The figure shows a $60^o$ angle in which are placed $2007$ numbered circles (only the first three are shown in the figure). The circles are numbered according to size. The circles are tangent to the sides of the angle and to each other as shown. Circle number one has radius $1$. Determine the radius of circle number $2007$. [img]https://1.bp.blogspot.com/-1bsLIXZpol4/Xzb-Nk6ospI/AAAAAAAAMWk/jrx1zVYKbNELTWlDQ3zL9qc_22b2IJF6QCLcBGAsYHQ/s0/2007%2BMohr%2Bp4.png[/img]

Let $\omega$ be the incircle of triangle $ABC$ touches $BC$ at point $K$ . Draw a circle passing through points $B$ and $C$ , and touching $\omega$ at the point $S$ . Prove that $S K$ passes through the center of the exscribed circle of triangle $A B C$ , tangent to side $B C$ .

Let $ABC$ be a non isosceles triangle with circumcircle $(O)$ and incircle $(I)$. Denote $(O_1)$ as the circle that external tangent to $(O)$ at $A'$ and also tangent to the lines $AB,AC$ at $A_b,A_c$ respectively. Define the circles $(O_2), (O_3)$ and the points $B', C', B_c , B_a, C_a, C_b$ similarly. 1. Denote J as the radical center of $(O_1), (O_2), (O_3) $and suppose that $JA'$ intersects $(O_1)$ at the second point $X, JB'$ intersects $(O_2)$ at the second point Y , JC' intersects $(O_3)$ at the second point $Z$. Prove that the circle $(X Y Z)$ is tangent to $(O_1), (O_2), (O_3)$. 2. Prove that $AA', BB', CC'$ are concurrent at the point $M$ and $3$ points $I,M,O$ are collinear.

Let $ABC$ be acute-angled triangle with circumcircle $\Gamma$. Points $H$ and $M$ are the orthocenter and the midpoint of $BC$ respectively. The line $HM$ meets the circumcircle $\omega$ of triangle $BHC$ at point $N\not= H$. Point $P$ lies on the arc $BC$ of $\omega$ not containing $H$ in such a way that $\angle HMP = 90^\circ$. The segment $PM$ meets $\Gamma$ at point $Q$. Points $B'$ and $C'$ are the reflections of $A$ about $B$ and $C$ respectively. Prove that the circumcircles of triangles $AB'C'$ and $PQN$ are tangent.

The quadrangle $ ABCD $ contains two circles of radii $ R_1 $ and $ R_2 $ tangent externally. The first circle touches the sides of $ DA $,$ AB $ and $ BC $, moreover, the sides of $ AB $ at the point $ E $. The second circle touches sides $ BC $, $ CD $ and $ DA $, and sides $ CD $ at $ F $. Diagonals of the quadrangle intersect at $ O $. Prove that $ OE + OF \leq 2 (R_1 + R_2) $. (F. Bakharev, S. Berlov)

Bisectors $AA_1$, $BB_1$, and $CC_1$ of triangle $ABC$ meet at point $I$. The perpendicular bisector to $BB_1$ meets $AA_1,CC_1$ at points $A_0,C_0$ respectively. Prove that the circumcircles of triangles $A_0IC_0$ and $ABC$ touch.

Let $\alpha_1,\alpha_2,\alpha_3$ be three congruent circles that are tangent to each other. A third circle $\beta$ is tangent to them at points $A_1,A_2,A_3$ respectively. Let $P$ be a point on $\beta$ which is different from $A_1,A_2,A_3$. For $i=1,2,3$, let $B_i$ be the second intersection point of the line $PA_i$ with circle $\alpha_i$. Prove that $\Delta B_1B_2B_3$ is equilateral.

Let $ABC$ be a triangle with right angle $ACB$. Denote by $F$ the projection of $C$ on $AB$. A circle $\omega$ touches $FB$ at point $P$, touches $CF$ at point $Q$, and the circumcircle of $ABC$ at point $R$. Prove that the points $A, Q, R$ all lie on the same line and $AP=AC$.

Circles $K_1$ and $K_2$ are externally tangent to each other at $A$ and are internally tangent to a circle $K$ at $A_1$ and $A_2$ respectively. The common tangent to $K_1$ and $K_2$ at $A$ meets $K$ at point $P$. Line $PA_1$ meets $K_1$ again at $B_1$ and $PA_2$ meets $K_2$ again at $B_2$. Show that $B_1B_2$ is a common tangent of $K_1$ and $K_2$.

As shown in the figure, $\odot O$ is the circumcircle of $\vartriangle ABC$, $\odot J$ is inscribed in $\odot O$ and is tangent to $AB$, $AC$ at points $D$ and E respectively, line segment $FG$ and $\odot O$ are tangent to point $A$, and $AF =AG=AD$, the circumscribed circle of $\vartriangle AFB$ intersects $\odot J$ at point $S$. Prove that the circumscribed circle of $\vartriangle ASG$ is tangent to $\odot J$. [img]https://cdn.artofproblemsolving.com/attachments/a/a/62d44e071ea9903ebdd68b43943ba1d93b4138.png[/img]

In the figure shown, the small circles have radius $1$. Calculate the area of the gray part of the figure. [img]https://1.bp.blogspot.com/-oy-WirJ6u9o/XzcFc3roVDI/AAAAAAAAMX8/qxNy5I_0RWUOxl-ZE52fnrwo0v0T7If9QCLcBGAsYHQ/s0/1998%2BMohr%2Bp1.png[/img]

Given an isosceles triangle $\triangle ABC$, $AB=AC$. A line passes through $M$, the midpoint of $BC$, and intersects segment $AB$ and ray $CA$ at $D$ and $E$, respectively. Let $F$ be a point of $ME$ such that $EF=DM$, and $K$ be a point on $MD$. Let $\Gamma_1$ be the circle passes through $B,D,K$ and $\Gamma_2$ be the circle passes through $C,E,K$. $\Gamma_1$ and $\Gamma_2$ intersect again at $L \neq K$. Let $\omega_1$ and $\omega_2$ be the circumcircle of $\triangle LDE$ and $\triangle LKM$. Prove that, if $\omega_1$ and $\omega_2$ are symmetric wrt $L$, then $BF$ is perpendicular to $BC$.

