Two equal chess-boards ($8\times 8$) have the same centre, but one is rotated by $45$ degrees with respect to another. Find the total area of black fields intersection, if the fields have unit length sides.

Let $ABC$ be a triangle and $O$ be its circumcircle. Let $A', B', C'$ be the midpoints of minor arcs $AB$, $BC$ and $CA$ respectively. Let $I$ be the center of incircle of $ABC$. If $AB = 13$, $BC = 14$ and $AC = 15$, what is the area of the hexagon $AA'BB'CC'$? Suppose $m \angle BAC = \alpha$ , $m \angle CBA = \beta$, and $m \angle ACB = \gamma$. [b]p10.[/b] Let the incircle of $ABC$ be tangent to $AB, BC$, and $AC$ at $J, K, L$, respectively. Compute the angles of triangles $JKL$ and $A'B'C'$ in terms of $\alpha$, $\beta$, and $\gamma$, and conclude that these two triangles are similar. [b]p11.[/b] Show that triangle $AA'C'$ is congruent to triangle $IA'C'$. Show that $AA'BB'CC'$ has twice the area of $A'B'C'$. [b]p12.[/b] Let $r = JL/A'C'$ and the area of triangle $JKL$ be $S$. Using the previous parts, determine the area of hexagon $AA'BB'CC'$ in terms of $ r$ and $S$. [b]p13.[/b] Given that the circumradius of triangle $ABC$ is $65/8$ and that $S = 1344/65$, compute $ r$ and the exact value of the area of hexagon $AA'BB'CC'$. PS. You had better use hide for answers.

Let $ABCD$ be a square of side $a$. On side $AD$ consider points $E$ and $Z$ such that $DE=a/3$ and $AZ=a/4$. If the lines $BZ$ and $CE$ intersect at point $H$, calculate the area of the triangle $BCH$ in terms of $a$.

In a convex hexagon $ABCDEF, AB$ is parallel to $DE, BC$ is parallel to $EF$ and $CD$ is parallel to $FA$. Prove that the triangles $ACE$ and $BDF$ have the same area.

Let $A$ be a vertex of a regular star-shaped pentagon, the angle at $A$ being less than $180^o$ and the broken line $AA_1BB_1CC_1DD_1EE_1$ being its contour. Lines $AB$ and $DE$ meet at $F$. Prove that polygon $ABB_1CC_1DED_1$ has the same area as the quadrilateral $AD_1EF$. Note: A regular star pentagon is a figure formed along the diagonals of a regular pentagon.

Let $P$ the intersection point of the diagonals of a convex quadrilateral $ABCD$. It is known that the area of triangles $ABC$, $BCD$ and $DAP$ is equal to $8 cm^2$, $9 cm^2$ and $10 cm^2$. Find the area of the quadrilateral $ABCD$.

$(NET 1)$ The vertices of an $(n + 1)-$gon are placed on the edges of a regular $n-$gon so that the perimeter of the $n-$gon is divided into equal parts. How does one choose these $n + 1$ points in order to obtain the $(n + 1)-$gon with $(a)$ maximal area; $(b)$ minimal area?

On hyperbola $y=\frac{1}{x}$ points $A_1,\ldots,A_{10}$ are chosen such that $(A_i)_x=2^{i-1}a$, where $a$ is some positive constant. Find the area of $A_1A_2 \ldots A_{10}$

The point $M$ inside a convex quadrilateral $ABCD$ is equidistant from the lines $AB$ and $CD$ and is equidistant from the lines $BC$ and $AD$. The area of $ABCD$ occurred to be equal to $MA\cdot MC +MB \cdot MD$. Prove that the quadrilateral $ABCD$ is a) tangential (circumscribed), b) cyclic (inscribed). (Nairi Sedrakyan)

Let be a point $ P $ in the interior of a triangle $ ABC. $ The lines $ AP,BP,CP $ meet $ BC,AC, $ respectively, $ AB $ at $ A_1,B_1, $ respectively, $ C_1. $ If $$ \mathcal{A}_{PBA_1} +\mathcal{A}_{PCB_1} +\mathcal{A}_{PAC_1} =\frac{1}{2}\mathcal{A}_{ABC} , $$ show that $ P $ lies on a median of $ ABC. $ $ \mathcal{A} $ [i]denotes area.[/i]

Sides $AB$ and $CD$ of quadrilateral $ABCD$ intersect at point $E$. On the diagonals$ AC$ and $BD$ points $M$ and $N$ are taken, respectively, so that $AM / AC = BN / BD = k$. Find the area of ​​a triangle $EMN$ if the area of ​​$ABCD$ is $S$.

