An acute angle $XAY$ and a point $P$ inside the angle are given. Construct (using a ruler and a compass) a line that passes through $P$ and intersects the rays $AX$ and $AY$ at $B$ and $C$ such that the area of the triangle $ABC$ equals $AP^2$. [i]Greece[/i]

A convex quadrilateral $ABCD$ is such that $\angle B = \angle D$ and are both acute angles. $E$ is on $AB$ such that $CB = CE$ and $F$ is on $AD$ such that $CF = CD$. If the circumcenter of $CEF$ is $O_1$ and the circumcenter of $ABD$ is $O_2$. Prove that $C,O_1,O_2$ are collinear. [i]Proposed by Kapil Pause[/i]

Triangle $\vartriangle ABC$ has $AB = 13$, $BC = 14$, and $CA = 15$. $\vartriangle ABC$ has incircle $\gamma$ and circumcircle $\omega$. $\gamma$ has center at $I$. Line $AI$ is extended to hit $\omega$ at $P$. What is the area of quadrilateral $ABPC$?

The four congruent circles below touch one another and each has radius 1. [asy] unitsize(30); fill(box((-1,-1), (1, 1)), gray); filldraw(circle((1, 1), 1), white); filldraw(circle((1, -1), 1), white); filldraw(circle((-1, 1), 1), white); filldraw(circle((-1, -1), 1), white); [/asy] What is the area of the shaded region?

Let $ABCD$ be a trapezoid with $\angle A = \angle B = 90^o$ and a point $E$ lies on the segment $CD$. Denote $(\omega)$ as incircle of triangle $ABE$ and it is tangent to $AB,AE,BE$ respectively at $P, F,K$. Suppose that $KF$ cuts $BC,AD$ at $M,N$ and $PM,PN$ cut $(\omega)$ at $H, T$. Prove that $PH = PT$.

We consider a game on an infinite chessboard similar to that of solitaire: If two adjacent fields are occupied by pawns and the next field is empty (the three fields lie on a vertical or horizontal line), then we may remove these two pawns and put one of them on the third field. Prove that if in the initial position pawns fill a $3k \times n$ rectangle, then it is impossible to reach a position with only one pawn on the board.

A half-circle of diameter $EF$ is placed on the side $BC$ of a triangle $ABC$ and it is tangent to the sides $AB$ and $AC$ in the points $Q$ and $P$ respectively. Prove that the intersection point $K$ between the lines $EP$ and $FQ$ lies on the altitude from $A$ of the triangle $ABC$. [i]Albania[/i]

A triangle $ABC$ and a positive number $k$ are given. Find the locus of a point $M$ inside the triangle such that the projections of $M$ on the sides of $\Delta ABC$ form a triangle of area $k$.

Let $\triangle ABC$ be an acute-angled triangle with its incenter $I$. Suppose that $N$ is the midpoint of the arc $\overarc{BAC}$ of the circumcircle of triangle $\triangle ABC$, and $P$ is a point such that $ABPC$ is a parallelogram.Let $Q$ be the reflection of $A$ over $N$ and $R$ the projection of $A$ on $\overline{QI}$. Show that the line $\overline{AI}$ is tangent to the circumcircle of triangle $\triangle PQR$ [i]Proposed by Patrik Bak - Slovakia[/i]

There are $N{}$ friends and a round pizza. It is allowed to make no more than $100{}$ straight cuts without shifting the slices until all cuts are done; then the resulting slices are distributed among all the friends so that each of them gets a share off pizza having the same total area. Is there a cutting which gives the above result if a) $N=201$ and b) $N=400$?

Let $ABC$ be a triangle with circumcenter $O$, incenter $I$, $\angle B=45^\circ$, and $OI\parallel BC$. Find $\cos\angle C$.

In a triangle $ABC$ the point $D$ is the intersection of the interior angle bisector of $\angle BAC$ and side $BC$. Let $P$ be the second intersection point of the exterior angle bisector of $\angle BAC$ with the circumcircle of $\angle ABC$. A circle through $A$ and $P$ intersects line segment $BP$ internally in $E$ and line segment $CP$ internally in $F$. Prove that $\angle DEP = \angle DFP$.

