The mean of the nine consecutive integers is the fifth number in the series, 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 also a perfect square. For the options given, the fifth numbers would be 114, 124, 134, 144, 154 respectively. Of these, only 144 is a perfect square, so the answer is
This problem is taken from the UKMT Mathematical Challenges.