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1994 AMC 12 #3

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December 30, 2011
modular arithmeticAMCAMC 12

Problem Statement

How many of the following are equal to xx+xxx^x+x^x for all x>0x>0?
<spanclass=′latex−bold′>I:</span> 2xx<spanclass=′latex−bold′>II:</span> x2x<spanclass=′latex−bold′>III:</span> (2x)x<spanclass=′latex−bold′>IV:</span> (2x)2x<span class='latex-bold'>I:</span>\ 2x^x \qquad<span class='latex-bold'>II:</span>\ x^{2x} \qquad<span class='latex-bold'>III:</span>\ (2x)^x \qquad<span class='latex-bold'>IV:</span>\ (2x)^{2x}
<spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 1<spanclass=′latex−bold′>(C)</span> 2<spanclass=′latex−bold′>(D)</span> 3<spanclass=′latex−bold′>(E)</span> 4 <span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ 3 \qquad<span class='latex-bold'>(E)</span>\ 4
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