Let $n$ and $m$ be fixed positive integers. The hexagon $ABCDEF$ with vertices $A = (0,0)$, $B = (n,0)$, $C = (n,m)$, $D = (n-1,m)$, $E = (n-1,1)$, $F = (0,1)$ has been partitioned into $n+m-1$ unit squares. Find the number of paths from $A$ to $C$ along grid lines, passing through every grid node at most once.

et $ABCDEF$ be a convex hexagon which has an inscribed circle and a circumcribed. Denote by $\omega_{A}, \omega_{B},\omega_{C},\omega_{D},\omega_{E}$ and $\omega_{F}$ the inscribed circles of the triangles $FAB, ABC, BCD, CDE, DEF$ and $EFA$, respecitively. Let $l_{AB}$, be the external of $\omega_{A}$ and $\omega_{B}$; lines $l_{BC}$, $l_{CD}$, $l_{DE}$, $l_{EF}$, $l_{FA}$ are analoguosly defined. Let $A_1$ be the intersection point of the lines $l_{FA}$ and $l_{AB}$, $B_1, C_1, D_1, E_1, F_1$ are analogously defined. Prove that $A_1D_1, B_1E_1, C_1F_1$ are concurrent.

Show that if equiangular hexagon has sides $a, b, c, d, e, f$ in order then $a - d = e - b = c - f$.

$ABCDEF$ is a convex hexagon with all its sides equal. Also $\angle A + \angle C + \angle E = \angle B + \angle D + \angle F$. Show that $\angle A = \angle D$, $\angle B = \angle E$ and $\angle C = \angle F$.

In a regular hexagon, segments with lengths from $1$ to $6$ were drawn as shown in the right figure (the segments go sequentially in increasing length, all the angles between them are right). Find the side length of this hexagon. [img]https://cdn.artofproblemsolving.com/attachments/3/1/82e4225b56d984e897a43ba1f403d89e5f4736.png[/img]

The hexagon $ABCDEF$ is inscribed in a circle. The sides $AB$, $CD$ and $EF$ are all equal in length to the radius. Prove that the midpoints of the other three sides determine an equilateral triangle.

Decide if exists a convex hexagon with all sides longer than $1$ and all nine of its diagonals are less than $2$ in length.

A football is made by sewing together some black and white leather patches. The black patches are regular pentagons of the same size. The white patches are regular hexagons of the same size. Each pentagon is bordered by 5 hexagons. Each hexagons is bordered by $3$ pentagons and $3$ hexagons. We need $12$ pentagons to make one football. How many hexagons are needed to make one football?

Given an equilateral triangle with sidelength $k$ cm. With lines parallel to it's sides, we split it into $k^2$ small equilateral triangles with sidelength $1$ cm. This way, a triangular grid is created. In every small triangle of sidelength $1$ cm, we place exactly one integer from $1$ to $k^2$ (included), such that there are no such triangles having the same numbers. With vertices the points of the grid, regular hexagons are defined of sidelengths $1$ cm. We shall name as [i]value [/i] of the hexagon, the sum of the numbers that lie on the $6$ small equilateral triangles that the hexagon consists of . Find (in terms of the integer $k>4$) the maximum and the minimum value of the sum of the values of all hexagons .

The diagonals $AC$ and $CE$ of the regular hexagon $ABCDEF$ are divided by inner points $M$ and $N$ respectively, so that \[ {AM\over AC}={CN\over CE}=r. \] Determine $r$ if $B,M$ and $N$ are collinear.

Let $ABCDEF$ be a hexagon having all interior angles equal to $120^o$ each. Let $P,Q,R, S, T, V$ be the midpoints of the sides of the hexagon $ABCDEF$. Prove the inequality $$p(PQRSTV ) \ge \frac{\sqrt3}{2} p(ABCDEF)$$, where $p(.)$ denotes the perimeter of the polygon. Nguyễn Tiến Lâm

An acute triangle $ ABC$ is given. Points $ A_1$ and $ A_2$ are taken on the side $ BC$ (with $ A_2$ between $ A_1$ and $ C$), $ B_1$ and $ B_2$ on the side $ AC$ (with $ B_2$ between $ B_1$ and $ A$), and $ C_1$ and $ C_2$ on the side $ AB$ (with $ C_2$ between $ C_1$ and $ B$) so that \[ \angle AA_1A_2 \equal{} \angle AA_2A_1 \equal{} \angle BB_1B_2 \equal{} \angle BB_2B_1 \equal{} \angle CC_1C_2 \equal{} \angle CC_2C_1.\] The lines $ AA_1,BB_1,$ and $ CC_1$ bound a triangle, and the lines $ AA_2,BB_2,$ and $ CC_2$ bound a second triangle. Prove that all six vertices of these two triangles lie on a single circle.

