IMC 2019 Q19
19. Which of these expressions has the largest value?
- A \(\dfrac{1}{2}\)
- B \(\dfrac{1}{3} + \dfrac{1}{4}\)
- C \(\dfrac{1}{4} + \dfrac{1}{5} + \dfrac{1}{6}\)
- D \(\dfrac{1}{5} + \dfrac{1}{6} + \dfrac{1}{7} + \dfrac{1}{8}\)
- E \(\dfrac{1}{6} + \dfrac{1}{7} + \dfrac{1}{8} + \dfrac{1}{9} + \dfrac{1}{10}\)
Intermediate Mathematical Challenge 2019, Question 19. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E \(\dfrac{1}{6} + \dfrac{1}{7} + \dfrac{1}{8} + \dfrac{1}{9} + \dfrac{1}{10}\)
Note that \(\dfrac{1}{6} + \dfrac{1}{7} + \dfrac{1}{8} + \dfrac{1}{9} + \dfrac{1}{10} > \dfrac{1}{6} + \dfrac{1}{7} + \dfrac{1}{8} + \dfrac{1}{10} + \dfrac{1}{10} = \dfrac{1}{5} + \dfrac{1}{6} + \dfrac{1}{7} + \dfrac{1}{8}\).
Similarly \(\dfrac{1}{5} + \dfrac{1}{6} + \dfrac{1}{7} + \dfrac{1}{8} > \dfrac{1}{5} + \dfrac{1}{6} + \dfrac{1}{8} + \dfrac{1}{8} = \dfrac{1}{4} + \dfrac{1}{5} + \dfrac{1}{6}\). Also \(\dfrac{1}{4} + \dfrac{1}{5} + \dfrac{1}{6} > \dfrac{1}{4} + \dfrac{1}{6} + \dfrac{1}{6} = \dfrac{1}{3} + \dfrac{1}{4}\).
Finally, \(\dfrac{1}{3} + \dfrac{1}{4} > \dfrac{1}{4} + \dfrac{1}{4} = \dfrac{1}{2}\). Therefore the expression with the largest value is that in option E.
Solution © UK Mathematics Trust, from the IMC 2019 solutions. Longer versions with investigations: Solutions and investigations (PDF).