Difference between revisions of "1988 AIME Problems/Problem 13"

Revision as of 11:05, 30 June 2021

Problem

Find $a$ if $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^{17} + bx^{16} + 1$ .

Solution 1 (Fibonacci Version 1)

Let $F_n$ represent the $n$ th number in the Fibonacci sequence. Therefore,

$x^2 - x - 1 = 0\Longrightarrow x^n = F_n(x),\ n\in N\Longrightarrow x^{n + 2} = F_{n + 1}\cdot x + F_n,\ n\in N\ .$

The above uses the similarity between the Fibonacci recursion|recursive definition, $F_{n+2} - F_{n+1} - F_n = 0$ , and the polynomial $x^2 - x - 1 = 0$ .

$0 = ax^{17} + bx^{16} + 1 = a(F_{17}\cdot x + F_{16}) + b(F_{16}\cdot x + F_{15}) + 1\Longrightarrow$

$(aF_{17} + bF_{16})\cdot x + (aF_{16} + bF_{15} + 1) = 0,\ x\not\in Q\Longrightarrow$

$aF_{17} + bF_{16} = 0$ and $aF_{16} + bF_{15} + 1 = 0\Longrightarrow$

$a = F_{16},\ b = - F_{17}\Longrightarrow a = \boxed {987}\ .$

Solution 2 (Fibonacci Version 2)

We can long divide and search for a pattern; then the remainder would be set to zero to solve for $a$ . Writing out a few examples quickly shows us that the remainders after each subtraction follow the Fibonacci sequence. Carrying out this pattern, we find that the remainder is $(F_{16}b + F_{17}a)x + F_{15}b + F_{16}a + 1 = 0.$ Since the coefficient of $x$ must be zero, this gives us two equations, $F_{16}b + F_{17}a = 0$ and $F_{15}b + F_{16}a + 1 = 0$ . Solving these two as above, we get that $a = \boxed{987}$ .

There are various similar solutions which yield the same pattern, such as repeated substitution of $x^2 = x + 1$ into the polynomial with a higher degree, as shown in Solution 6.

Solution 3 (Fibonacci Version 3: For Beginners, Less Technical)

Trying to divide $ax^{17} + bx^{16} + 1$ by $x^2-x-1$ would be very tough, so let's try to divide using smaller degrees of $x$ . Doing $\frac{ax^3+bx^2+1}{x^2-x-1}$ , we get the following systems of equations: $\begin{align*} a+b &= -1, \\ 2a+b &= 0. \end{align*}$ Continuing with $\frac{ax^4+bx^3+1}{x^2-x-1}$ : $\begin{align*} 2a+b &= -1, \\ 3a+2b &= 0. \end{align*}$ There is somewhat of a pattern showing up, so let's try $\frac{ax^5+bx^4+1}{x^2-x-1}$ to verify. We get: $\begin{align*} 3a+2b &= -1, \\ 5a+3b &= 0. \end{align*}$ Now we begin to see that our pattern is actually the Fibonacci Numbers! Using the previous equations, we can make a general statement about $\frac{ax^n+bx^{n-1}+1}{x^2-x-1}$ : $\begin{align*} af_{n-1}+bf_{n-2} &= -1, \\ af_n+bf_{n-1} &= 0. \end{align*}$ Also, noticing our solutions from the previous systems of equations, we can create the following statement:

If $ax^n+bx^{n-1}+1$ has $x^2-x-1$ as a factor, then $a=f_{n-1}$ and $b = f_n.$

Thus, if $ax^{17}+bx^{16}+1$ has $x^2-x-1$ as a factor, we get that $a = 987$ and $b = -1597,$ so $a = \boxed {987}$ .

Solution 4 (Fibonacci Version 4: Not Rigorous)

Let's work backwards! Let $F(x) = ax^{17} + bx^{16} + 1$ and let $P(x)$ be the polynomial such that $P(x)(x^2 - x - 1) = F(x)$ .

Clearly, the constant term of $P(x)$ must be $- 1$ . Now, we have $(x^2 - x - 1)(c_1x^{15} + c_2x^{14} + \cdots + c_{15}x - 1)$ , where $c_{i}$ is some coefficient. However, since $F(x)$ has no $x$ term, it must be true that $c_{15} = 1$ .

Let's find $c_{14}$ now. Notice that all we care about in finding $c_{14}$ is that $(x^2 - x - 1)(\cdots + c_{14}x^2 + x - 1) = \text{something} + 0x^2 + \text{something}$ . Therefore, $c_{14} = - 2$ . Undergoing a similar process, $c_{13} = 3$ , $c_{12} = - 5$ , $c_{11} = 8$ , and we see a nice pattern. The coefficients of $P(x)$ are just the Fibonacci sequence with alternating signs! Therefore, $a = c_1 = F_{16}$ , where $F_{16}$ denotes the 16th Fibonnaci number and $a = \boxed{987}$ .

