$\triangle ABC$ is an acute angled triangle. Perpendiculars drawn from its vertices on the opposite sides are $AD$, $BE$ and $CF$. The line parallel to $ DF$ through $E$ meets $BC$ at $Y$ and $BA$ at $X$. $DF$ and $CA$ meet at $Z$. Circumcircle of $XYZ$ meets $AC$ at $S$. Given, $\angle B=33 ^\circ.$ find the angle $\angle FSD $ with proof.

The points $A,B,C,D$ lie, in this order, on a circle $\omega$, where $AD$ is a diameter of $\omega$. Furthermore, $AB=BC=a$ and $CD=c$ for some relatively prime integers $a$ and $c$. Show that if the diameter $d$ of $\omega$ is also an integer, then either $d$ or $2d$ is a perfect square.

Let $ABCD$ be an isosceles trapezoid with $AB\parallel CD$. Let $E$ be the midpoint of $AC$. Denote by $\omega$ and $\Omega$ the circumcircles of the triangles $ABE$ and $CDE$, respectively. Let $P$ be the crossing point of the tangent to $\omega$ at $A$ with the tangent to $\Omega$ at $D$. Prove that $PE$ is tangent to $\Omega$. [i]Jakob Jurij Snoj, Slovenia[/i]

Given $8$ unit cubes, $24$ of their faces are painted in blue and the remaining $24$ faces in red. Show that it is always possible to assemble these cubes into a cube of edge $2$ on whose surface there are equally many blue and red unit squares.

A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$) and square tiles composed by four unit squares. Let $n\ge 0$ be the number of the $2 \times 2 $ tiles which can be used in such a tiling. Find all the values of $n$.

Let $A_1 = (0, 0)$, $B_1 = (1, 0)$, $C_1 = (1, 1)$, $D_1 = (0, 1)$. For all $i > 1$, we recursively define $$A_i =\frac{1}{2020} (A_{i-1} + 2019B_{i-1}),B_i =\frac{1}{2020} (B_{i-1} + 2019C_{i-1})$$ $$C_i =\frac{1}{2020} (C_{i-1} + 2019D_{i-1}), D_i =\frac{1}{2020} (D_{i-1} + 2019A_{i-1})$$ where all operations are done coordinate-wise. [img]https://cdn.artofproblemsolving.com/attachments/8/7/9b6161656ed2bc70510286dc8cb75cc5bde6c8.png[/img] If $[A_iB_iC_iD_i]$ denotes the area of $A_iB_iC_iD_i$, there are positive integers $a, b$, and $c$ such that $\sum_{i=1}^{\infty}[A_iB_iC_iD_i] = \frac{a^2b}{c}$, where $b$ is square-free and $c$ is as small as possible. Compute the value of $a + b + c$

A paper convex quadrilateral will be called [b]folding[/b] if there are points $P,Q,R,S$ on the interiors of segments $AB,BC,CD,DA$ respectively so that if we fold in the triangles $SAP, PBQ, QCR, RDS$, they will exactly cover the quadrilateral $PQRS$. In other words, if the folded triangles will cover the quadrilateral $PQRS$ but won't cover each other. Prove that if quadrilateral $ABCD$ is folding, then $AC\perp BD$ or $ABCD$ is a trapezoid.

Let $\vartriangle ABC$ be an acute triangle with altitudes $AD$, $BE$, $CF$ and orthocenter $H$. Circle $\odot V$ is the circumcircle of $\vartriangle DE F$. Let segments $FD$, $BH$ intersect at point $P$. Let segments $ED$, $HC$ intersect at point $Q$. Let $K$ be a point on $AC$ such that $VK \perp CF$. a) Prove that $\vartriangle PQH \sim \vartriangle AKV$. b) Let line $PQ$ intersect $\odot V$ at points $I,G$. Prove that points $B,I,H,G,C$ are concyclic [hide]with center the symmetric point $X$ of circumcenter $O$ of $\vartriangle ABC$ wrt $BC$.[/hide] [hide=PS.] There is a chance that those in the hide were not wanted in the problem, as I tried to understand the wording from a solutions' video. I couldn't find the original wording pdf or picture.[/hide] [img]https://cdn.artofproblemsolving.com/attachments/c/3/0b934c5756461ff854d38f51ef4f76d55cbd95.png[/img] [url=https://www.geogebra.org/m/cjduebke]geogebra file[/url]

Prove the following theorem: If two triangles have a common angle, then the sum of the sines of the angles will be larger in that triangle where the difference of the remaining two angles is smaller. On the basis of this theorem, determine the shape of that triangle for which the sum of the sines of its angles is a maximum.

