Assume $A,B,C$ are three collinear points that $B \in [AC]$. Suppose $AA'$ and $BB'$ are to parrallel lines that $A'$, $B'$ and $C$ are not collinear. Suppose $O_1$ is circumcenter of circle passing through $A$, $A'$ and $C$. Also $O_2$ is circumcenter of circle passing through $B$, $B'$ and $C$. If area of $A'CB'$ is equal to area of $O_1CO_2$, then find all possible values for $\angle CAA'$

Consider the configuration of six squares as shown on the picture. Prove that the sum of the area of the three outer squares ($ I,II$ and $ III$) equals three times the sum of the areas of the three inner squares ($ IV,V$ and $ VI$).

Let $ABCDEF$ be a convex hexagon. The diagonal $AC$ is cut by $BF$ and $BD$ at points $P$ and $Q$, respectively. The diagonal $CE$ is cut by $DB$ and $DF$ at points $R$ and $S$, respectively. The diagonal $EA$ is cut by $FD$ and $FB$ at points $T$ and $U$, respectively. It is known that each of the seven triangles $APB, PBQ, QBC, CRD, DRS, DSE$ and $AUF$ has area $1$. Find the area of the hexagon $ABCDEF$.

Given triangle $ ABC $ with points $ M $ and $ N $ are in the sides $ AB $ and $ AC $ respectively. If $ \dfrac{BM}{MA} +\dfrac{CN}{NA} = 1 $ , then prove that the centroid of $ ABC $ lies on $ MN $ .

In the adjoining figure, the circle meets the sides of an equilateral triangle at six points. If $AG=2$, $GF=13$, $FC=1$, and $HJ=7$, then $DE$ equals [asy] size(200); defaultpen(fontsize(10)); real r=sqrt(22); pair B=origin, A=16*dir(60), C=(16,0), D=(10-r,0), E=(10+r,0), F=C+1*dir(120), G=C+14*dir(120), H=13*dir(60), J=6*dir(60), O=circumcenter(G,H,J); dot(A^^B^^C^^D^^E^^F^^G^^H^^J); draw(Circle(O, abs(O-D))^^A--B--C--cycle, linewidth(0.7)); label("$A$", A, N); label("$B$", B, dir(210)); label("$C$", C, dir(330)); label("$D$", D, SW); label("$E$", E, SE); label("$F$", F, dir(170)); label("$G$", G, dir(250)); label("$H$", H, SE); label("$J$", J, dir(0)); label("2", A--G, dir(30)); label("13", F--G, dir(180+30)); label("1", F--C, dir(30)); label("7", H--J, dir(-30));[/asy] $\textbf {(A) } 2\sqrt{22} \qquad \textbf {(B) } 7\sqrt{3} \qquad \textbf {(C) } 9 \qquad \textbf {(D) } 10 \qquad \textbf {(E) } 13$

A simple polygon in the plane is a figure bounded by a closed nonself-intersecting broken line. (a) Do there exist two congruent simple $7$-gons in the plane such that all the seven vertices of one of the $7$-gons are the vertices of the other one and yet these two $7$-gons have no common sides? (b) Do there exist three such $7$-gons? (V Proizvolov)

Given three parallel straight lines. Construct a square three of whose vertices belong to these lines.

Points $P_1, P_2, P_3,$ and $P_4$ are $(0,0), (10, 20), (5, 15),$ and $(12, -6)$, respectively. For what point $P \in \mathbb{R}^2$ is the sum of the distances from $P$ to the other $4$ points minimal?

$(YUG 3)$ Let four points $A_i (i = 1, 2, 3, 4)$ in the plane determine four triangles. In each of these triangles we choose the smallest angle. The sum of these angles is denoted by $S.$ What is the exact placement of the points $A_i$ if $S = 180^{\circ}$?

Let $ABC$ be an acute triangle with $AB<AC$. Let then • $D$ be the foot of the bisector of the angle in $A$, • $E$ be the point on segment $BC$ (different from $B$) such that $AB=AE$, • $F$ be the point on segment $BC$ (different from $B$) such that $BD=DF$, • $G$ be the point on segment $AC$ such that $AB=AG$. Prove that the circumcircle of triangle $EFG$ is tangent to line $AC$.

Let $\omega$ be the circumcircle of acute triangle $ABC$ where $\angle A<\angle B$ and $M,N$ be the midpoints of minor arcs $BC,AC$ of $\omega$ respectively. The line $PC$ is parallel to $MN$, intersecting $\omega$ at $P$ (different from $C$). Let $I$ be the incentre of $ABC$ and let $PI$ intersect $\omega$ again at the point $T$. 1) Prove that $MP\cdot MT=NP\cdot NT$; 2) Let $Q$ be an arbitrary point on minor arc $AB$ and $I,J$ be the incentres of triangles $AQC,BCQ$. Prove that $Q,I,J,T$ are concyclic.

