In trapezoid $ABCD$, the diagonals intersect at $E$, the area of $\triangle ABE$ is 72 and the area of $\triangle CDE$ is 50. What is the area of trapezoid $ABCD$?

In a rectangular coordinate system we call a horizontal line parallel to the $x$ -axis triangular if it intersects the curve with equation \[y = x^4 + px^3 + qx^2 + rx + s\] in the points $A,B,C$ and $D$ (from left to right) such that the segments $AB, AC$ and $AD$ are the sides of a triangle. Prove that the lines parallel to the $x$ - axis intersecting the curve in four distinct points are all triangular or none of them is triangular.

A given tetrahedron $ ABCD$ is isoceles, that is, $ AB\equal{}CD$, $ AC\equal{}BD$, $ AD\equal{}BC$. Show that the faces of the tetrahedron are acute-angled triangles.

A regular hexagon and an equilateral triangle have equal perimeter. If the area of the triangle is $4\sqrt3$ square units, the area of the hexagon is (A): $5\sqrt3$, (B): $6\sqrt3$, (C): $7\sqrt3$, (D): $8\sqrt3$, (E): None of the above.

The diagonals of a trapezoid $ ABCD$ intersect at point $ P$. Point $ Q$ lies between the parallel lines $ BC$ and $ AD$ such that $ \angle AQD \equal{} \angle CQB$, and line $ CD$ separates points $ P$ and $ Q$. Prove that $ \angle BQP \equal{} \angle DAQ$. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]

Let $ABC$ be a triangle. Prove that there exists a unique point $P$ for which one can find points $D$, $E$ and $F$ such that the quadrilaterals $APBF$, $BPCD$, $CPAE$, $EPFA$, $FPDB$, and $DPEC$ are all parallelograms. [i]Proposed by Lewis Chen[/i]

We have an $n \times n$ board, filled with $n$ rectangles aligned to the grid. The $n$ rectangles cover all the board and are never superposed. Find, in terms of $n$, the smallest value the sum of the $n$ diagonals of the rectangles can take.

Prove this proposition: Center the sphere circumscribed around a tetrahedron which coincides with the center of a sphere inscribed in that tetrahedron if and only if the skew edges of the tetrahedron are equal.

With the medians of an acute-angled triangle another triangle is constructed. If $R$ and $R_m$ are the radii of the circles circumscribed about the first and the second triangle, respectively, prove that \[R_m>\frac{5}{6}R\]

The sides of a triangle are three consecutive integers and its inradius is $4$. Find the circumradius.

In a trapezoid $ABCD$ with $AB // CD$, the diagonals $AC$ and $BD$ intersect at point $E$. Let $F$ and $G$ be the orthocenters of the triangles $EBC$ and $EAD$. Prove that the midpoint of $GF$ lies on the perpendicular from $E$ to $AB$.

Tetris is a Soviet block game developed in $1984$, probably to torture misbehaving middle school children. Nowadays, Tetris is a game that people play for fun, and we even have a mini-event featuring it, but it shall be used on this test for its original purpose. The $7$ Tetris pieces, which will be used in various problems in this theme, are as follows: [img]https://cdn.artofproblemsolving.com/attachments/b/c/f4a5a2b90fcf87968b8f2a1a848ad32ef52010.png[/img] [b]p1.[/b] Each piece has area $4$. Find the sum of the perimeters of each of the $7$ Tetris pieces. [b]p2.[/b] In a game of Tetris, Qinghan places $4$ pieces every second during the first $2$ minutes, and $2$ pieces every second for the remainder of the game. By the end of the game, her average speed is $3.6$ pieces per second. Find the duration of the game in seconds. [b]p3.[/b] Jeff takes all $7$ different Tetris pieces and puts them next to each other to make a shape. Each piece has an area of $4$. Find the least possible perimeter of such a shape. [b]p4.[/b] Qepsi is playing Tetris, but little does she know: the latest update has added realistic physics! She places two blocks, which form the shape below. Tetrominoes $ABCD$ and $EFGHI J$ are both formed from $4$ squares of side length $1$. Given that $CE = CF$, the distance from point $I$ to the line $AD$ can be expressed as $\frac{A\sqrt{B}-C}{D}$ . Find $1000000A+10000B +100C +D$. [img]https://cdn.artofproblemsolving.com/attachments/9/a/5e96a855b9ebbfd3ea6ebee2b19d7c0a82c7c3.png[/img] [b]p5.[/b] Using the following tetrominoes: [img]https://cdn.artofproblemsolving.com/attachments/3/3/464773d41265819c4f452116c1508baa660780.png[/img] Find the number of ways to tile the shape below, with rotation allowed, but reflection disallowed: [img]https://cdn.artofproblemsolving.com/attachments/d/6/943a9161ff80ba23bb8ddb5acaf699df187e07.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let A and B be fixed points in the plane with distance AB = 1. An ant walks on a straight line from point A to some point C in the plane and notices that the distance from itself to B always decreases at any time during this walk. Compute the area of the region in the plane containing all points where point C could possibly be located.

