Maclaurin 2019 Q3
3. Two numbers are such that the sum of their reciprocals is equal to 1. Each of these numbers is then reduced by 1 to give two new numbers.
Prove that these two new numbers are reciprocals of each other.
Maclaurin Olympiad 2019, Question 3. © UK Mathematics Trust. Original paper on ukmt.org.uk
Let the original numbers be \(x\) and \(y\). We have
Multiplying both sides by \(xy\), we obtain
Rearranging, we have
so that
Since their product is 1 neither \(x - 1\) nor \(y - 1\) is 0. Therefore
and so the numbers reduced by 1 are reciprocals of each other.
Note
It is essential that the argument here is presented in the order shown, going from what we are given to what we have to prove.
Solution © UK Mathematics Trust, from the Maclaurin Olympiad 2019 solutions.