Prove that all quadrilaterals $ABCD$ where $\angle B = \angle D = 90^o$, $|AB| = |BC|$ and $|AD| + |DC| = 1$, have the same area. [img]https://1.bp.blogspot.com/-55lHuAKYEtI/XzRzDdRGDPI/AAAAAAAAMUk/n8lYt3fzFaAB410PQI4nMEz7cSSrfHEgQCLcBGAsYHQ/s0/2016%2Bmohr%2Bp3.png[/img]

Prove that the volume $V$ and the lateral area $S$ of a right circular cone satisfy the inequality \[\left( \frac{6V}{\pi}\right)^2 \leq \left( \frac{2S}{\pi \sqrt 3}\right)^3\] When does equality occur?

A triangle with a parallelogram inside was placed in a square. Prove that the area of a parallelogram is not more than a quarter of a square.

In triangle $\vartriangle ABC$, $D$ is the midpoint of $BC$. Points $E$ and $F$ are chosen on side $AC$ so that $AF = F E = EC$. Let $AD$ intersect $BE$ and $BF$ and $G$ and $H$, respectively. Find the ratio of the areas of $\vartriangle BGH$ and $\vartriangle ABC$.

Let be given triangle $ABC$. Find all points $M$ such that area of $\vartriangle MAB$= area of $\vartriangle MAC$

Show that if $ A$ is a shape in the Cartesian coordinate plane with area greater than $ 1$, then there are distinct points $(a, b)$, $(c, d)$ in $A$ where $a - c = 2x + 5y$ and $b - d = x + 3y$ where $x, y$ are integers.

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)

Prove that if all four vertices of a parallelogram are lattice points and there are some other lattice points in or on the parallelogram, then its area exceeds $1$.

In the star shaped figure below, if all side lengths are equal to $3$ and the three largest angles of the figure are $210$ degrees, its area can be expressed as $\frac{a \sqrt{b}}{c}$ , where $a, b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime and that $b$ is square-free. Compute $a + b + c$. [img]https://cdn.artofproblemsolving.com/attachments/a/f/d16a78317b0298d6894c6bd62fbcd1a5894306.png[/img]

$H$ 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 $ABH$.

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 $X$.

Two squares, both with side length $1$, are arranged so that one has one vertex in the center of the other. Determine the area of the gray area. [img]https://1.bp.blogspot.com/-xt3pe0rp1SI/XzcGLgEw1EI/AAAAAAAAMYM/vFKxvvVuLvAJ5FO_yX315X3Fg_iFaK2fACLcBGAsYHQ/s0/1997%2BMohr%2Bp2.png[/img]

We are given trapezoid $ABCD$ and point $M$ on the intersection of its diagonals. The parallel sides are $AD$ and $BC$ and it is known that $AB$ is perpendicular to $AD$ and that the trapezoid can have an inscribed circle. If the radius of this inscribed circle is $R$ find the area of triangle $DCM$ .

Prove that any rigid flat triangle $T$ of area less than $4$ can be inserted through a triangular hole $Q$ with area $3$.

Find the smallest real number $A$, such that there are two different triangles, with integer sidelengths and so that the area of each be $A$.

$(NET 1)$ 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?

In a triangle $A B C$, let $E$ be the midpoint of $A C$ and $F$ be the midpoint of $A B$. The medians $B E$ and $C F$ intersect at $G$. Let $Y$ and $Z$ be the midpoints of $B E$ and $C F$ respectively. If the area of triangle $A B C$ is 480 , find the area of triangle $G Y Z$.

Let $BC$ be a chord of a circle $(O)$ such that $BC$ is not a diameter. Let $AE$ be the diameter perpendicular to $BC$ such that $A$ belongs to the larger arc $BC$ of $(O)$. Let $D$ be a point on the larger arc $BC$ of $(O)$ which is different from $A$. Suppose that $AD$ intersects $BC$ at $S$ and $DE$ intersects $BC$ at $T$. Let $F$ be the midpoint of $ST$ and $I$ be the second intersection point of the circle $(ODF)$ with the line $BC$. 1. Let the line passing through $I$ and parallel to $OD$ intersect $AD$ and $DE$ at $M$ and $N$, respectively. Find the maximum value of the area of the triangle $MDN$ when $D$ moves on the larger arc $BC$ of $(O)$ (such that $D \ne A$). 2. Prove that the perpendicular from $D$ to $ST$ passes through the midpoint of $MN$

Rosa and Sara play with a triangle $ABC$, right at $B$. Rosa begins by marking two interior points of the hypotenuse $AC$, then Sara marks an interior point of the hypotenuse $AC$ different from those of Rosa. Then, from these three points the perpendiculars to the sides $AB$ and $BC$ are drawn, forming the following figure. [img]https://cdn.artofproblemsolving.com/attachments/9/9/c964bbacc4a5960bee170865cc43902410e504.png[/img] Sara wins if the area of the shaded surface is equal to the area of the unshaded surface, in other case wins Rosa. Determine who of the two has a winning strategy.

In the triangle $ABC$, the median $BD$ is drawn and through its midpoint and vertex $A$ the line $\ell$. Thus the triangle $ABC$ is divided into three triangles and one quadrilateral. Determine the areas of these figures if the area of ​​triangle $ABC$ is equal to $S$.

In a right triangle with legs $a$, $b$ and hypotenuse $c$, draw semicircles with diameters on the sides of the triangle as indicated in the figure. The purple areas have values $X,Y$ . Calculate $X + Y$. [img]https://cdn.artofproblemsolving.com/attachments/1/a/5086dc7172516b0a986ef1af192c15eba4d6fc.png[/img]

The figure shows a large circle with radius $2$ m and four small circles with radii $1$ m. It is to be painted using the three shown colours. What is the cost of painting the figure? [img]https://1.bp.blogspot.com/-oWnh8uhyTIo/XzP30gZueKI/AAAAAAAAMUY/GlC3puNU_6g6YRf6hPpbQW8IE8IqMP3ugCLcBGAsYHQ/s0/2018%2BMohr%2Bp2.png[/img]

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 $S$.

Angle $C$ of an isosceles triangle $ABC$ equals $120^o$. Each of two rays emitting from vertex $C$ (inwards the triangle) meets $AB$ at some point ($P_i$) reflects according to the rule the angle of incidence equals the angle of reflection" and meets lateral side of triangle $ABC$ at point $Q_i$ ($i = 1,2$). Given that angle between the rays equals $60^o$, prove that area of triangle $P_1CP_2$ equals the sum of areas of triangles $AQ_1P_1$ and $BQ_2P_2$ ($AP_1 < AP_2$).

This figure is cut out from a sheet of paper. Folding the sides upwards along the dashed lines, one gets a (non-equilateral) pyramid with a square base. Calculate the area of the base. [img]https://1.bp.blogspot.com/-lPpfHqfMMRY/XzcBIiF-n2I/AAAAAAAAMW8/nPs_mLe5C8srcxNz45Wg-_SqHlRAsAmigCLcBGAsYHQ/s0/2005%2BMohr%2Bp1.png[/img]