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by math_explorer, Feb 1, 2011, 1:37 AM

As everybody knows, either the first two perfect numbers or first three digits of tau.

2004.3. Find all $k \in \mathbb{R}^+$ such that a rectangle of dimensions $1 \times k$ can be divided into two similar but noncongruent polygons.

I still have to deal with the polynomial one though.
geo problem
geometry

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628 is the integer part of $200\pi$.

$0 < k < \frac{1}{2}$ and $2 < k$ works. For $2 < k$, $x + \dfrac{1}{x} = k$ always have distinct solutions, so we can divide it into $1 \times x$ and $1 \times \dfrac{1}{x}$ rectangles. For $0 < k < \frac{1}{2}$, $x + \dfrac{1}{x} = \dfrac{1}{k}$ always have distinct solutions, so we can divide it into rectangles that the dimensions I'm not currently bothering. Not sure about $\frac{1}{2} \leq k \leq 2$.
This post has been edited 1 time. Last edited by chaotic_iak, Feb 1, 2011, 3:35 PM

by chaotic_iak, Feb 1, 2011, 3:33 PM

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A staircase-shape of $s$ steps can prove possible $k > \frac{s+1}{s}$ and $0 < k < \frac{s}{s+1}$, so $k = 1$ is all that's left.

The square seems impossible to me despite its cute fractal solutions.

by math_explorer, Feb 2, 2011, 9:06 AM

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