1997 AIME Problem 9

Given a nonnegative real number x, let \langle x\rangle denote the fractional part of x; that is, \langle x\rangle=x-\lfloor x\rfloor, where \lfloor x\rfloor denotes the greatest integer less than or equal to x. Suppose that a is positive, \langle a^{-1}\rangle=\langle a^2\rangle, and 2<a^2<3. Find the value of a^{12}-144a^{-1}.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.