Juku built a robot that moves along the border of a regular octagon, passing each side in exactly $1$ minute. The robot starts in some vertex $A$ and upon reaching each vertex can either continue in the same direction, or turn around and continue in the opposite direction. In how many different ways can the robot move so that after $n$ minutes it will be in the vertex $B$ opposite to $A$?

i) $ABCD$ is a rhombus. A tangent to the inscribed circle meets $AB, DA, BC, CD$ at $M, N, P, Q$ respectively. Find a relationship between $BM$ and $DN$. ii) $ABCD$ is a rhombus and $P$ a point inside. The circles through $P$ with centers $A, B, C, D$ meet the four sides $AB, BC, CD, DA$ in eight points. Find a property of the resulting octagon. Use it to construct a regular octagon. iii) Rotate the figure about the line $AC$ to form a solid. State a similar result.

Quadrilaterals may be obtained from an octagon by cutting along its diagonals (in $8$ different ways) . Can it happen that among these $8$ quadrilaterals (a) four (b ) five possess an inscribed circle? (P. M . Sedrakyan , Yerevan)

$ABCDEF GH$ is a regular octagon with $10$ units side . The circle with center $A$ and radius $AC$ intersects the circle with center $D$ and radius $CD$ at point $ I$, different from $C$. What is the length of the segment $IF$?

The vertices of the square $ABCD$ are the centers of four circles, all of which pass through the center of the square. Prove that the intersections of the circles on the square $ABCD$ sides are vertices of a regular octagon.

From a square of side length $1$, four identical triangles are removed, one at each corner, leaving a regular octagon. What is the area of the octagon?

Let $A_1A_2\ldots A_8$ be a convex cyclic octagon, and for $i=1,2\ldots,8$ let $B_i=A_iA_{i+3}\cap A_{i+1}A_{i+4}$ (indices are meant modulo 8). Prove that points $B_1,\ldots, B_8$ lie on the same conic section.

Given a square $ABCD$ with side length $6$. We draw line segments from the midpoints of each side to the vertices on the opposite side. For example, we draw line segments from the midpoint of side $AB$ to vertices $C$ and $D$. The eight resulting line segments together bound an octagon inside the square. What is the area of this octagon?

The lengths of the four sides of an cyclic octagon are $4$ cm, the lengths of the other four sides are $6$ cm. Find the area of ​​the octagon.

Given a regular octagon $ABCDEFGH$ with side length $3$. By drawing the four diagonals $AF$, $BE$, $CH$, and $DG$, the octagon is divided into a square, four triangles, and four rectangles. Find the sum of the areas of the square and the four triangles.

Regular octagon $ABCDEFGH$ has area $n$. Let $m$ be the area of quadrilateral $ACEG$. What is $\tfrac{m}{n}?$ $\textbf{(A) } \frac{\sqrt{2}}{4} \qquad \textbf{(B) } \frac{\sqrt{2}}{2} \qquad \textbf{(C) } \frac{3}{4} \qquad \textbf{(D) } \frac{3\sqrt{2}}{5} \qquad \textbf{(E) } \frac{2\sqrt{2}}{3}$

The regular octagon of the following figure is inscribed in a circle of radius $1$ and $P$ is a arbitrary point of this circle. Calculate the value of $PA^2 + PB^2 +...+ PH^2$. [img]https://cdn.artofproblemsolving.com/attachments/4/c/85e8e48c45970556077ac09c843193959b0e5a.png[/img]

Each side of a regular octagon is colored blue or yellow. In each step, the sides are simultaneously recolored as follows: if the two neighbors of a side have different colors, the side will be recolored blue, otherwise it will be recolored yellow. Show that after a finite number of moves all sides will be colored yellow. What is the least value of the number $N$ of moves that always lead to all sides being yellow?

Two equal squares, one with red sides, another with blue ones, give an octagon in intersection. Prove that the sum of red octagon sides lengths is equal to the sum of blue octagon sides lengths.

Compute the area of an octagon inscribed in a circle, whose four sides have length $1$ and the other four sides have length $2$.

All internal angles of a convex octagon $ABCDEFGH$ are equal to each other and the edges are alternatively equal: $$AB = CD = EF = GH,BC = DE = FG = HA$$ (we call such an octagon semiregular). The diagonals $AD$, $BE$, $CF$, $DG$, $EH$, $FA$, $GB$ and $HC$ divide the inside of the octagon into certain parts. Consider the part containing the centre of the octagon. If that part is an octagon, then this central octagon is semiregular (this is obvious). In this case we construct similar diagonals in the central octagon and so on. If, after several steps, the central figure is not an octagon, then the process stops. Prove that if the process never stops, then the initial octagon was regular. (A. Tolpygo, Kiev)

In the illustration, a regular hexagon and a regular octagon have been tiled with rhombuses. In each case, the sides of the rhombuses are the same length as the sides of the regular polygon. (a) Tile a regular decagon ($10$-gon) into rhombuses in this manner. (b) Tile a regular dodecagon ($12$-gon) into rhombuses in this manner. (c) How many rhombuses are in a tiling by rhombuses of a $2002$-gon? Justify your answer. [img]https://cdn.artofproblemsolving.com/attachments/8/a/8413e4e2712609eba07786e34ba2ce4aa72888.png[/img]

Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a right octagon with center $O$ and $\lambda_1$,$\lambda_2$, $\lambda_3$, $\lambda_4$ be some rational numbers for which: $\lambda_1 \overrightarrow{OA_1}+\lambda_2 \overrightarrow{OA_2}+\lambda_3 \overrightarrow{OA_3}+\lambda_4 \overrightarrow{OA_4} =\overrightarrow{o}$. Prove that $\lambda_1=\lambda_2=\lambda_3=\lambda_4=0$.

In the octagon below all sides have the length $1$ and all angles are equal. Determine the distance between the corners $A$ and $B$. [img]https://1.bp.blogspot.com/-i6TAFDvcQ8w/XzXCRhnV_kI/AAAAAAAAMVw/rKrQMfPYYJIaCwl8hhdVHdqO4fIn8O7cwCLcBGAsYHQ/s0/2011%2BMogh%2Bp2.png[/img]

Prove that an octagon, whose all angles are equal and all sides have rational length, has a center of symmetry.

Prove that any centrally symmetric convex octagon has a diagonal passing through the center of symmetry that is not parallel to any of its sides.

Consider a rectangle with sides $2a$ and $2b$, where $a > b$. There should be four congruent right triangles (one triangle at each vertex of this rectangle , whose legs are on the sides of the rectangle lie) must be cut off so that the remaining figure forms an octagon with sides of equal length. The side of the octagon is to be expressed in terms of a and $b$ and constructed from $a$ and $b$. Besides that it must be stated under which conditions the problem can be solved.

We divided a regular octagon into parallelograms. Prove that there are at least $2$ rectangles between the parallelograms.

In the octagon below all sides have the length $1$ and all angles are equal. Determine the distance between the corners $A$ and $B$. [img]https://1.bp.blogspot.com/-i6TAFDvcQ8w/XzXCRhnV_kI/AAAAAAAAMVw/rKrQMfPYYJIaCwl8hhdVHdqO4fIn8O7cwCLcBGAsYHQ/s0/2011%2BMogh%2Bp2.png[/img]

Let $Q$ be any point on a circle and let $P_1P_2P_3...P_8$ be a regular inscribed octagon. Prove that the sum of the fourth powers of the distances from $Q$ to the diameters $P_1P_5$, $P_2P_6$, $P_3P_7$, $P_4P_8$ is independent of the position of $Q$.