2010 AMC 12B Problems/Problem 15

Revision as of 15:10, 7 November 2010 by Jhggins (talk | contribs) (Solution)

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 ${(1+i)^{y}=zneqi^x}$.

See also

2010 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
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All AMC 12 Problems and Solutions