Given two circles $\Gamma_1$ and $\Gamma_2$ which intersect at points $A$ and $B$. A line through $A$ intersects $\Gamma_1$ and $\Gamma_2$ at points $C$ and $D$, respectively. Let $M$ be the midpoint of arc $BC$ in $\Gamma_1$ ,which does not contains $A$, and $N$ is the midpoint of the arc $BD$ in $\Gamma_2$, which does not contain $A$. If $K$ is the midpoint of $CD$, prove that $\angle MKN = 90^o.$

Let $ABC$ be a triangle with $ \angle ACB = 90^o + \frac12 \angle ABC$ . The point $M$ is the midpoint of the side $BC$ . A circle with center at vertex $A$ intersects the line $BC$ at points $M$ and $D$. Prove that $MD = AB$.

In the plane, four points are marked. It is known that these points are the centers of four circles, three of which are pairwise externally tangent, and all these three are internally tangent to the fourth one. It turns out, however, that it is impossible to determine which of the marked points is the center of the fourth (the largest) circle. Prove that these four points are the vertices of a rectangle.

Inside triangle $ABC$ there are three circles $k_1, k_2, k_3$ each of which is tangent to two sides of the triangle and to its incircle $k$. The radii of $k_1, k_2, k_3$ are $1, 4$, and $9$. Determine the radius of $k.$

Three circles $k_1,k_2$ and $k_3$ go through the points $A$ and $B$. A secant through $A$ intersects the circles $k_1,k_2$ and $k_3$ again in the points $C,D$ resp. $E$. Prove that the ratio $|CD|:|DE|$ does not depend on the choice of the secant.

Let $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. A point $D$ is chosen on the internal bisector of $\angle ACB$ so that the points $D$ and $C$ are separated by $AB$. A circle $\omega$ centered at $D$ is tangent to the segment $AB$ at $E$. The tangents to $\omega$ through $C$ meet the segment $AB$ at $K$ and $L$, where $K$ lies on the segment $AL$. A circle $\Omega_1$ is tangent to the segments $AL, CL$, and also to $\Omega$ at point $M$. Similarly, a circle $\Omega_2$ is tangent to the segments $BK, CK$, and also to $\Omega$ at point $N$. The lines $LM$ and $KN$ meet at $P$. Prove that $\angle KCE = \angle LCP$. Poland

A circle cuts each side of a rhombus twice thus dividing each side into three segments. Let us go around the perimeter of the rhombus clockwise beginning at a vertex and paint these segments successively in red, white and blue. Prove that the sum of lengths of the blue segments equals that of the red ones. (V Proizvolov)

A quadrilateral $ABCD$ is inscribed into a circle $\omega$ with center $O$. Let $M_1$ and $M_2$ be the midpoints of segments $AB$ and $CD$ respectively. Let $\Omega$ be the circumcircle of triangle $OM_1M_2$. Let $X_1$ and $X_2$ be the common points of $\omega$ and $\Omega$ and $Y_1$ and $Y_2$ the second common points of $\Omega$ with the circumcircles of triangles $CDM_1$ and $ABM_2$. Prove that $X_1X_2 // Y_1Y_2$.

Circles $C$ and $C'$ intersect at $O$ and $X$. A circle center $O$ meets $C$ at $Q$ and $R$ and meets $C'$ at $P$ and $S$. $PR$ and $QS$ meet at $Y$ distinct from $X$. Show that $\angle YXO = 90^o$.

We put, forming a circumference of a circle, ${2004}$ bicolored files: white on one side of the file and black on the other. A movement consists in choosing a file with the black side upwards and flipping three files: the one chosen, the one to its right, and the one to its left. Suppose that initially there was only one file with its black side upwards. Is it possible, repeating the movement previously described, to get all of the files to have their white sides upwards? And if we were to have ${2003}$ files, between which exactly one file began with the black side upwards?

Let $ABCD$ be a quadrilateral and let $l$ be a line. Let $l$ intersect the lines $AB,CD,BC,DA,AC,BD$ at points $X,X',Y,Y',Z,Z'$ respectively. Given that these six points on $l$ are in the order $X,Y,Z,X',Y',Z'$, show that the circles with diameter $XX',YY',ZZ'$ are coaxal.

Let $ABC$ be an acute triangle. Suppose that circle $\Gamma_1$ has it's center on the side $AC$ and is tangent to the sides $AB$ and $BC$, and circle $\Gamma_2$ has it's center on the side $AB$ and is tangent to the sides $AC$ and $BC$. The circles $\Gamma_1$ and $ \Gamma_2$ intersect at two points $P$ and $Q$. Show that if $A, P, Q$ are collinear, then $AB = AC$.

Prove that you can always pose a circle of radius $S/P$ inside a convex polygon with the perimeter $P$ and area $S$.

