2004 AMC 12A Problems/Problem 6

Problem

Let $U=2\cdot 2004^{2005}$ , $V=2004^{2005}$ , $W=2003\cdot 2004^{2004}$ , $X=2\cdot 2004^{2004}$ , $Y=2004^{2004}$ and $Z=2004^{2003}$ . Which of the following is the largest?

$\mathrm {(A)} U-V \qquad \mathrm {(B)} V-W \qquad \mathrm {(C)} W-X \qquad \mathrm {(D)} X-Y \qquad \mathrm {(E)} Y-Z \qquad$

Solution

$\begin{eqnarray*} U-V&=&2004*2004^{2004}\\ V-W&=&1*2004^{2004}\\ W-X&=&2001*2004^{2004}\\ X-Y&=&1*2004^{2004}\\ Y-Z&=&2003*2004^{2003} \end{eqnarray*}$

After comparison, $U-V$ is the largest. $\mathrm {(A)}$

Solution 2

A quick check reveals the positive integers are in decreasing order. Then note $V = 2002^{2005}$ . $\newline$ $U - V = 2004^{2005} = V$ , and any of the other differences cannot be greater than or equal to $V$ , hence choose $\boxed{A}$ as the answer.

2004 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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2004 AMC 12A Problems/Problem 6

Contents

Problem

Solution

Solution 2

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