1974 IMO Problems/Problem 5
Determine all possible values of where are arbitrary positive numbers.
Note that We will now prove that can reach any range in between and .
Choose any positive number . For some variables such that and , let , , and . Plugging this back into the original fraction, we get The above equation can be further simplified to Note that is a continuous function and that is a strictly increasing function. We can now decrease and to make tend arbitrarily close to . We see , meaning can be brought arbitrarily close to . Now, set and for some positive real numbers . Then Notice that if we treat the numerator and denominator each as a quadratic in , we will get , where has a degree lower than . This means taking , which means can be brought arbitrarily close to . Therefore, we are done. ~Imajinary