IMC 2004 Q24
24. If \(p\), \(q\) and \(p - q\) are all positive integers, which of the following is least?
- A \(\dfrac{q^2}{p^2}\)
- B \(\dfrac{p^2}{q^2}\)
- C \(\dfrac{q}{p}\)
- D \(\sqrt{\dfrac{q}{p}}\)
- E \(\sqrt{\dfrac{p}{q}}\)
Intermediate Mathematical Challenge 2004, Question 24. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A \(\dfrac{q^2}{p^2}\)
We may deduce that \(p > q\), so, as \(p\) and \(q\) are both positive, \(\dfrac{p^2}{q^2} > \dfrac{\sqrt{p}}{\sqrt{q}} > 1\). We may also deduce that \(0 < \dfrac{q^2}{p^2} < \dfrac{q}{p} < \dfrac{\sqrt{q}}{\sqrt{p}} < 1\) and hence \(\dfrac{q^2}{p^2}\) is the least of the five numbers.
Solution © UK Mathematics Trust, from the IMC 2004 solutions.