2010 AMC 12B Problems/Problem 15

Problem 15

For how many ordered triples $(x,y,z)$ of nonnegative integers less than $20$ are there exactly two distinct elements in the set $\{i^x, (1+i)^y, z\}$ , where $i=\sqrt{-1}$ ?

$\textbf{(A)}\ 149 \qquad \textbf{(B)}\ 205 \qquad \textbf{(C)}\ 215 \qquad \textbf{(D)}\ 225 \qquad \textbf{(E)}\ 235$

Solution

We have either ${i^{x}=(1+i)^{y}neqz}$ , ${i^{x}=zneq(1+i)^{y}}$ , or ${(1+i)^{y}=zneqi^x}$ .

2010 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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2010 AMC 12B Problems/Problem 15

Problem 15

Solution

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