Let $ABCD$ be a circumscribed quadrilateral. Its incircle $\omega$ touches the sides $BC$ and $DA$ at points $E$ and $F$ respectively. It is known that lines $AB,FE$ and $CD$ concur. The circumcircles of triangles $AED$ and $BFC$ meet $\omega$ for the second time at points $E_1$ and $F_1$. Prove that $EF$ is parallel to $E_1 F_1$.

Let be a tetahedron $ ABCD, $ and $ E $ be the projection of $ D $ on the plane formed by $ ABC. $ If $ \mathcal{A}_{\mathcal{R}} $ denotes the area of the region $ \mathcal{R}, $ show that the following affirmations are equivalent: [b]a)[/b] $ C=E\vee CE\parallel AB $ [b]b)[/b] $ M\in\overline{CD}\implies\mathcal{A}_{ABM}^2=\frac{CM^2}{CD^2}\cdot\mathcal{A}_{ABD}^2 +\left( 1-\frac{CM^2}{CD^2}\right)\cdot\mathcal{A}_{ABC}^2 $

Let $ABC$ be a triangle in a plane $\pi$. Find the set of all points $P$ (distinct from $A,B,C$ ) in the plane $\pi$ such that the circumcircles of triangles $ABP$, $BCP$, $CAP$ have the same radii.

Let $P$ be a given (non-degenerate) polyhedron. Prove that there is a constant $c(P)>0$ with the following property: If a collection of $n$ balls whose volumes sum to $V$ contains the entire surface of $P,$ then $n>c(P)/V^2.$

Over each side of a cyclic quadrilateral erect a rectangle whose height is equal to the length of the opposite side. Prove that the centers of these rectangles form another rectangle.

Let $ABCD$ be a non-cyclic convex quadrilateral. The feet of perpendiculars from $A$ to $BC,BD,CD$ are $P,Q,R$ respectively, where $P,Q$ lie on segments $BC,BD$ and $R$ lies on $CD$ extended. The feet of perpendiculars from $D$ to $AC,BC,AB$ are $X,Y,Z$ respectively, where $X,Y$ lie on segments $AC,BC$ and $Z$ lies on $BA$ extended. Let the orthocenter of $\triangle ABD$ be $H$. Prove that the common chord of circumcircles of $\triangle PQR$ and $\triangle XYZ$ bisects $BH$.

(a) A point $A$ is marked inside a circle. Two perpendicular lines drawn through $A$ intersect the circle at four points. Prove that the centre of mass of these four points does not depend on the choice of the lines. (b) A regular $2n$-gon ($n \ge 2$) with centre $A$ is drawn inside a circle (A does not necessarily coincide with the centre of the circle). The rays going from $A$ to the vertices of the $2n$-gon mark $2n$ points on the circle. Then the $2n$-gon is rotated about $A$. The rays going from $A$ to the new locations of vertices mark new $2n$ points on the circle. Let $O$ and $N$ be the centres of gravity of old and new points respectively. Prove that $O = N$.

Let $ABC$ be an acute triangle and $D$ a variable point on side $AC$ . Point $E$ is on $BD$ such that $BE =\frac{BC^2-CD\cdot CA}{BD}$ . As $D$ varies on side $AC$ prove that the circumcircle of $ADE$ passes through a fixed point other than $A$ .

$ n > 2 $ points are chosen on a circle and each of them is connected to every other by a segment. Is it possible to draw all of these segments in one sequence, i.e. so that the end of the first segment is the beginning of the second, the end of the second - the beginning of the third, etc., and so that the end of the last segment is the beginning of the first?

Let $ABC$ be an isosceles triangle with apex $A$ and altitude $AD$. On $AB$, choose a point $F$ distinct from $B$ such that $CF$ is tangent to the incircle of $ABD$. Suppose that $\vartriangle BCF$ is isosceles. Show that those conditions uniquely determine: a) which vertex of $BCF$ is its apex, b) the size of $\angle BAC$

Let M and N be the distinct points in the plane of the triangle ABC such that AM : BM : CM = AN : BN : CN. Prove that the line MN contains the circumcenter of △ABC.

The base of an inclined prism is a triangle $ABC$. The perpendicular projection of $B_1$, one of the top vertices, is the midpoint of $BC$. The dihedral angle between the lateral faces through $BC$ and $AB$ is $\alpha$, and the lateral edges of the prism make an angle $\beta$ with the base. If $r_1, r_2, r_3$ are exradii of a perpendicular section of the prism, assuming that in $ABC, \cos^2 A + \cos^2 B + \cos^2 C = 1, \angle A < \angle B < \angle C,$ and $BC = a$, calculate $r_1r_2 + r_1r_3 + r_2r_3.$

Let $I$ be the incenter and let $\Gamma$ be the incircle of $\vartriangle ABC$, and let $P = \Gamma \cap BC$. Let $Q$ denote the intersection of $\Gamma$ and the line passing through $P$ parallel to $AI$. Let $\ell$ be the tangent line to $\Gamma$ at $Q$ and let $\ell \cap AB = S$, $\ell \cap AC = R$. If $AB = 7$, $BC = 6$, $AC = 5$, what is $RS$?

