Difference between revisions of "1997 AIME Problems/Problem 9"

Revision as of 11:27, 27 March 2022

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 $2<a^2<3$ . 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:

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

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

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

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

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

Another way to factor $a^3 - 2a - 1 = 0$ :

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

~isabelchen

@@ Line 31: / Line 31: @@
 Therefore, <math>a^{12} - 144 a^{-1} = 144a + 89 - 144(a-1) = 89 + 144 = \boxed{233}</math>
+Another way to factor <math>a^3 - 2a - 1 = 0</math>:
+<math>a^3 - 2a - 1 = a^3 + 1 -2a -2 = (a+1)(a^2 - a + 1) - 2(a+1) = (a + 1)(a^2 - a - 1)</math>
 ~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]

1997 AIME (Problems • Answer Key • Resources)
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