Solution 5 (Fibonacci Version 5)

The roots of $x^2-x-1$ are $\phi$ (the Golden Ratio) and $1-\phi$ . These two must also be roots of $ax^{17}+bx^{16}+1$ . Thus, we have two equations: $\begin{align*} a\phi^{17}+b\phi^{16}+1=0, \\ a(1-\phi)^{17}+b(1-\phi)^{16}+1=0. \end{align*}$ Subtract these two and divide by $\sqrt{5}$ to get $\frac{a(\phi^{17}-(1-\phi)^{17})}{\sqrt{5}}+\frac{b(\phi^{16}-(1-\phi)^{16})}{\sqrt{5}}=0.$ Noting that the formula for the $n$ th Fibonacci number is $\frac{\phi^n-(1-\phi)^n}{\sqrt{5}}$ , we have $1597a+987b=0$ . Since $1597$ and $987$ are coprime, the solutions to this equation under the integers are of the form $a=987k$ and $b=-1597k$ , of which the only integral solutions for $a$ on $[0,999]$ are $0$ and $987$ . $(a,b)=(0,0)$ cannot work since $x^2-x-1$ does not divide $1$ , so the answer must be $\boxed{987}$ . (Note that this solution would not be valid on an Olympiad or any test that does not restrict answers as integers between $000$ and $999$ ).

Solution 6 (Decreases the Powers)

We are given that $x^2 - x - 1$ is a factor of $ax^{17} + bx^{16} + 1,$ so the roots of $x^2 - x - 1$ are roots of $ax^{17} + bx^{16} + 1.$

Let $x=t$ be a root of $x^2 - x - 1$ so that $t^2 - t - 1 = 0,$ or $t^2 = t + 1.$ Note that $\begin{align*} t^4 &= (t+1)^2 \\ &= t^2 + 2t + 1 \\ &= (t+1) + 2t + 1 \\ &= 3t + 2, \\ t^8 &= (3t+2)^2 \\ &= 9t^2 + 12t + 4 \\ &= 9(t+1) + 12t + 4 \\ &= 21t + 13, \\ t^{16} &= (21t + 13)^2 \\ &= 441t^2 + 546t + 169 \\ &= 441(t+1) +546t + 169 \\ &= 987t + 610. \end{align*}$

@@ Line 57: / Line 57: @@
 == Solution 5 (Fibonacci Version 5) ==
-The roots of <math>x^2-x-1</math> are <math>\phi</math> (the Golden Ratio) and <math>1-\phi</math>. These two must also be roots of <math>ax^{17}+bx^{16}+1</math>. Thus, we have two equations: <math>a\phi^{17}+b\phi^{16}+1=0</math> and <math>a(1-\phi)^{17}+b(1-\phi)^{16}+1=0</math>. Subtract these two and divide by <math>\sqrt{5}</math> to get <math>\frac{a(\phi^{17}-(1-\phi)^{17})}{\sqrt{5}}+\frac{b(\phi^{16}-(1-\phi)^{16})}{\sqrt{5}}=0</math>. Noting that the formula for the <math>n</math>th Fibonacci number is <math>\frac{\phi^n-(1-\phi)^n}{\sqrt{5}}</math>, we have <math>1597a+987b=0</math>. Since <math>1597</math> and <math>987</math> are coprime, the solutions to this equation under the integers are of the form <math>a=987k</math> and <math>b=-1597k</math>, of which the only integral solutions for <math>a</math> on <math>[0,999]</math> are <math>0</math> and <math>987</math>. <math>(a,b)=(0,0)</math> cannot work since <math>x^2-x-1</math> does not divide <math>1</math>, so the answer must be <math>\boxed{987}</math>. (Note that this solution would not be valid on an Olympiad or any test that does not restrict answers as integers between <math>000</math> and <math>999</math>).
+The roots of <math>x^2-x-1</math> are <math>\phi</math> (the Golden Ratio) and <math>1-\phi</math>. These two must also be roots of <math>ax^{17}+bx^{16}+1</math>. Thus, we have two equations:
+<cmath>\begin{align*}
+a\phi^{17}+b\phi^{16}+1=0, \\
+a(1-\phi)^{17}+b(1-\phi)^{16}+1=0.
+\end{align*}</cmath>
+Subtract these two and divide by <math>\sqrt{5}</math> to get <cmath>\frac{a(\phi^{17}-(1-\phi)^{17})}{\sqrt{5}}+\frac{b(\phi^{16}-(1-\phi)^{16})}{\sqrt{5}}=0.</cmath> Noting that the formula for the <math>n</math>th Fibonacci number is <math>\frac{\phi^n-(1-\phi)^n}{\sqrt{5}}</math>, we have <math>1597a+987b=0</math>. Since <math>1597</math> and <math>987</math> are coprime, the solutions to this equation under the integers are of the form <math>a=987k</math> and <math>b=-1597k</math>, of which the only integral solutions for <math>a</math> on <math>[0,999]</math> are <math>0</math> and <math>987</math>. <math>(a,b)=(0,0)</math> cannot work since <math>x^2-x-1</math> does not divide <math>1</math>, so the answer must be <math>\boxed{987}</math>. (Note that this solution would not be valid on an Olympiad or any test that does not restrict answers as integers between <math>000</math> and <math>999</math>).
 == Solution 6 (Decreases the Powers) ==

1988 AIME (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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