Let $ABCD$ be a cyclic quadrilateral an let $P$ be a point on the side $AB.$ The diagonals $AC$ meets the segments $DP$ at $Q.$ The line through $P$ parallel to $CD$ mmets the extension of the side $CB$ beyond $B$ at $K.$ The line through $Q$ parallel to $BD$ meets the extension of the side $CB$ beyond $B$ at $L.$ Prove that the circumcircles of the triangles $BKP$ and $CLQ$ are tangent .

Given a convex pentagon $ABCDE$ with $\angle ABC= \angle ADE$ and $\angle AEC= \angle ADB$ . Prove that $\angle BAC = \angle DAE$ .

Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $30^\circ$. The area of face $ABC$ is $120$, the area of face $BCD$ is $80$, and $BC=10$. Find the volume of the tetrahedron.

Determine the values of $n \in N$ such that a square of side $n$ can be split into a square of side $1$ and five rectangles whose side measures are $10$ distinct natural numbers and all greater than $1$.

Let $\alpha,\beta,\gamma$ be positive real numbers such that $\alpha+\beta+\gamma < \pi$, $\alpha+\beta > \gamma$,$ \beta+\gamma > \alpha$, $\gamma + \alpha > \beta.$ Prove that with the segments of lengths $\sin \alpha, \sin \beta, \sin \gamma $ we can construct a triangle and that its area is not greater than \[A=\dfrac 18\left( \sin 2\alpha+\sin 2\beta+ \sin 2\gamma \right).\] [i]Proposed by Soviet Union[/i]

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?

Given $n$ lines on the plane, they divide the plane onto several bounded or bounded polygonal regions. Define the rank of a region as the number of vertices on its boundary (a vertex is a point which belongs to at least two lines). Prove that the sum of squares of ranks of all regions does not exceed $10n^2$. (D. Fomin)

[b]p1.[/b] Consider a parallelogram $ABCD$ with sides of length $a$ and $b$, where $a \ne b$. The four points of intersection of the bisectors of the interior angles of the parallelogram form a rectangle $EFGH$. A possible configuration is given below. Show that $$\frac{Area(ABCD)}{Area(EFGH)}=\frac{2ab}{(a - b)^2}$$ [img]https://cdn.artofproblemsolving.com/attachments/e/a/afaf345f2ef7c8ecf4388918756f0b56ff20ef.png[/img] [b]p2.[/b] A metal wire of length $4\ell$ inches (where $\ell$ is a positive integer) is used as edges to make a cardboard rectangular box with surface area $32$ square inches and volume $8$ cubic inches. Suppose that the whole wire is used. (i) Find the dimension of the box if $\ell= 9$, i.e., find the length, the width, and the height of the box without distinguishing the different orders of the numbers. Justify your answer. (ii) Show that it is impossible to construct such a box if $\ell = 10$. [b]p3.[/b] A Pythagorean n-tuple is an ordered collection of counting numbers $(x_1, x_2,..., x_{n-1}, x_n)$ satisfying the equation $$x^2_1+ x^2_2+ ...+ x^2_{n-1} = x^2_{n}.$$ For example, $(3, 4, 5)$ is an ordinary Pythagorean $3$-tuple (triple) and $(1, 2, 2, 3)$ is a Pythagorean $4$-tuple. (a) Given a Pythagorean triple $(a, b, c)$ show that the $4$-tuple $(a^2, ab, bc, c^2)$ is Pythagorean. (b) Extending part (a) or using any other method, come up with a procedure that generates Pythagorean $5$-tuples from Pythagorean $3$- and/or $4$-tuples. Few numerical examples will not suffice. You have to find a method that will generate infinitely many such $5$-tuples. (c) Find a procedure to generate Pythagorean $6$-tuples from Pythagorean $3$- and/or $4$- and/or $5$-tuples. Note. You can assume without proof that there are infinitely many Pythagorean triples. [b]p4.[/b] Consider the recursive sequence defined by $x_1 = a$, $x_2 = b$ and $$x_{n+2} =\frac{x_{n+1} + x_n - 1}{x_n - 1}, n \ge 1 .$$ We call the pair $(a, b)$ the seed for this sequence. If both $a$ and $b$ are integers, we will call it an integer seed. (a) Start with the integer seed $(2, 2019)$ and find $x_7$. (b) Show that there are infinitely many integer seeds for which $x_{2020} = 2020$. (c) Show that there are no integer seeds for which $x_{2019} = 2019$. [b]p5.[/b] Suppose there are eight people at a party. Each person has a certain amount of money. The eight people decide to play a game. Let $A_i$, for $i = 1$ to $8$, be the amount of money person $i$ has in his/her pocket at the beginning of the game. A computer picks a person at random. The chosen person is eliminated from the game and their money is put into a pot. Also magically the amount of money in the pockets of the remaining players goes up by the dollar amount in the chosen person's pocket. We continue this process and at the end of the seventh stage emerges a single person and a pot containing $M$ dollars. What is the expected value of $M$? The remaining player gets the pot and the money in his/her pocket. What is the expected value of what he/she takes home? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$ABC$ is a triangle with a circumscribed circle $k$, center $I$ of its inscribed circle $\omega$, and center $I_a$ of its excircle $\omega _a$, opposite to $A$. $\omega$ and $\omega _a$ are tangent to $BC$ in points $P$ and $Q$, respectively, and $S$ is the middle point of the arc $\widehat{BC}$ that doesn’t contain $A$. Consider a circle that is tangent to $BC$ in point $P$ and to $k$ in point $R$. Let $RI$ intersect $k$ for a second time in point $L$. Prove that, $LI_a$ and $SQ$ intersect in a point that lies on $k$.

