$H$ is the orthocentre of an acute–angled triangle $ABC$. A point $E$ is taken on the line segment $CH$ such that $ABE$ is a right–angled triangle. Prove that the area of the triangle $ABE$ is the geometric mean of the areas of triangles $ABC$ and $ABH$.

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)

Is there a planar polygon whose vertices have integer coordinates and whose area is $1/2$, such that this polygon is (a) a triangle with at least two sides longer than $1000$? (b) a triangle whose sides are all longer than $1000$? (c) a quadrangle?

A circle centered at a point $(0, 1)$ on the coordinate plane intersects the parabola $y = x^2$ at four points: $A, B, C, D.$ Find the largest possible value of the area of ​​the quadrilateral $ABCD$.

Let $ M,N,P,Q $ be points on the sides $ AB,BC,CD,DA $ (respectively) of a convex quadrilateral $ ABCD $ so that: $$ \frac{MA}{MB} =\frac{NB}{NC} =\frac{PD}{PC} =\frac{QA}{QD}\neq 1 $$ Show that the area of $ MNPQ $ is half the area of $ ABCD $ if and only if $ ABD $ and $ BCD $ have equal areas. [i]Petre Asaftei[/i]

Let $D, E$ and $F{}$ be the midpoints of the sides $BC, CA$ and $AB{}$ of the acute-angled triangle $ABC$ and let $H_a, H_b$ and $H_c{}$ be the orthocenters of the triangles $ABD, BCE$ and $CAF{}$ respectively. Prove that the triangles $H_aH_bH_c$ and $DEF$ have equal areas. [i]Proposed by Tran Quang Hung[/i]

Benedek draws circles with the same center in the following way. The first circle he draws has radius $1$. Next, he draws a second circle such that the ring between the first and second circles has twice the area of the first circle. Next, he draws a third circle such that the ring between the second and third circles is three times the area of the first circle, and so on (see the diagram). What is the smallest $n$ fow which the radius of the $n$-th circle is an integer greater than $1$? [img]https://cdn.artofproblemsolving.com/attachments/e/2/afa6d5ead6f2252aa821028370a3768912e674.png[/img]

A convex quadrilateral $ABCD$ is given. Let $O_a$ be the circumcenter of the triangle $DBC$, and define $O_b,O_c$ and $O_d$ similarly. The points $O_a, O_b, O_c, O_d$ are the vertices of a convex quadrilateral. Prove that its area is equal to half of the absolute value of the difference between the areas of $AO_bCO_d$ and $BO_cDO_a$. [i]Proposed by V. Dubrovsky[/i]

The figure shows an arbitrary (green) triangle in the center. White squares were built on its sides to the outside. Some of their vertices were connected by segments, white squares were built on them again to the outside, and so on. In the spaces between the squares, triangles and quadrilaterals were formed, which were painted in different colors. Prove that [list=a] [*]all colored quadrilaterals are trapezoids; [*]the areas of all polygons of the same color are equal; [*]the ratios of the bases of one-color trapezoids are equal; [*]if $S_0=1$ is the area of the original triangle, and $S_i$ is the area of the colored polygons at the $i^{\text{th}}$ step, then $S_1=1$, $S_2=5$ and for $n\geqslant 3$ the equality $S_n=5S_{n-1}-S_{n-2}$ is satisfied. [/list] [i]Proposed by F. Nilov[/i] [center][img width="40"]https://i.ibb.co/n8gt0pV/Screenshot-2023-03-09-174624.png[/img][/center]

Inside a given convex quadrilateral, find a point such that the segments connecting this point with the midpoints of the quadrilateral's sides divide the quadrilateral into four parts with equal areas.

Given a rectangle $ABCD$. Points $E$ and $F$ lie on sides $BC$ and $CD$ respectively so that the area of triangles $ABE$, $ECF$, $FDA$ is equal to $1$. Determine the area of triangle $AEF$.

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 $S$.

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?

In the plane are given $2000$ congruent triangles of area $1$, which are all images of one triangle under translations. Each of these triangles contains the centroid of every other triangle. Prove that the union of these triangles has area less than $22/9$.

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.

Find the ratio between the sum of the areas of the circles and the area of the fourth circle that are shown in the figure Each circle passes through the center of the previous one and they are internally tangent. [img]https://cdn.artofproblemsolving.com/attachments/d/2/29d2be270f7bcf9aee793b0b01c2ef10131e06.jpg[/img]

Given the points $O = (0, 0)$ and $A = (2024, -2024)$ in the plane. For any positive integer $n$, Damian draws all the points with integer coordinates $B_{i,j} = (i, j)$ with $0 \leq i, j \leq n$ and calculates the area of each triangle $OAB_{i,j}$. Let $S(n)$ denote the sum of the $(n+1)^2$ areas calculated above. Find the following limit: \[ \lim_{n \to \infty} \frac{S(n)}{n^3}. \]

Let $ABCD$ be a cyclic quadrilateral, $O$ be its circumcenter, $P$ be a common points of its diagonals, and $M , N$ be the midpoints of $AB$ and $CD$ respectively. A circle $OPM$ meets for the second time segments $AP$ and $BP$ at points $A_1$ and $B_1$ respectively and a circle $OPN$ meets for the second time segments $CP$ and $DP$ at points $C_1$ and $D_1$ respectively. Prove that the areas of quadrilaterals $AA_1B_1B$ and $CC_1D_1D$ are equal.

Let $\mathbf T$ be a triangle with $[\mathbf T]=1.$ Show that there is a right triangle $\mathbf R$ such that $[\mathbf R]\le\sqrt3$ and $\mathbf T\subseteq\mathbf R.$ ($[-]$ denotes area of a triangle.)

Is there a polyhedron whose area ratio of any two faces is at least $2$ ?

An interior point in a rectangle is connected by line segments to the midpoints of its four sides. Thus four domains (polygons) with the areas $a, b, c$ and $d$ appear (see the figure). Prove that $a + c = b + d$. [img]https://1.bp.blogspot.com/-BipDNHELjJI/XzcCa68P3HI/AAAAAAAAMXY/H2Iqya9VItMLXrRqsdyxHLTXCAZ02nEtgCLcBGAsYHQ/s0/2002%2BMohr%2Bp1.png[/img]

Given a convex pentagon $ABCDE$ with $AB=BC, CD=DE, \angle ABC=150^o, \angle CDE=30^o, BD=2$. Find the area of $ABCDE$. (I.Voronovich)

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 $M$ is the midpoint of the median $BD$ of the triangle $ABC$, the area of ​​which is $S$. The line $AM$ intersects the side $BC$ at the point $K$. Determine the area of ​​the triangle $BKM$.

The circle with the center of $ O $ touches the sides of the angle $ A $ at the points of $ K $ and $ M $. The tangent to the circle intersects the segments $ AK $ and $ AM $ at points $ B $ and $ C $ respectively, and the line $ KM $ intersects the segments $ OB $ and $ OC $ at the points $ D $ and $ E $. Prove that the area of the triangle $ ODE $ is equal to a quarter of the area of a triangle $ BOC $ if and only if the angle $ A $ is $ 60^\circ $.