In the Jordan Building (the Olympiad building of High School Mandegar Alborz), Ali and Khosro are playing a game. First, Ali selects 2025 points on the plane such that no three points are collinear and no four points are concyclic. Then, Khosro selects a point, followed by Ali selecting another point, and then Khosro selects one more point. The circumcircle of these three points is drawn, and the number of points inside the circle is denoted by \( t \). If Khosro's goal is to maximize \( t \) and Ali's goal is to minimize \( t \), and both play optimally, determine the value of \( t \). Proposed by Reza Tahernejad Karizi

On triangle $ABC$ the $AM$ ($M\in BC$) is median and $BB_1$ and $CC_1$ ($B_1 \in AC,C_1 \in AB$) are altitudes. The stright line $d$ is perpendicular to $AM$ at the point $A$ and intersect the lines $BB_1$ and $CC_1$ at the points $E$ and $F$ respectively. Let denoted with $\omega$ the circle passing through the points $E, M$ and $F$ and with $\omega_1$ and with $\omega_2$ the circles that are tangent to segment $EF$ and with $\omega$ at the arc $EF$ which is not contain the point $M$. If the points $P$ and $Q$ are intersections points for $\omega_1$ and $\omega_2$ then prove that the points $P, Q$ and $M$ are collinear.

A triangle $ABC$ is inscribed in the circle $(O,R)$. A circle $(O',R')$ is internally tangent to $(O)$ at $I$ such that $R < R'$. $P$ is a point on the circle $(O)$. Rays $PA, PB, PC$ meet $(O')$ at $A_1,B_1,C_1$. Let $A_2B_2C_2$ be the triangle formed by the intersections of the line symmetric to $B_1C_1$ about $BC$, the line symmetric to $C_1A_1$ about $CA$ and the line symmetric to $A_1B_1$ about $AB$. Prove that the circumcircle of $A_2B_2C_2$ is tangent to $(O)$. Nguyễn Văn Linh

Circles ${{w} _ {1}}$ and ${{w} _ {2}}$ with centers ${{O} _ {1}}$ and ${{O} _ {2}}$ intersect at points $A$ and $B$, respectively. The line ${{O} _ {1}} {{O} _ {2}}$ intersects ${{w} _ {1}}$ at the point $Q$, which does not lie inside the circle ${{w} _ {2}}$, and ${{w} _ {2}}$ at the point $X$ lying inside the circle ${{w} _ {1} }$. Around the triangle ${{O} _ {1}} AX$ circumscribe a circle ${{w} _ {3}}$ intersecting the circle ${{w} _ {1}}$ for the second time in point $T$. The line $QT$ intersects the circle ${{w} _ {3}}$ at the point $K$, and the line $QB$ intersects ${{w} _ {2}}$ the second time at the point $H$. Prove that a) points $T, \, \, X, \, \, B$ lie on one line; b) points $K, \, \, X, \, \, H$ lie on one line. (Vadym Mitrofanov)

Let $(O_1), (O_2)$ be given two circles intersecting at $A$ and $B$. The tangent lines of $(O_1)$ at $A, B$ intersect at $O$. Let $I$ be a point on the circle $(O_1)$ but outside the circle $(O_2)$. The lines $IA, IB$ intersect circle $(O_2)$ at $C, D$. Denote by $M$ the midpoint of $C D$. Prove that $I, M, O$ are collinear.

Let $ABC$ be an acute-angled, nonisosceles triangle. Point $A_1, A_2$ are symmetric to the feet of the internal and the external bisectors of angle $A$ wrt the midpoint of $BC$. Segment $A_1A_2$ is a diameter of a circle $\alpha$. Circles $\beta$ and $\gamma$ are defined similarly. Prove that these three circles have two common points.

Two circles intersect at points $P$ and $Q$. The straight line intersects these circles at points $A$, $B$, $C$, $D$, as shown in fig. . Prove that $\angle APB = \angle CQD$. [img]https://cdn.artofproblemsolving.com/attachments/1/a/a581e11be68bbb628db5b5b8e75c7ff6e196c5.png[/img]

On circle there are points $A$, $B$ and $C$ such that they divide circle in ratio $3:5:7$. Find angles of triangle $ABC$

Find the length of the longer side of the rectangle on the picture, if the shorter side has length $1$ and the circles touch each other and the sides of the rectangle as shown. [img]https://cdn.artofproblemsolving.com/attachments/b/8/3986683247293bd089d8e83911309308ce0c3a.png[/img]

Consider a triangle $ABC$ with $BC>AC$. The circle with center $C$ and radius $AC$ intersects the segment $BC$ in $D$. Let $I$ be the incenter of triangle $ABC$ and $\Gamma$ be the circle that passes through $I$ and is tangent to the line $CA$ at $A$. The line $AB$ and $\Gamma$ intersect at a point $F$ with $F \neq A$. Prove that $BF=BD$.

Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$. Let the internal angle bisectors at $A$ and $B$ meet at $X$, the internal angle bisectors at $B$ and $C$ meet at $Y$, the internal angle bisectors at $C$ and $D$ meet at $Z$, and the internal angle bisectors at $D$ and $A$ meet at $W$. Further, let $AC$ and $BD$ meet at $P$. Suppose that the points $X$, $Y$, $Z$, $W$, $O$, and $P$ are distinct. Prove that $O$, $X$, $Y$, $Z$, $W$ lie on the same circle if and only if $P$, $X$, $Y$, $Z$, and $W$ lie on the same circle.

To two circles $r_1$ and $r_2$, intersecting at points $A$ and $B$, their common tangent $CD$ is drawn ($C$ and $D$ are tangency points, respectively, point $B$ is closer to line $CB$ than $A$). Line passing through $A$ , intersects $r_1$ and $r_2$ for second time at points $K$ and $L$, respectively ($A$ lies between $K$ and $L$). Lines $KC$ and $LD$ intersect at point $P$. Prove that $PB$ is the symmedian of triangle $KPL$. (Yu. Blinkov)