A regular hexagon in the plane is called sweet if its area is equal to $1$. Is it possible to place $2000000$ sweet hexagons in the plane such that the union of their interiors is a convex polygon of area at least $1900000$? Remark: A subset $S$ of the plane is called convex if for every pair of points in $S$, every point on the straight line segment that joins the pair of points also belongs to $S$. The hexagons may overlap.
Problem
Source: European mathematical cup 2017
Tags: combinatorics
03.01.2018 21:03
I posted that already and nobody answered. I think the answer should be yes.
10.01.2018 00:13
Any solution?
22.01.2018 17:04
Bump .
22.01.2018 19:02
We use $\frac{n(n+1)}{2}$ such hexagons to stack them in triangular pattern ($n$ to be determine later), each of three "edges" of the figure obtained consist of $n-1$ triangle area needed to be fill in order to make the whole figure convex. We hope to fill all such triangle area with some hexagons. Note that we do not need to (and will not) cover endpoints of "edges" of our almost-triangle figure. It's not hard to see that we can cover each of such area using two more hexagons so that no area appear exterior of the figure we hoped. So we need, in total, $\frac{n(n+1)}{2}+3\times (2\times (n-1))$ hexagons. Also, the area of obtained figure is at least of those non-overlapped $\frac{n(n+1)}{2}$ hexagons, which is equal to $\frac{n(n+1)}{2}$. Finally, use $n=1949$ and Wolfram guarantee it's satisfy both constraints.
23.01.2018 18:23
Is there any other approach, different from the official solution?
25.01.2018 00:05
The problem is that the problem is so easy that it becomes hard just because of that fact. But another thing it is to keep in mind that yes/no questions on EMC are 95% yes
25.01.2018 16:51
JANMATH111 wrote: The problem is that the problem is so easy that it becomes hard just because of that fact. But another thing it is to keep in mind that yes/no questions on EMC are 95% yes Yes, since problems 1 and 4 were easy, I supposed that P2 must be hard, so i didn't try to solve it more than five minutes. *facepalm*