Suppose four coplanar points $A, B, C$, and $D$ satisfy $AB = 3$, $BC = 4$, $CA = 5$, and $BD = 6$. Determine the maximal possible area of $\vartriangle ACD$.

Given points $A' ,B' ,C' ,D',$ on the extension of the $[AB], [BC], [CD], [DA]$ sides of the convex quadrangle $ABCD$, such, that the following pairs of vectors are equal: $$[BB']=[AB], [CC']=[BC], [DD']=[CD], [AA']=[DA].$$ Prove that the quadrangle $A'B'C'D'$ area is five times more than the quadrangle $ABCD$ area.

a) Prove that the line dividing the triangle onto two polygons with equal perimeters and equal areas passes through the centre of the inscribed circle. b) Prove the same statement for the arbitrary tangential polygon. c) Prove that all the lines halving its perimeter and area simultaneously, intersect in one point.

A right-angled triangle made of paper is folded along a straight line so that the vertex at the right angle coincides with one of the other vertices of the triangle and a quadrilateral is obtained . (a) What is the ratio into which the diagonals of this quadrilateral divide each other? (b) This quadrilateral is cut along its longest diagonal. Find the area of the smallest piece of paper thus obtained if the area of the original triangle is $1$ . (A Shapovalov)

Let $ABCDE$ be a convex pentagon inscribed in a circle $\Gamma$, such that $AB = BC = CD$. Let $F$ and $G$ be the intersections of $BE$ with $AC$ and of $CE$ with $BD$, respectively. Show that: a) $[ABC] = [FBCG]$ b) $\frac{[EFG]}{[EAD]} = \frac{BC}{AD}$ [b]Note: [/b] $[X]$ denotes the area of polygon $X$.

Triangle $ABC$ of area $1$ is given. Point $A'$ lies on the extension of side $BC$ beyond point $C$ with $BC = CA'$. Point $B'$ lies on extension of side $CA$ beyond $A$ and $CA = AB'$. $C'$ lies on extension of $AB$ beyond $B$ with $AB = BC'$. Find the area of triangle $A'B'C'$.

In the triangle $ABC$ on the sides $AB$ and $AC$, points $D$ and E are chosen, respectively. Can the segments $CD$ and $BE$ divide $ABC$ into four parts of the same area? [img]https://cdn.artofproblemsolving.com/attachments/1/c/3bbadab162b22530f1b254e744ecd068dea65e.png[/img]

Points $A_0,A_1,\ldots,A_{2k}$, in this order, divide a circumference into $2k+1$ equal arcs. Point $A_0$ is connected by chords to all the other points. These $2k$ chords divide the interior of the circle into $2k+1$ parts. These parts are alternately painted red and blue so that there are $k+1$ red and $k$ blue parts. Show that the blue area is larger than the red area.

The sides of the parallelogram serve as the diagonals of the four squares. The vertices of the squares lying in the part of the plane external to the parallelogram (the sides of the squares emerging from these vertices do not have common points with the parallelogram) serve as the vertices of a quadrilateral of area $a$, the four vertices opposite to them form a quadrilateral of area $b$. Find the area of the parallelogram.

A trapezoid $ABCD$ with perpendicular diagonals $AC$ and $BD$ is inscribed in a circle $k$. Let $k_a$ and $k_c$ respectively be the circles with diameters $AB$ and $CD$. Compute the area of the region which is inside the circle $k$, but outside the circles $k_a$ and $k_c$.

The Order "For Faithful Service" of the $7$th degree in shape is a combination of a semicircle with a diameter $AB = 2$ and a triangle $AM B$. The sides$ AM$ and $BM$ intersect the semicircle (the border of the semicircle). The part of the circle outside the triangle and the part of the triangle outside the circle are made of pure copper. What should the side of the triangle be equal to in order for the area of the copper part to be the smallest?

An infinite rectangular stripe of width $3$ cm is folded along a line. What is the minimum possible area of the region of overlapping?

Five square carpets have been bought for a square hall with a side of $6$ m , two with the side $2$ m, one with the side $2.1$ m and two with the side $2.5$ m. Is it possible to place the five carpets so that they do not overlap in any way each other? The edges of the carpets do not have to be parallel to the cradles in the hall.

We choose a point on each side of a parallelogram $ABCD$, let these four points be $P, Q, R$ and $S$. Then we divide the parallelogram into several regions using line segments as shown in the figure. The areas of the grey regions are given, except for one (see the figure). Find the area of the region marked with a question mark. [img]https://cdn.artofproblemsolving.com/attachments/4/7/dbd009042dabdb2eafc8fc74960e9011038dae.png[/img]