(a) Let $ABC$ be any triangle. Let $X, Y, Z$ be points on the sides $BC, CA, AB$ respectively. Suppose that $BX \leq XC, CY \leq YA, AZ \leq ZB$. Show that the area of the triangle $XYZ$ $\geq 1\slash 4$ times the area of $ABC.$ (b) Let $ABC$ be any triangle, and let $X, Y, Z$ be points on the sides $BC, CA, AB$ respectively. Using (a) or by any other method, show: One of the three corner triangles $AZY, BXZ, CYX$ has an area $\leq$ area of the triangle $XYZ.$

A triangle and a square are circumscribed around the unit circle. Prove that the intersection area is more than $3.4$. Is it possible to assert that it is more than $3.5$?

Let $K$ be a convex polygon in the plane and suppose that $K$ is positioned in the coordinate system in such a way that \[\text{area } (K \cap Q_i) =\frac 14 \text{area } K \ (i = 1, 2, 3, 4, ),\] where the $Q_i$ denote the quadrants of the plane. Prove that if $K$ contains no nonzero lattice point, then the area of $K$ is less than $4.$

Let $ABC$ be a triangle with a circumcircle $(K)$. A circle touching the sides $AB,AC$ is internally tangent to $(K)$ at $K_a$; two other points $K_b,K_c$ are defined in the same manner. Prove that the area of triangle $K_aK_bK_c$ does not exceed that of triangle $ABC$. Nguyen Minh Ha, Hanoi University of Education, Xuan Thuy, Cau Giay, Hanoi.

$ ABC$ is a triangle, the bisector of angle $ A$ meets the circumcircle of triangle $ ABC$ in $ A_1$, points $ B_1$ and $ C_1$ are defined similarly. Let $ AA_1$ meet the lines that bisect the two external angles at $ B$ and $ C$ in $ A_0$. Define $ B_0$ and $ C_0$ similarly. Prove that the area of triangle $ A_0B_0C_0 \equal{} 2 \cdot$ area of hexagon $ AC_1BA_1CB_1 \geq 4 \cdot$ area of triangle $ ABC$.

Three sides of a tetrahedron are right-angled triangles having the right angle at their common vertex. The areas of these sides are $A, B$, and $C$. Find the total surface area of the tetrahedron.

Let $ABC$ be an equilateral triangle. $N$ is a point on the side $AC$ such that $\vec{AC} = 7\vec{AN}$, $M$ is a point on the side $AB$ such that $MN$ is parallel to $BC$ and $P$ is a point on the side $BC$ such that $MP$ is parallel to $AC$. Find the ratio of areas $\frac{ (MNP)}{(ABC)}$

Show that if a circumscribable quadrilateral of sides $a,b,c,d$ has area $A= \sqrt{abcd},$ then it is also inscribable.

In the following diagram two sides of a square are tangent to a circle with diameter $8$. One corner of the square lies on the circle. There are positive integers $m$ and $n$ so that the area of the square is $m +\sqrt{n}$. Find $m + n$. [asy] import graph; size(4.4cm); real labelscalefactor = 0.5; pen dotstyle = black; filldraw((-0.707106781,0.707106781)--(-0.707106781,-1)--(1,-1)--(1,0.707106781)--cycle,gray, linewidth(1.4)); draw(circle((0,0),1), linewidth(1.4)); [/asy]

Prove that for every tetrahedron there exist two planes such that the projection areas on those planes ratio is not less than $\sqrt 2$.

In triangle $ABC$, points $D, E$, and $F$ intersect one-third of the respective sides. Show that the sum of the areas of the three gray triangles is equal to the area of middle triangle. [img]https://1.bp.blogspot.com/-KWrhwMHXfDk/XzcIkhWnK5I/AAAAAAAAMYk/Tj6-PnvTy9ksHgke8cDlAjsj2u421Xa9QCLcBGAsYHQ/s0/1993%2BMohr%2Bp4.png[/img]

In the plane, given an angle $Axy$. a) Given a triangle $MNP$ of area $T$, describe how to construct a triangle of given area $T$ and altitude $h$. Using this, describe how to construct parallelogram A$BCD$ with two sides lying on $Ax$ and $Ay$, the area $T$ and the distance between the two opposite sides equal to d given. b) From an arbitrary point $I$ on the line $CD$, construct a line that intersects the lines $A$B, $BC$, $AD$ at $E$, $G$ and $F$ respectively so that the area of triangle $AEF$ is equal to the area of parallelogram $ABCD$. c) Apply the above two sentences: Given any point $O$ in the plane. From $O$, construct a line that intersects two rays $Ax$ and $Ay$ at $E$ and $F$ respectively so that the area of triangle $AEF$ is equal to the area of any given triangle.

