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
i.e., they are multiples of each other. The reason behind this is that the roots of the first equation are , which implies that
is one of two certain multiples of
, and when
,
can only assume two distinct values. Q.E.D.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.