
is obtained.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
1959 IMO (Problems) • Resources | ||
Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 4 |
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