2007 JBMO Problems/Problem 1
Let be positive real number such that . Prove that the equation has no real solution.
The discriminant of the equation is In order for the quadratic equation to have no real solution, the discriminant must be less than zero, so we need to show that That means we need to show that
Assume that Rearranging the equation results in If then would be negative, making the equality fail. If then , making However, that means so the equality also fails.
Thus, by proof by contradiction, must be greater than , so the discriminant of the equation is negative. That means the equation has no real solution.
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