4. Let $n$ be a positive integer. A regular hexagon with side $n$ is divided into equilateral triangles with side 1 by lines parallel to its sides. Find the number of regular hexagons all of whose vertices are among the vertices of equilateral triangles.

(United Kingdom)

Time allowed: 270 minutes.

Each problem is worth 10 points.
4. Denote the given hexagon by $P$. We call the vertices of the equilateral triangles simply nodes. For every regular hexagon $Q$ with the vertices in nodes, consider the hexagon $\bar{Q}$ circumscribed about $Q$ with sides parallel to the sides of $P$. The vertices of $\bar{Q}$ are nodes as well.

For $0 \leqslant m<n$, hexagon $\bar{Q}$ of side $n-m$

![](https://cdn.mathpix.com/cropped/2024_06_04_6f458ef4962c28de5a12g-13.jpg?height=332&width=393&top_left_y=1597&top_left_x=1129)
can be chosen in $3 m^{2}+3 m+1=(m+1)^{3}-m^{3}$ ways. Also, given $\bar{Q}$, hexagon $Q$ can be chosen in $n-m$ ways. Thus the total number of hexagons $Q$ equals

$$
\begin{aligned}
& \sum_{m=0}^{n-1}(n-m)\left((m+1)^{3}-m^{3}\right)=\sum_{m=0}^{n-1}(n-m)(m+1)^{3}-\sum_{m=0}^{n-1}(n-m) m^{3} \\
& \quad=\sum_{m=1}^{n}(n-m+1) m^{3}-\sum_{m=0}^{n-1}(n-m) m^{3}=\sum_{m=1}^{n} m^{3}=\frac{n^{2}(n+1)^{2}}{4}
\end{aligned}
$$

The SMO was sponsored by

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![](https://cdn.mathpix.com/cropped/2024_06_04_6f458ef4962c28de5a12g-16.jpg?height=1040&width=1426&top_left_y=222&top_left_x=215)

Mathematical Competitions in Serbia

$h t t p: / /$ srb.imomath.com/

$\sim \sim \sim \sim \sim \sim \sim$

The IMO Compendium Olympiad Archive http://www.imocompendium.com/
Mathematical Society of Serbia $h t t p: / / w w w . d m s . r s /$

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## The IMO Compendium - 2nd Edition: 1959-2009

Until the first edition of this book appearing in 2006, it has been almost impossible to obtain a complete collection of the problems proposed at the IMO in book form. "The IMO Compendium" is the result of a collaboration between four former IMO participants from Yugoslavia, now Serbia, to rescue these problems from old and scattered manuscripts, and produce the ultimate source of IMO practice problems. This book attempts to gather all the problems and solutions appearing on the IMO through 2009. This second edition contains 143 new problems, picking up where the 1959-2004 edition has left off, accounting for 2043 problems in total.

Publisher: Springer (2011); Hardcover, 823 pages; Language: English; ISBN: 1441998535

For information on how to order, visit http://www.imocompendium.com/