Diameter $ACE$ is divided at $C$ in the ratio $2:3$. The two semicircles, $ABC$ and $CDE$, divide the circular region into an upper (shaded) region and a lower region. The ratio of the area of the upper region to that of the lower region is [asy]pair A,B,C,D,EE; A = (0,0); B = (2,2); C = (4,0); D = (7,-3); EE = (10,0); fill(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW)--arc((5,0),EE,A,CCW)--cycle,gray); draw(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW)); draw(circle((5,0),5)); dot(A); dot(B); dot(C); dot(D); dot(EE); label("$A$",A,W); label("$B$",B,N); label("$C$",C,E); label("$D$",D,N); label("$E$",EE,W); [/asy] $\textbf{(A)}\ 2:3 \qquad \textbf{(B)}\ 1:1 \qquad \textbf{(C)}\ 3:2 \qquad \textbf{(D)}\ 9:4 \qquad \textbf{(E)}\ 5:2$

A given truncated pyramid has triangular bases. The areas of the bases are $B_1$ and $B_2$ and the area of the surface is $S$. Prove that if there exists a plane parallel to the bases whose intersection divides the pyramid to two truncated pyramids in which may be inscribed by spheres then $$S=(\sqrt{B_1}+\sqrt{B_2})(\sqrt[4]{B_1}+\sqrt[4]{B_2})^2$$ [i]G. Gantchev[/i]

Through a point $O$ on the diagonal $BD$ of a parallelogram $ABCD$, segments $MN$ parallel to $AB$, and $PQ$ parallel to $AD$, are drawn, with $M$ on $AD$, and $Q$ on $AB$. Prove that diagonals $AO,BP,DN$ (extended if necessary) will be concurrent.

The triangles $ABG$ and $AEF$ are in the same plane. Between them the following conditions hold: (a) $E$ is the mid-point of $AB$; (b) points $A,G$ and $F$ are on the same line; (c) there is a point $C$ at which $BG$ and $EF$ intersect; (d) $|CE|=1$ and $|AC|=|AE|=|FG|$. Show that if $|AG|=x$, then $|AB|=x^3$.

$PQ, CD$ are parallel chords of a circle. The tangent at $D$ cuts $PQ$ at $T$ and $B$ is the point of contact of the other tangent from $T$ (Fig. ). Prove that $BC$ bisects $PQ$. [img]https://cdn.artofproblemsolving.com/attachments/2/f/22f69c03601fbb8e388e319cd93567246b705c.png[/img]

In a circle with center $M$, two chords $AC$ and $BD$ intersect perpendicularly. The circle of diameter $AM$ intersects the circle of diameter $BM$ besides $M$ also in point $P$. The circle of diameter $BM$ intersects the circle with diameter $CM$ besides $M$ also in point $Q$. The circle of diameter $CM$ intersects the circle of diameter $DM$ besides $M$ also in point $R$. The circle of diameter $DM$ intersects the circle of diameter $AM$ besides $M$ also in point $S$. Prove that quadrilateral $PQRS$ is a rectangle. [asy] unitsize (3 cm); pair A, B, C, D, M, P, Q, R, S; M = (0,0); A = dir(170); C = dir(10); B = dir(120); D = dir(240); draw(Circle(M,1)); draw(A--C); draw(B--D); draw(Circle(A/2,1/2)); draw(Circle(B/2,1/2)); draw(Circle(C/2,1/2)); draw(Circle(D/2,1/2)); P = (A + B)/2; Q = (B + C)/2; R = (C + D)/2; S = (D + A)/2; dot("$A$", A, A); dot("$B$", B, B); dot("$C$", C, C); dot("$D$", D, D); dot("$M$", M, E); dot("$P$", P, SE); dot("$Q$", Q, SE); dot("$R$", R, NE); dot("$S$", S, NE); [/asy]

Let $B = (20, 14)$ and $C = (18, 0)$ be two points in the plane. For every line $\ell$ passing through $B$, we color red the foot of the perpendicular from $C$ to $\ell$. The set of red points enclose a bounded region of area $\mathcal{A}$. Find $\lfloor \mathcal{A} \rfloor$ (that is, find the greatest integer not exceeding $\mathcal A$). [i]Proposed by Yang Liu[/i]

Let $ABCD$ be a convex quadrilateral, $I_1$ and $I_2$ be the incenters of triangles $ABC$ and $DBC$ respectively. The line $I_1I_2$ intersects the lines $AB$ and $DC$ at points $E$ and $F$ respectively. Given that $AB$ and $CD$ intersect in $P$, and $PE=PF$, prove that the points $A$, $B$, $C$, $D$ lie on a circle.

In triangle $ABC$, $AB = 100$, $BC = 120$, and $CA = 140$. Points $D$ and $F$ lie on $\overline{BC}$ and $\overline{AB}$, respectively, such that $BD = 90$ and $AF = 60$. Point $E$ is an arbitrary point on $\overline{AC}$. Denote the intersection of $\overline{BE}$ and $\overline{CF}$ as $K$, the intersection of $\overline{AD}$ and $\overline{CF}$ as $L$, and the intersection of $\overline{AD}$ and $\overline{BE}$ as $M$. If $[KLM] = [AME] + [BKF] + [CLD]$, where $[X]$ denotes the area of region $X$, compute $CE$. [i]Proposed by Lewis Chen [/i]

We will say that two pyramids touch each other by faces if they have no common interior points and if the intersection of a face of one of them with a face of the other is either a triangle or a polygon. Is it possible to place $8$ tetrahedra in such a way that every two of them touch each other by faces? (A Andjans, Riga)

Points $M$ and $N$ are taken on the hypotenuse $AB$ of a right triangle $ABC$ so that $BC = BM$ and $AC = AN$. Prove that the angle $MCN$ is equal to $45$ degrees. (Folklore)

$4$ equilateral triangles of side length $1$ are drawn on the interior of a unit square, each one of which shares a side with one of the $4$ sides of the unit square. What is the common area enclosed by all $4$ equilateral triangles?