Let $ABC$ be a triangle with integral side lengths. The angle bisector drawn from $B$ and the altitude drawn from $C$ meet at point $P$ inside the triangle. Prove that the ratio of areas of triangles $APB$ and $APC$ is a rational number.

Consider a cyclic quadrilateral such that the midpoints of its sides form another cyclic quadrilateral. Prove that the area of the smaller circle is less than or equal to half the area of the bigger circle

Three circles that touch each other externally have all their centers on one fourth circle with radius $R$. Show that the total area of the three circle disks is smaller than $4\pi R^2$.

The circle touches the square and goes through its two vertices as shown in the figure. Find the area of the square. (Distance in the picture is measured horizontally from the midpoint of the side of the square.) [img]https://cdn.artofproblemsolving.com/attachments/7/5/ab4b5f3f4fb4b70013e6226ce5189f3dc2e5be.png[/img]

Let $[ABC]$ be an equilateral triangle and $P$ be a point on $AC$ such that $\overline{PC}= 7$. The straight line that passes through $P$ and is perpendicular to $AC$, intersects $CB$ at point $M$ and intersects $AB$ at point $Q$. The midpoint $N$ of $[MQ]$ is such that $\overline{BN} = 14$. Determine the side of the 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 $ABC$.

The diagonals $AC$ and $BD$ in the convex quadrilateral $ABCD$ intersect in $S$. Let $F_1$ and $F_2$ be the areas of $\vartriangle ABS$ and $\vartriangle CSD$. and let $F$ be the area of the quadrilateral $ABCD$. Show that $\sqrt{ F_1 }+\sqrt{ F_2}\le \sqrt{ F}$

Three cevians are drawn in a triangle that do not intersect at one point. In this case, $4$ triangles and $3$ quadrangles were formed. Find the sum of the areas of the quadrilaterals if the area of each of the four triangles is $8$.

The following is known about the quadrilateral $ABCD$: triangles $ABC$ and $CDA$ are equal in area, the area of triangle $BCD$ is $k$ times greater than the area of triangle $DAB$, the bisectors of angles $ABC$ and $CDA$ intersect on the diagonal $AC$, straight lines $AC$ and $BD$ are not perpendicular. Find the ratio $AC/BD$.

A triangle of area $1$ is cut out of paper. Prove that it can be bent along a straight segment so that the area of the resulting figure is less than $s_0$, where $s_0=\frac{\sqrt5-1}{2}$. Note. The value $s_0$ specified in the condition can be reduced (the smallest value of$s_0$ is unknown to the authors of the problem). If you manage to do this (and justify it), write.

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 $BQ_2P_2$ ($AP_1 < AP_2$).

A part of a forest park which is located between two roads has caught fire. The fire is spreading at a speed of $10$ kilometers per hour. If the distance between the starting point of the fire and both roads is $10$ kilometers, what is the area of the burned region after two hours in kilometers squared? (Assume that the roads are long, straight parallel lines and the fire does not enter the roads) $\textbf{(A)}\ 200\sqrt 3\qquad\textbf{(B)}\ 100 \sqrt 3\qquad\textbf{(C)}\ 400\sqrt 3 + 400 \frac{\pi}{3} \qquad\textbf{(D)}\ 200\sqrt 3 + 400 \frac{\pi}{3} \qquad\textbf{(E)}\ 400\sqrt 3 $

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 $MN$

The incircle of the triangle $ABC$, with the center $I$ , touches the sides $AB, AC$ and $BC$ in the points $K, L$ and $M$ respectively. Points $P$ and $Q$ are taken on the sides $AC$ and $BC$ respectively, such that $|AP| = |CL|$ and $|BQ| = |CM|$. Prove that the difference of areas of the figures $APIQB$ and $CPIQ$ is equal to the area of the quadrangle $CLIM$

In convex hexagon $ABCDEF, AB$ is parallel to $CF, CD$ is parallel to $BE$ and $EF$ is parallel to $AD$. Prove that the areas of triangles $ACE$ and $BDF$ are equal .

Consider the points $A_1$ and $A_2$ on the side $AB$ of the square $ABCD$ taken in such a way that $|AB| = 3 |AA_1| $ and $|AB| = 4 |A_2B|$, similarly consider points $B_1$ and $B_2, C_1$ and $C_2, D_1$ and $D_2$ respectively on the sides $BC$, $CD$ and $DA$. The intersection point of straight lines $D_2A_1$ and $A_2B_1$ is $E$, the intersection point of straight lines $A_2B_1$ and $B_2C_1$ is $F$, the intersection point of straight lines $B_2C_1$ and $C_2D_1$ is $G$ and the intersection point of straight lines $C_2D_1$ and $D_2A_1$ is $H$. Find the area of the square $EFGH$, knowing that the area of $ABCD$ is $1$.

Let $A$, $B$, $C$, $D$ be four points in the plane, with $C$ and $D$ on the same side of the line $AB$, such that $AC \cdot BD = AD \cdot BC$ and $\angle ADB = 90^{\circ}+\angle ACB$. Find the ratio \[\frac{AB \cdot CD}{AC \cdot BD}, \] and prove that the circumcircles of the triangles $ACD$ and $BCD$ are orthogonal. (Intersecting circles are said to be orthogonal if at either common point their tangents are perpendicuar. Thus, proving that the circumcircles of the triangles $ACD$ and $BCD$ are orthogonal is equivalent to proving that the tangents to the circumcircles of the triangles $ACD$ and $BCD$ at the point $C$ are perpendicular.)

Let $\vartriangle ABC$ be an equilateral triangle with side $1$. $1011$ points $P_1$, $P_2$, $P_3$, $...$, $P_{1011}$ on the side $AC$ and $1011$ points $Q_1$, $Q_2$, $Q_3$, $...$ ,$ Q_{1011}$ on side AB (see figure) in such a way as to generate $2023$ triangles of equal area. Find the length of the segment $AP_{1011}$. [img]https://cdn.artofproblemsolving.com/attachments/f/6/fea495c16a0b626e0c3882df66d66011a1a3af.png[/img] PS. Harder version of [url=https://artofproblemsolving.com/community/c4h3323135p30741470]2023 Chile NMO L1 P3[/url]

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

In the plane, a triangle is given. Determine all points $P$ in the plane such that each line through $P$ that divides the triangle into two parts with the same area must pass through one of the vertices of the triangle.

Let $ ABCD $ be a quadrilateral, whose diagonals intersect at $ O $. The triangles $ \triangle AOB $, $ \triangle BOC $, $ \triangle COD $ have areas $1, 2, 4$, respectively. Find the area of $ \triangle AOD $ and prove that $ ABCD $ is a trapezoid.

Three line segments, all of length $1$, form a connected figure in the plane. Any two different line segments can intersect only at their endpoints. Find the maximum area of the convex hull of the figure.

Let $n \ge 4$ be an integer. Find the maximum value of the area of a $n$-gon which is inscribed in the circle of radius $1$ and has two perpendicular diagonals.

Let $SABCDE$ be a pyramid whose base $ABCDE$ is a regular pentagon and whose other faces are acute triangles. The altitudes from $S$ to the base sides dissect them into ten triangles, colored red and blue alternatingly. Prove that the sum of the squared areas of the red triangles is equal to the sum of the squared areas of the blue triangles.

For any set $S$ of five points in the plane, no three of which are collinear, let $M(S)$ and $m(S)$ denote the greatest and smallest areas, respectively, of triangles determined by three points from $S$. What is the minimum possible value of $M(S)/m(S)$ ?