The point $D$ is the midpoint of the side $[AB]$ of the triangle $ABC$ . The points $E$ and $F$ belong to $[AC]$ and $[BC]$ respectively. Prove that the area of triangle $DEF$ area does not exceed the sum of the areas of triangles $ADE$ and $BDF$.

A triangular piece of sheet metal weighs $900$ g. Prove that by cutting this sheet metal along a straight line passing through the center of gravity of the triangle, it is impossible to cut off a piece weighing less than $400$ g.

Each of the sides $AB$ and $CD$ of a convex quadrilateral $ABCD$ is divided into three equal parts, $|AE| = |EF| = |F B|$ , $|DP| = |P Q| = |QC|$. The diagonals of $AEPD$ and $FBCQ$ intersect at $M$ and $N$, respectively. Prove that the sum of the areas of $\vartriangle AMD$ and $\vartriangle BNC$ is equal to the sum of the areas of $\vartriangle EPM$ and $\vartriangle FNQ$.

Let $ABCD$ be a rectangle. A line $r$ moves parallel to $AB$ and intersects diagonal $AC$ , forming two triangles opposite the vertex, inside the rectangle. Prove that the sum of the areas of these triangles is minimal when $r$ passes through the midpoint of segment $AD$ .

In the plane, the segments $AB$ and $CD$ are given, while the lines $AB$ and $CD$ intersect. Prove that the set of all points $P$ in the plane such that triangles $ABP$ and $CDP$ have equal areas , form two lines intersecting at the intersection of the lines $AB$ and $CD$.

Let $ABCD$ be a unit square and let $E$ and $F$ be points inside $ABCD$ such that the line containing $\overline{EF}$ is parallel to $\overline{AB}$. Point $E$ is closer to $\overline{AD}$ than point $F$ is to $\overline{AD}$. The line containing $\overline{EF}$ also bisects the square into two rectangles of equal area. Suppose $[AEF B] = [DEFC] = 2[AED] = 2[BFC]$. The length of segment $\overline{EF}$ can be expressed as $m/n$ , where m and $n$ are relatively prime positive integers. Compute $m + n$.

Given a rectangle, circumscribe a rectangle of maximum area.

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 $ABCD$.

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.

Point $P$ is $\sqrt3$ units away from plane $A$. Let $Q$ be a region of $A$ such that every line through $P$ that intersects $A$ in $Q$ intersects $A$ at an angle between $30^o$ and $60^o$ . What is the largest possible area of $Q$?

In the triangle $ ABC $, the point $ A' $ on the side $ BC $, the point $ B' $ on the side $ AC $, the point $ C' $ on the side $ AB $ are chosen so that the straight lines $ AA' $, $ CC' $ intersect at one point, i.e. equivalently $ |BA'| \cdot |CB'| \cdot |AC'| = |CA'| \cdot |AB'| \cdot |BC'| $. Prove that the area of triangle $ A'B'C' $ is not greater than $ 1/4 $ of the area of triangle $ ABC $.

The points $X, Y,Z$ are marked on the sides $AB, BC,AC$ of the triangle $ABC$, respectively. Points $A',B', C'$ are on the $XZ, XY, YZ$ sides of the triangle $XYZ$, respectively, so that $\frac{AB}{A'B'} = \frac{AB}{A'B'} =\frac{BC}{B'C'}= 2$ and $ABB'A',BCC'B',ACC'A'$ are trapezoids in which the sides of the triangle $ABC$ are bases. a) Determine the ratio between the area of the trapezium $ABB'A'$ and the area of the triangle $A'B'X$. b) Determine the ratio between the area of the triangle $XYZ$ and the area of the triangle $ABC$.

Let $ABCD$ be a convex quadrilateral. We assume that there exists a point $P$ inside the quadrilateral such that the areas of the triangles $ABP, BCP, CDP$, and $DAP$ are equal. Show that at least one of the diagonals of the quadrilateral bisects the other diagonal.