IMC 2011 Q14
14. The number 6 lies exactly halfway between 3 and 32. Which of the following is not halfway between a positive integer and its square?
- A 3
- B 10
- C 15
- D 21
- E 30
Intermediate Mathematical Challenge 2011, Question 14. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E 30
\(3 = \dfrac{2 + 2^2}{2}\); \(10 = \dfrac{4 + 4^2}{2}\); \(15 = \dfrac{5 + 5^2}{2}\); \(21 = \dfrac{6 + 6^2}{2}\). However \(\dfrac{7 + 7^2}{2} = 28\) and \(\dfrac{8 + 8^2}{2} = 36\), so 30 is not exactly halfway between a positive integer and its square.
(Note that every number which is exactly halfway between a positive integer and its square is a triangle number. Can you explain why this is so?)
Solution © UK Mathematics Trust, from the IMC 2011 solutions. Longer versions with investigations: Solutions and investigations (PDF).