1959 IMO Problems/Problem 3

Problem

Let $a,b,c$ be real numbers. Consider the quadratic equation in $\cos{x}$ :

Using the numbers $a,b,c$ , form a quadratic equation in $\cos{2x}$ , whose roots are the same as those of the original equation. Compare the equations in $\cos{x}$ and $\cos{2x}$ for $a=4, b=2, c=-1$ .

Let the original equation be satisfied only for $\cos{x}=m, \cos{x}=n$ . Then we wish to construct a quadratic with roots $2m^2 -1, 2n^2 -1$ .

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 $a=4, b=2, c= -1$ , our equations are

and

The roots of the first equation are $\cos {x} = \frac{-1 \pm \sqrt{5}}{4}$ , which implies that $x$ is one of two certain multiples of $\frac{\pi}{5}$ . The roots of the second equation are $\cos(2x) = \frac{1 \pm \sqrt{5}}{4}$ . It is straightforward to verify that they result in the same values of $x$ .

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

1959 IMO (Problems)
Preceded by Problem 2	1 • 2 • 3 • 4 • 5 • 6	Followed by Problem 4