Let us call a convex hexagon $ABCDEF$ [i]boring [/i] if $\angle A+ \angle C + \angle E = \angle B + \angle D + \angle F$. a) Is every cyclic hexagon boring? b) Is every boring hexagon cyclic?

The hexagon $ABCDEF$ is inscribed in a circle, $AB = BC = a, CD = DE = b$, and $EF = FA = c$. Prove that the area of triangle $BDF$ equals half the area of the hexagon. (I.P. Nagel, Yevpatoria).

Call a convex polyhedron a [i]footballoid [/i] if it has the following properties. (1) Any face is either a regular pentagon or a regular hexagon. (2) All neighbours of a pentagonal face are hexagonal (a [i]neighbour [/i] of a face is a face that has a common edge with it). Find all possibilities for the number of pentagonal and hexagonal faces of a footballoid.

Let $ABCDEF$ be a regular hexagon with sidelength s. The points $P$ and $Q$ are on the diagonals $BD$ and $DF$, respectively, such that $BP = DQ = s$. Prove that the three points $C$, $P$ and $Q$ are on a line. [i](Walther Janous)[/i]

Given a regular hexagon, whose sidelength is $ 1$ . What is the largest number of circles of radius $\frac{\sqrt3}{4}$ can be placed without overlapping inside such a hexagon? (Circles can touch each other and the sides of the hexagon.)

Given convex hexagon $ABCDEF$, inscribed in the circle. Prove that $AC*BD*DE*CE*EA*FB \geq 27 AB * BC * CD * DE * EF * FA$

The vertices of an equilateral triangle lie on sides $ AB$, $CD$ and $EF$ of a regular hexagon $ABCDEF$. Prove that the triangle and the hexagon have a common centre. (N Sedrakyan, Yerevan )

The diagonals $AD, BE$ and $CF$ of a convex hexagon concur at a point $M$. Suppose the six triangles $ABM, BCM, CDM, DEM, EFM$ and $FAM$ are all acute-angled and the circumcentre of all these triangles lie on a circle. Prove that the quadrilaterals $ABDE, BCEF$ and $CDFA$ have equal areas.

Prove that in every convex hexagon of area $S$ one can draw a diagonal that cuts off a triangle of area not exceeding $\frac{1}{6}S.$

Let $ABCDEF$ be a hexagon with sides $AB,CD,EF$ being equal to $m$ units, sides $BC,DE, FA$ being equal to $n$ units. The diagonals $AD,BE,CF$ have lengths $x, y$, and $z$ units. Prove the inequality $$\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx} \ge \frac{3}{(m+ n)^2}$$ Nguyễn Văn Quý

Denote the midpoints of the convex hexagon $A_1A_2A_3A_4A_5A_6$ diagonals $A_6A_2$, $A_1A_3$, $A_2A_4$, $A_3A_5$, $A_4A_6$, $A_5A_1$ as $B_1, B_2, B_3, B_4, B_5, B_6$ respectively. Prove that if the hexagon $B_1B_2B_3B_4B_5B_6$ is convex, than its area equals to the quarter of the initial hexagon.

Let $ABCDEF$ be a regular hexagon with sides $1m$ and $O$ as its center. Suppose that $OPQRST$ is a regular hexagon, so that segments $OP$ and $AB$ intersect at $X$ and segments $OT$ and $CD$ intersect at $Y$, as shown in the figure below. Determine the area of the pentagon $OXBCY$.

Prove that if opposite sides of a hexagon are parallel and the diagonals connecting opposite vertices have equal lengths, a circle can be circumscribed around the hexagon.