Let $\Omega$ and $O$ be the circumcircle and the circumcentre of an acute-angled triangle $ABC$ with $AB > BC$. The angle bisector of $\angle ABC$ intersects $\Omega$ at $M \ne B$. Let $\Gamma$ be the circle with diameter $BM$. The angle bisectors of $\angle AOB$ and $\angle BOC$ intersect $\Gamma$ at points $P$ and $Q,$ respectively. The point $R$ is chosen on the line $P Q$ so that $BR = MR$. Prove that $BR\parallel AC$. (Here we always assume that an angle bisector is a ray.) [i]Proposed by Sergey Berlov, Russia[/i]

The diagram below shows a triangle divided into sections by three horizontal lines which divide the altitude of the triangle into four equal parts, and three lines connecting the top vertex with points that divide the opposite side into four equal parts. If the shaded region has area $100$, find the area of the entire triangle. [asy] import graph; size(5cm); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; filldraw((-1,2.5)--(-1,1.75)--(0.5,1.75)--(0,2.5)--cycle,grey); draw((-1,4)--(-2,1)); draw((-1,4)--(2,1)); draw((-2,1)--(2,1)); draw((-1,4)--(-1,1)); draw((-1,4)--(-0.5,2.5)); draw((-0.25,1.75)--(0,1)); draw((-1,2.5)--(-1,1.75)); draw((-1,1.75)--(0.5,1.75)); draw((0.5,1.75)--(0,2.5)); draw((0,2.5)--(-1,2.5)); draw((-1.25,3.25)--(-0.25,3.25)); draw((-1.5,2.5)--(0.5,2.5)); draw((1.25,1.75)--(-1.75,1.75)); draw((-1,4)--(0,2.5)); draw((0.47,1.79)--(1,1)); dot((-1,1),dotstyle); dot((0,1),dotstyle); dot((1,1),dotstyle); dot((-1.25,3.25),dotstyle); dot((-1.5,2.5),dotstyle); dot((-1.75,1.75),dotstyle); dot((1.25,1.75),dotstyle); dot((0.5,2.5),dotstyle); dot((-0.25,3.25),dotstyle); [/asy]

Let $ABC$ be a triangle and $m$ a line which intersects the sides $AB$ and $AC$ at interior points $D$ and $F$, respectively, and intersects the line $BC$ at a point $E$ such that $C$ lies between $B$ and $E$. The parallel lines from the points $A$, $B$, $C$ to the line $m$ intersect the circumcircle of triangle $ABC$ at the points $A_1$, $B_1$ and $C_1$, respectively (apart from $A$, $B$, $C$). Prove that the lines $A_1E$ , $B_1F$ and $C_1D$ pass through the same point. [i]Greece[/i]

Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$