Let $ABCC_1B_1A_1$ be a convex hexagon such that $AB=BC$, and suppose that the line segments $AA_1, BB_1$, and $CC_1$ have the same perpendicular bisector. Let the diagonals $AC_1$ and $A_1C$ meet at $D$, and denote by $\omega$ the circle $ABC$. Let $\omega$ intersect the circle $A_1BC_1$ again at $E \neq B$. Prove that the lines $BB_1$ and $DE$ intersect on $\omega$.

Let $ABC$ be a triangle, $O{}$ be its circumcenter, $I{}$ its incenter and $I_A,I_B,I_C$ the excenters. Let $M$ be the midpoint of $BC$ and $H_1$ and $H_2$ be the orthocenters of the triangles $MII_A$ and $MI_BI_C$. Prove that the parallel to $BC$ through $O$ passes through the midpoint of the segment $H_1H_2$. [i]Proposed by David Anghel[/i]

Let $ABCD$ be a square. On the sides $BC$ and $CD$ the points $M$ and $K$ respectively, so that $MC = KD$. Let $P$ the intersection point of of segments $MD$ and $BK$. Prove that $AP \perp MK$.

Circle $S_1$ has radius $5$. Circle $S_2$ has radius $7$ and has its center lying on $S_1$. Circle $S_3$ has an integer radius and has its center lying on $S_2$. If the center of $S_1$ lies on $S_3$, how many possible values are there for the radius of $S_3$? [i]Ray Li[/i]

Quadrilateral $ABCD$ with $AC = 800$ is inscribed in a circle, and $E, W, X, Y, Z$ are the midpoints of segments $BD$, $AB$, $BC$, $CD$, $DA$, respectively. If the circumcenters of $EW Z$ and $EXY$ are $O_1$ and $O_2$, respectively, determine $O_1O_2$.

Given triangle $ABC$ with $|AC| > |BC|$. The point $M$ lies on the angle bisector of angle $C$, and $BM$ is perpendicular to the angle bisector. Prove that the area of triangle AMC is half of the area of triangle $ABC$. [img]https://cdn.artofproblemsolving.com/attachments/4/2/1b541b76ec4a9c052b8866acbfea9a0ce04b56.png[/img]

Prove that the sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space.

Let $ABC$ be a triangle such that $AB < AC$. Let $P$ lie on a line through $A$ parallel to line $BC$ such that $C$ and $P$ are on the same side of line $AB$. Let $M$ be the midpoint of segment $BC$. Define $D$ on segment $BC$ such that $\angle BAD = \angle CAM$, and define $T$ on the extension of ray $CB$ beyond $B$ so that $\angle BAT = \angle CAP$. Given that lines $PC$ and $AD$ intersect at $Q$, that lines $PD$ and $AB$ intersect at $R$, and that $S$ is the midpoint of segment $DT$, prove that if $A$,$P$,$Q$, and $R$ lie on a circle, then $Q$, $R$, and $S$ are collinear. [i]David Rush[/i]

Given a triangle $ABC$, with $I$ as its incenter and $\Gamma$ as its circumcircle, $AI$ intersects $\Gamma$ again at $D$. Let $E$ be a point on the arc $BDC$, and $F$ a point on the segment $BC$, such that $\angle BAF=\angle CAE < \dfrac12\angle BAC$. If $G$ is the midpoint of $IF$, prove that the meeting point of the lines $EI$ and $DG$ lies on $\Gamma$. [i]Proposed by Tai Wai Ming and Wang Chongli, Hong Kong[/i]

Let $ABC$ be an isosceles triangle ($AB = AC$) with its circumcenter $O$. Point $N$ is the midpoint of the segment $BC$ and point $M$ is the reflection of the point $N$ with respect to the side $AC$. Suppose that $T$ is a point so that $ANBT$ is a rectangle. Prove that $\angle OMT = \frac{1}{2} \angle BAC$. [i]Proposed by Ali Zamani[/i]

How many $2 \times 2 \times 2$ cubes must be added to a $8 \times 8 \times 8$ cube to form a $12 \times 12 \times 12$ cube? [i]Proposed by Evan Chen[/i]

Consider three different planes and consider also one point on each of them. Give necessary and sufficient conditions for the existence of a quadratic which passes through the given points and whose tangent-plane at each of these points is the respective given plane.

Let $ABCD$ be a cyclic quadrilateral with center $O$. Suppose the circumcircles of triangles $AOB$ and $COD$ meet again at $G$, while the circumcircles of triangles $AOD$ and $BOC$ meet again at $H$. Let $\omega_1$ denote the circle passing through $G$ as well as the feet of the perpendiculars from $G$ to $AB$ and $CD$. Define $\omega_2$ analogously as the circle passing through $H$ and the feet of the perpendiculars from $H$ to $BC$ and $DA$. Show that the midpoint of $GH$ lies on the radical axis of $\omega_1$ and $\omega_2$. [i]Proposed by Yang Liu[/i]

Find the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length $3$ and the remaining four sides of length $2$. Give the answer in the form $r+s\sqrt{t}$ with $r,s, t$ positive integers.