2004 AIME I Problems/Problem 12

Problem

Let $S$ be the set of ordered pairs $(x, y)$ such that $0 < x \le 1, 0<y\le 1,$ and $\left[\log_2{\left(\frac 1x\right)}\right]$ and $\left[\log_5{\left(\frac 1y\right)}\right]$ are both even. Given that the area of the graph of $S$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$ The notation $[z]$ denotes the greatest integer that is less than or equal to $z.$

Solution

$\left\lfloor\log_2\left(\frac{1}{x}\right)\right\rfloor$ is even when

$x \in \left(\frac{1}{2},1\right) \cup \left(\frac{1}{8},\frac{1}{4}\right) \cup \left(\frac{1}{32},\frac{1}{16}\right) \cup...$

Likewise: $\left\lfloor\log_2\left(\frac{1}{y}\right)\right\rfloor$ is even when

$y \in \left(\frac{1}{5},1\right) \cup \left(\frac{1}{125},\frac{1}{25}\right) \cup \left(\frac{1}{3125},\frac{1}{625}\right) \cup...$

Graphing this creates a series of rectangles which become smaller as you move toward the origin.

The $x$ interval of each box is given by the sequence $\frac{1}{2} , \frac{1}{8}, \frac{1}{32} ...$

The $y$ interval is given by $\frac{4}{5} , \frac{4}{125}, \frac{4}{3125}...$

Each box is the product of one term of each sequence. The sum of these boxes is simply the product of the sum of each sequence or:

$\left(\frac{1}{2} + \frac{1}{8} + \frac{1}{32} ...\right)\left(\frac{4}{5} + \frac{4}{125} + \frac{4}{3125}...\right)$

Geometric sums are taken:

$\left(\frac{\frac{1}{2}}{1 - \frac{1}{4}}\right)\left(\frac{\frac{4}{5}}{1-\frac{1}{25}}\right)$

$\frac{2}{3} \cdot \frac{5}{6} = \frac{5}{9}$

$m+n = 5 + 9 = \boxed{014}$

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2004 AIME I Problems/Problem 12

Problem

Solution

See also