The hexagon $ABCDEF$ has all angles equal . We know that four consecutive sides of the hexagon have lengths $7, 6, 3$ and $5$ in this order. What is the sum of the lengths of the two remaining sides?

In a convex cyclic hexagon $ABCDEF$, $BC=EF$ and $CD=AF$. Diagonals $AC$ and $BF$ intersect at point $Q,$ and diagonals $EC$ and $DF$ intersect at point $P.$ Points $R$ and $S$ are marked on the segments $DF$ and $BF$ respectively so that $FR=PD$ and $BQ=FS.$ [b]The segments[/b] $RQ$ and $PS$ intersect at point $T.$ Prove that the line $TC$ bisects the diagonal $DB$.

a) Points $A_1, A_2, A_3, A_4, A_5, A_6$ divide a circle of radius $1$ into six equal arcs. Ray $\ell_1$ from $A_1$ connects $A_1$ with $A_2$, ray $\ell_2$ from $A_2$ connects $A_2$ with $A_3$, and so on, ray $\ell_6$ from $A_6$ connects $A_6$ with $A_1$. From a point $B_1$ on $\ell_1$ the perpendicular is drawn on $\ell_6$, from the foot of this perpendicular another perpendicular is drawn on $\ell_5$, and so on. Let the foot of the $6$-th perpendicular coincide with $B_1$. Find the length of segment $A_1B_1$. b) Find points $B_1, B_2,... , B_n$ on the extensions of sides $A_1A_2, A_2A_3,... , A_nA_1$ of a regular $n$-gon $A_1A_2...A_n$ such that $B_1B_2 \perp A_1A_2$, $B_2B_3 \perp A_2A_3$,$ . . . $, $B_nB_1 \perp A_nA_1$.

What are the possible areas of a hexagon with all angles equal and sides $1, 2, 3, 4, 5$, and $6$, in some order?

In a regular hexagon $ABCDEF$ with center $O$, points $M$ and $N$ are the midpoints of the sides $CD$ and $DE$, and $L$ the intersection point of $AM$ and $BN$. Prove that: (a) $ABL$ and $DMLN$ have equal areas; (b) $\angle ALD=\angle OLN=60^\circ$; (c) $\angle OLD=90^\circ$.

The distances from a certain point inside a regular hexagon to three of its consecutive vertices are equal to $1, 1$ and $2$, respectively. Determine the length of this hexagon's side. (Mikhail Evdokimov)

The midpoints of the opposite sides of the hexagon are connected by segments. It turned out that the points of pairwise intersection of these segments form an equilateral triangle. Prove that the drawn segments are equal. (M. Volchkevich)

In the convex $ABCDEF$ (has all interior angles less than $180^o$) with the perimeter $s$ the triangles $ACE$ and $BDF$ have perimeters $u$ and $v$ respectively. a) Show the inequalities $\frac{1}{2} \le \frac{s}{u+v}\le 1$ b) Check whether $1$ is replaced by a smaller number or $1/2$ by a larger number can the inequality remains valid for all convex hexagons.

The surface of a soccer ball is made up of black pentagons and white hexagons together. On the sides of each pentagon are nothing but hexagons, while on the sides of each border of hexagons alternately pentagons and hexagons. Determine from this information about the soccer ball , the number of its pentagons and its hexagons.

A regular hexagon of side $5$ is cut into unit equilateral triangles by lines parallel to the sides of the hexagon. We call the vertices of these triangles knots. If more than half of all knots are marked, show that there exist five marked knots that lie on a circle.

Let $ABCDEF$ be a convex hexagon inscribed in a circle . Prove that the diagonals $AD, BE$ and $CF$ intersect at one point if and only if $$\frac{AB}{BC} \cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$$

What are the possible areas of a hexagon with all angles equal and sides $1, 2, 3, 4, 5$, and $6$, in some order?

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

a) Given a circle with two inscribed triangles $T_1$ and $T_2$. The vertices of $T_1$ are the midpoints of the arcs with the ends in the vertices of $T_2$. Consider a hexagon -- the intersection of $T_1$ and $T_2$. Prove that its main diagonals are parallel to $T_1$ sides and are intersecting in one point. b) The segment, that connects the midpoints of the arcs $AB$ and $AC$ of the circle circumscribed around the $ABC$ triangle, intersects $[AB]$ and $[AC]$ sides in $D$ and $K$ points. Prove that the points $A,D,K$ and $O$ -- the centre of the circle -- are the vertices of a diamond.

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 .

A hexagon is inscribed in a circle of radius $r$. Two of the sides of the hexagon have length $1$, two have length $2$ and two have length $3$. Show that $r$ satisfies the equation $2r^3 - 7r - 3 = 0$.

A paper square is bent along the line $\ell$, which passes through its center, so that a non-convex hexagon is formed. Investigate the question of the circle of largest radius that can be placed in such a hexagon.

Consider the hexagon which is formed by the vertices of two equilateral triangles (not necessarily equal) when the triangles intersect. Prove that the area of the hexagon is unchanged when one of the triangles is translated (without rotating) relative to the other in such a way that the hexagon continues to be defined. (V Proizvolov)

The lengths of the sides of a convex hexagon $ ABCDEF$ satisfy $ AB \equal{} BC$, $ CD \equal{} DE$, $ EF \equal{} FA$. Prove that: \[ \frac {BC}{BE} \plus{} \frac {DE}{DA} \plus{} \frac {FA}{FC} \geq \frac {3}{2}. \]

A plane is tiled with regular hexagons of side $1$. $A$ is a fixed hexagon vertex. Find the number of paths $P$ such that: (1) one endpoint of $P$ is $A$, (2) the other endpoint of $P$ is a hexagon vertex, (3) $P$ lies along hexagon edges, (4) $P$ has length $60$, and (5) there is no shorter path along hexagon edges from $A$ to the other endpoint of $P$.

Let $ABCDEF$ be a convex hexagon with $AB=DE$, $BC=EF$, $CD=FA$, and $\angle A-\angle D = \angle C -\angle F = \angle E -\angle B$. Prove that the diagonals $AD$, $BE$, and $CF$ are concurrent.

Let $ABCDEF$ be a hexagon inscribed in a circle (with vertices in that order) with $\angle B + \angle C > 180^o$ and $\angle E + \angle F > 180^o$. Let the lines $AB$ and $CD$ intersect at $X$ and the lines $AF$ and $DE$ intersect at $S$. Let $XY$ and $ST$ be the diameters of the circumcircles of $\vartriangle BCX$ and $\vartriangle EFS$ respectively. If $U$ is the intersection point of the lines $BX$ and $ES$ and $V$ is the intersection point of the lines $BY$ and $ET,$ prove that the lines $UV, XY$ and $ST$ are all parallel.

Given convex hexagon $ABCDEF$ with $AB \parallel DE$, $BC \parallel EF$, and $CD \parallel FA$ . The distance between the lines $AB$ and $DE$ is equal to the distance between the lines $BC$ and $EF$ and to the distance between the lines $CD$ and $FA$. Prove that the sum $AD+BE+CF$ does not exceed the perimeter of hexagon $ABCDEF$.

The height of a truncated cone is equal to the radius of its base. The perimeter of a regular hexagon circumscribing its top is equal to the perimeter of an equilateral triangle inscribed in its base. Find the angle $\phi$ between the cone’s generating line and its base.

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