Let SABC be a regular triangular pyramid. Find the set of all points $ D (D! \equal{} S)$ in the space satisfing the equation $ |cos ASD \minus{} 2cosBSD \minus{} 2 cos CSD| \equal{} 3$.

The diagonals of a convex quadrilateral $ABCD$ are orthogonal and intersect at a point $E$. Prove that the projections of $E$ on $AB,BC,CD,DA$ are concyclic.

a) Let $AB CD$ be a regular tetrahedron. On the sides $AB$, $AC$ and $AD$, the points $M$, $N$ and $P$, are considered. Determine the volume of the tetrahedron $AMNP$ in terms of $x, y, z$, where $x=AM$, $y=AN$, $z=AP$. b) Show that for any real numbers $x, y, z, t, u, v \in (0, 1)$ : $$xyz + uv(1- x) + (1- y)(1- v)t + (1- z)(1- w)(1- t) < 1.$$

In $\triangle ABC$ let $I$ be the center of the inscribed circle, and let the bisector of $\angle ACB$ intersect $AB$ at $L$. The line through $C$ and $L$ intersects the circumscribed circle of $\triangle ABC$ at the two points $C$ and $D$. If $LI = 2$ and $LD = 3$, then $IC = \tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

Let $ ABCD$ be a cyclic quadrilateral and $ r$ and $ s$ the lines obtained reflecting $ AB$ with respect to the internal bisectors of $ \angle CAD$ and $ \angle CBD$, respectively. If $ P$ is the intersection of $ r$ and $ s$ and $ O$ is the center of the circumscribed circle of $ ABCD$, prove that $ OP$ is perpendicular to $ CD$.

Let $ABC$ be an acute angled triangle. Let $D,E,F$ be points on $BC, CA, AB$ such that $AD$ is the median, $BE$ is the internal bisector and $CF$ is the altitude. Suppose that $\angle FDE=\angle C, \angle DEF=\angle A$ and $\angle EFD=\angle B.$ Show that $ABC$ is equilateral.

Points $X$ and $Y$ on the unit circle centered at $O = (0,0)$ are at $(-1,0)$ and $(0,-1)$ respectively. Points $P$ and $Q$ are on the unit circle such that $\angle P XO = \angle QY O = 30^o$. Let $Z$ be the intersection of line $X P$ and line $Y Q$. The area bounded by segment $Z P$, segment $ZQ$, and arc $PQ$ can be expressed as $a\pi -b$ where $a$ and $b$ are rational numbers. Find $\frac{1}{ab}$ .

Let $T$ be an isosceles right triangle. Let $S$ be the circle such that the difference in the areas of $T \cup S$ and $T \cap S$ is the minimal. Prove that the centre of $S$ divides the altitude drawn on the hypotenuse of $T$ in the golden ratio (i.e., $\frac{(1 + \sqrt{5})}{2}$)

For a scalene triangle $ABC$ with an incenter $I$, let its incircle meets the sides $BC, CA, AB$ at $D, E, F$, respectively. Denote by $P$ the intersection of the lines $AI$ and $DF$, and $Q$ the intersection of the lines $BI$ and $EF$. Prove that $\overline{PQ}=\overline{CD}$.

Let $ABC$ be a triangle with $AB < BC$, and let $E$ and $F$ be points in $AC$ and $AB$ such that $BF = BC = CE$, both on the same halfplane as $A$ with respect to $BC$. Let $G$ be the intersection of $BE$ and $CF$. Let $H$ be a point in the parallel through $G$ to $AC$ such that $HG = AF$ (with $H$ and $C$ in opposite halfplanes with respect to $BG$). Show that $\angle EHG = \frac{\angle BAC}{2}$.

In a two-dimensional cave with a parallel floor and ceiling, two stalactites of lengths $16$ and $36$ hang perpendicularly from the ceiling, while two stalagmites of heights $25$ and $49$ grow perpendicularly from the ground. If the tips of these four structures form the vertices of a square in some order, compute the height of the cave.

Let $ABC$ be a scalene acute triangle, where $AL$ $(L \in BC)$ is the internal bisector of $\angle BAC$ and $M$ is the midpoint of $BC$. Let the internal bisectors of $\angle AMB$ and $\angle CMA$ intersect $AB$ and $AC$ in $P$ and $Q$, respectively. Prove that the circumcircle of $APQ$ is tangent to $BC$ if and only if $L$ belongs to it.