IMC 2005 Q20
20. One of the following is the largest of nine consecutive positive integers whose sum is a perfect square. Which one is it?
- A 118
- B 128
- C 138
- D 148
- E 158
Intermediate Mathematical Challenge 2005, Question 20. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 148
The mean of the nine consecutive positive integers is the fifth of the numbers, so their sum is nine times the fifth number. As nine is itself a perfect square, the sum will be a perfect square if and only if the fifth number is a perfect square. For the options given, the fifth numbers are 114, 124, 134, 144 and 154 respectively.
Solution © UK Mathematics Trust, from the IMC 2005 solutions.