2009 AIME I Problems/Problem 2

Problem

There is a complex number $z$ with imaginary part $164$ and a positive integer $n$ such that

\[\frac {z}{z + n} = 4i.\]

Find $n$.

Solution 1

Let $z = a + 164i$.

Then \[\frac {a + 164i}{a + 164i + n} = 4i\] and \[a + 164i = \left (4i \right ) \left (a + n + 164i \right ) = 4i \left (a + n \right ) - 656.\]

By comparing coefficients, equating the real terms on the leftmost and rightmost side of the equation,

we conclude that \[a = -656.\]

By equating the imaginary terms on each side of the equation,

we conclude that \[164i = 4i \left (a + n \right ) = 4i \left (-656 + n \right ).\]

We now have an equation for $n$: \[4i \left (-656 + n \right ) = 164i,\]

and this equation shows that $n = \boxed{697}.$


Solution 2

\[\frac {z}{z+n}=4i\]

\[1-\frac {n}{z+n}=4i\]

\[1-4i=\frac {n}{z+n}\]

\[\frac {1}{1-4i}=\frac {z+n}{n}\]

\[\frac {1+4i}{17}=\frac {z}{n}+1\]

Since their imaginary part has to be equal,

\[\frac {4i}{17}=\frac {164i}{n}\]

\[n=\frac {(164)(17)}{4}=697\]

\[n = \boxed{697}.\]

See also

2009 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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