The angle formed by the rays $y=x$ and $y=2x$ ($x \ge 0$) cuts off two arcs from a given parabola $y=x^2+px+q$. Prove that the projection of one arc onto the $x$-axis is shorter by $1$ than that of the second arc.

Let positive integers $a,\ b,\ c$ be pairwise coprime. Denote by $g(a, b, c)$ the maximum integer not representable in the form $xa+yb+zc$ with positive integral $x,\ y,\ z$. Prove that \[ g(a, b, c)\ge \sqrt{2abc}\] [i](M. Ivanov)[/i] [hide="Remarks (containing spoilers!)"] 1. It can be proven that $g(a,b,c)\ge \sqrt{3abc}$. 2. The constant $3$ is the best possible, as proved by the equation $g(3,3k+1,3k+2)=9k+5$. [/hide]

Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}.\] Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

A point $E$ lies on the extension of the side $AD$ of the rectangle $ABCD$ over $D$. The ray $EC$ meets the circumcircle $\omega$ of $ABE$ at the point $F\ne E$. The rays $DC$ and $AF$ meet at $P$. $H$ is the foot of the perpendicular drawn from $C$ to the line $\ell$ going through $E$ and parallel to $AF$. Prove that the line $PH$ is tangent to $\omega$. (A. Kuznetsov)

[color=darkred]Let $a$ , $b$ and $c$ be three complex numbers such that $a+b+c=0$ and $|a|=|b|=|c|=1$ . Prove that: \[3\le |z-a|+|z-b|+|z-c|\le 4,\] for any $z\in\mathbb{C}$ , $|z|\le 1\, .$[/color]

What is the area of the letter $O$ made by Dürer? The two circles have a unit radius. Their centers, or the angle of a triangle formed by an intersection point of the circles is $30^o$. [img]https://cdn.artofproblemsolving.com/attachments/b/c/fe052393871a600fc262bd60047433972ae1be.png[/img]

Let $\triangle ABC$ be a right-angled triangle with $\angle A = 90^{\circ}$, and $AB < AC$. Let points $D, E, F$ be located on side $BC$ such that $AD$ is the altitude, $AE$ is the internal angle bisector, and $AF$ is the median. Prove that $3AD + AF > 4AE$.

The circle $\omega$ passes through the vertices $B$ and $C$ of triangle $ABC$ and intersects its sides $AC,AB$ at points $A,E$, respectively. On the ray $BD$, a point $K$ such that $BK = AC$ is chosen , and on the ray $CE$, a point $L$ such that $CL = AB$ is chosen. Prove that the center $O$ of the circumscribed circle of the triangle $AKL$ lies on the circle $\omega$.

$ABCD$ is convex quadrilateral with $AB=CD$. $AC$ and $BD$ intersect in $O$. $X,Y,Z,T$ are midpoints of $BC,AD,AC,BD$. Prove, that circumcenter of $OZT$ lies on $XY$.

$ABCD$ is circumscribed in a circle $k$, such that $[ACB]=s$, $[ACD]=t$, $s<t$. Determine the smallest value of $\frac{4s^2+t^2}{5st}$ and when this minimum is achieved.

Let $ABCD$ be a square with circumcircle $\Gamma_1$. Let $P$ be a point on the arc $AC$ that also contains $B$. A circle $\Gamma_2$ touches $\Gamma_1$ in $P$ and also touches the diagonal $AC$ in $Q$. Let $R$ be a point on $\Gamma_2$ such that the line $DR$ touches $\Gamma_2$. Proof that $|DR| = |DA|$.

Describe all closed bounded figures $\Phi$ in the plane any two points of which are connectable by a semicircle lying in $\Phi$.

Convex hexagon $ABCDEF$ is drawn in the plane such that $ACDF$ and $ABDE$ are parallelograms with area 168. $AC$ and $BD$ intersect at $G$. Given that the area of $AGB$ is 10 more than the area of $CGB$, find the smallest possible area of hexagon $ABCDEF$.

Let $ABCD$ be a circumscribed quadrilateral and $M$ the centre of the incircle. There are points $P$ and $Q$ on the lines $MA$ and $MC$ such that $\angle CBA= 2\angle QBP.$ Prove that $\angle ADC = 2 \angle PDQ.$

While doing her homework for a Momentum Learning class, Valencia draws two intersecting segments $AB = 10$ and $CD = 7$ on a plane. Across all possible configurations of those two segments, determine the maximum possible area of quadrilateral $ACBD$.

Given an isosceles triangle. Find the set of the points inside the triangle such, that the distance from that point to the base equals to the geometric mean of the distances to the sides.