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Given a covering of the plane by closed unit discs
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{F}$$ \end{document}
and two points A and B in the region doubly covered by
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, what is the length of the shortest path connecting them that stays within the doubly covered region? This is a problem of G. Fejes-Tóth and he conjectured that if the distance between A and B is d, then the length of this path is at most
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sqrt {2d} + O(1)$$ \end{document}
. In this paper we give a bound of 2.78d + O(1).
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We consider packing a triangle with a number of equal positive homothetical copies. In particular, we show that every triangle can be packed with 7 copies of ratio
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, with 8 copies of ratio
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, and with 9 and 10 copies of ratio
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. All these ratios cannot be enlarged. We also present hypothetically best packings by greater number of copies.
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Any sequence of triangles homothetic to a fixed triangle T whose total area does not exceed one-half of the area of T can be packed translatively in T .

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