# 1997 AIME Problems/Problem 9

## Problem

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. Find the value of $a^{12}-144a^{-1}$.

## Solution 1

Looking at the properties of the number, it is immediately guess-able that $a = \phi = \frac{1+\sqrt{5}}2$ (the golden ratio) is the answer. The following is the way to derive that:

Since $\sqrt{2} < a < \sqrt{3}$, $0 < \frac{1}{\sqrt{3}} < a^{-1} < \frac{1}{\sqrt{2}} < 1$. Thus $\langle a^2 \rangle = a^{-1}$, and it follows that $a^2 - 2 = a^{-1} \Longrightarrow a^3 - 2a - 1 = 0$. Noting that $-1$ is a root, this factors to $(a+1)(a^2 - a - 1) = 0$, so $a = \frac{1 + \sqrt{5}}{2}$ (we discard the negative root).

Our answer is $(a^2)^{6}-144a^{-1} = \left(\frac{3+\sqrt{5}}2\right)^6 - 144\left(\frac{2}{1 + \sqrt{5}}\right)$. Complex conjugates reduce the second term to $-72(\sqrt{5}-1)$. The first term we can expand by the binomial theorem to get $\frac 1{2^6}\left(3^6 + 6\cdot 3^5\sqrt{5} + 15\cdot 3^4 \cdot 5 + 20\cdot 3^3 \cdot 5\sqrt{5} + 15 \cdot 3^2 \cdot 25 + 6 \cdot 3 \cdot 25\sqrt{5} + 5^3\right)$ $= \frac{1}{64}\left(10304 + 4608\sqrt{5}\right) = 161 + 72\sqrt{5}$. The answer is $161 + 72\sqrt{5} - 72\sqrt{5} + 72 = \boxed{233}$.

Note that to determine our answer, we could have also used other properties of $\phi$ like $\phi^3 = 2\phi + 1$.

## Solution 2

Find $a$ as shown above. Note that, since $a$ is a root of the equation $a^3 - 2a - 1 = 0$, $a^3 = 2a + 1$, and $a^{12} = (2a + 1)^4$. Also note that, since $a$ is a root of $a^2 - a - 1 = 0$, $\frac{1}{a} = a - 1$. The expression we wish to calculate then becomes $(2a + 1)^4 - 144(a - 1)$. Plugging in $a = \frac{1 + \sqrt{5}}{2}$, we plug in to get an answer of $(161 + 72\sqrt{5}) + 72 - 72\sqrt{5} = 161 + 72 = \boxed{233}$.

## Solution 3

Find $a$ as shown above. Note that $a$ satisfies the equation $a^2 = a+1$ (this is the equation we solved to get it). Then, we can simplify $a^{12}$ as follows using the fibonacci numbers:

$a^{12} = a^{11}+a^{10}= 2a^{10} + a^{9} = 3a^9+ 2a^8 = ... = 144a^1+89a^0 = 144a+89$

So we want $144(a-\frac1a)+89 = 144(1)+89 = \boxed{233}$ since $a-\frac1a = 1$ is equivalent to $a^2 = a+1$.

## Solution 4

As Solution 1 stated, $a^3 - 2a - 1 = 0$. $a^3 - 2a - 1 = a^3 - a^2 -a + a^2 -a -1 = (a+1)(a^2 - a - 1)$. So, $a^2 - a - 1 = 0$, $1 = a^2 - a$, $\frac1a = a-1$, $a^3 = 2a+1$, $a^2 = a+1$.

$a^6 = (a^3)^2 = (2a+1)^2= 4a^2 + 4a +1= 4(a+1) + 4a + 1= 8a+5$

$a^{12} = (a^6)^2 = (8a+5)^2 = 64a^2 + 80a + 25 = 64(a+1) + 80a + 25 = 144a + 89$

Therefore, $a^{12} - 144 a^{-1} = 144a + 89 - 144(a-1) = 89 + 144 = \boxed{233}$

Another way to factor $a^3 - 2a - 1$:

$a^3 - 2a - 1 = a^3 + 1 -2a -2 = (a+1)(a^2 - a + 1) - 2(a+1) = (a + 1)(a^2 - a - 1)$