Mock AIME 1 2007-2008 Problems/Problem 7

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Problem

Consider the following function $g(x)$ defined as $(x^{2^{2008}-1}-1)g(x) = (x+1)(x^2+1)(x^4+1)\cdots (x^{2^{2007}}+1) - 1$ Find $g(2)$ .

Solution

Multiply both sides by $x-1$ ; the right hand side collapses by the reverse of the difference of squares.

$\begin{align*}(x-1)(x^{2^{2008}-1}-1)g(x) &= (x-1)(x+1)(x^2+1)(x^4+1)\cdots (x^{2^{2007}}+1) - (x-1)\\&= (x^2-1) (x^2+1)(x^4+1)\cdots (x^{2^{2007}}+1) - (x-1)\\&= \cdots\\&= \left(x^{2^{2008}}-1\right) - (x-1) = x^{2^{2008}} - x \end{align*}$ Substituting $x = 2$ , we have $\left(2^{2^{2008}-1}-1\right) \cdot g(2) = 2^{2^{2008}}-2 = 2\left(2^{2^{2008}-1}-1\right)$ Dividing both sides by $2^{2^{2008}-1}$ , we find $g(2) = \boxed{002}$ .

Mock AIME 1 2007-2008 (Problems, Source)
Preceded by Problem 6	Followed by Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

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Mock AIME 1 2007-2008 Problems/Problem 7

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