A semicircle of radius $5$ and a quarter of a circle of radius $8$ touch each other and are located inside the square as shown in the figure. Find the length of the part of the common tangent, enclosed in the same square. [img]https://cdn.artofproblemsolving.com/attachments/f/2/010f501a7bc1d34561f2fe585773816f168e93.png[/img]

Let $ABC$ be a triangle such that $\measuredangle BAC = 60^{\circ}$. Let $D$ and $E$ be the feet of the perpendicular from $A$ to the bisectors of the external angles of $B$ and $C$ in triangle $ABC$, respectively. Let $O$ be the circumcenter of the triangle $ABC$. Prove that circumcircle of the triangle $BOC$ has exactly one point in common with the circumcircle of $ADE$.

Let $ABCD$ be a quadrilateral inscribed in the circle $(O)$. Let $(K)$ be an arbitrary circle passing through $B,C$. Circle $(O_1)$ tangent to $AB,AC$ and is internally tangent to $(K)$. Circle $(O_2)$ touches $DB,DC$ and is internally tangent to $(K)$. Prove that one of the two external common tangents of $(O_1)$ and $(O_2)$ is parallel to $AD$. Trần Quang Hùng

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. The circle with center $X_A$ passes through the points $A$ and $H$ and is tangent to the circumcircle of the triangle $ABC$. Similarly, define the points $X_B$ and $X_C$. Let $O_A$, $O_B$ and $O_C$ be the reflections of $O$ with respect to sides $BC$, $CA$ and $AB$, respectively. Prove that the lines $O_AX_A$, $O_BX_B$ and $O_CX_C$ are concurrent.

Let $ABC$ be a triangle with $AB\neq AC$. Let the excircle $\omega$ opposite $A$ touch $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. Suppose $X$ and $Y$ are points on the segments $AC$ and $AB$, respectively, such that $XY$ and $BC$ are parallel, and let $\Gamma$ be a circle through $X$ and $Y$ which is externally tangent to $\omega$ at $Z$. Prove that the lines $EF$, $DZ$, and $XY$ are concurrent.

Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other

Three circles $\omega_1$, $\omega_2$, $\omega_3$ of radius $r$ pass through the point$ S$ and internally touch the circle $\omega$ of radius $R$ ($R > r$) at points $T_1$, $T_2$, $T_3$ respectively. Prove that the line $T_1T_2$ passes through the second (different from $S$) intersection point of the circles $\omega_1$ and $\omega_2$.

Two circles $A$ and $B$, both with radius $1$, touch each other externally. Four circles $P, Q, R$ and $S$, all four with the same radius $r$, lie such that $P$ externally touches on $A, B, Q$ and $S$, $Q$ externally touches on $P, B$ and $R$, $R$ externally touches on $A, B, Q$ and $S$, $S$ externally touches on $P, A$ and $R$. Calculate the length of $r.$ [asy] unitsize(0.3 cm); pair A, B, P, Q, R, S; real r = (3 + sqrt(17))/2; A = (-1,0); B = (1,0); P = intersectionpoint(arc(A,r + 1,0,180), arc(B,r + 1,0,180)); R = -P; Q = (r + 2,0); S = (-r - 2,0); draw(Circle(A,1)); draw(Circle(B,1)); draw(Circle(P,r)); draw(Circle(Q,r)); draw(Circle(R,r)); draw(Circle(S,r)); label("$A$", A); label("$B$", B); label("$P$", P); label("$Q$", Q); label("$R$", R); label("$S$", S); [/asy]

$\vartriangle ABC$ is a scalene triangle with circumcircle $\Omega$. For a arbitrary $X$ in the plane, define $D_x,E_x, F_x$ to be the intersection of tangent line of $X$ (with respect to $BXC$) and $BC,CA,AB$, respectively. Let the intersection of $AX$ with $\Omega$ be $S_x$ and $T_x = D_xS_x \cap \Omega$. Show that $\Omega$ and circumcircle of $\vartriangle T_xE_xF_x$ are tangent to each other. [img]https://2.bp.blogspot.com/-rTMODHbs5Ac/XnYNQYjYzBI/AAAAAAAALeg/576nGDQ6NDA0-W5XqiNczNtI07cEZxPeQCK4BGAYYCw/s1600/imoc2019g4.png[/img]

Two circles are tangent externally at one point. Common external tangents are drawn to them and the tangent points are connected. Prove that the sum of the lengths of the opposite sides of the quadrilateral obtained are equal.

Let $\Gamma_1$ and $\Gamma_2$ be two circles that tangents each other at point $N$, with $\Gamma_2$ located inside $\Gamma_1$. Let $A, B, C$ be distinct points on $\Gamma_1$ such that $AB$ and $AC$ tangents $\Gamma_2$ at $D$ and $E$, respectively. Line $ND$ cuts $\Gamma_1$ again at $K$, and line $CK$ intersects line $DE$ at $I$. (i) Prove that $CK$ is the angle bisector of $\angle ACB$. (ii) Prove that $IECN$ and $IBDN$ are cyclic quadrilaterals.

Altitudes $AA_1$ and $BB_1$ of triangle ABC meet in point $H$. Line $CH$ meets the semicircle with diameter $AB$, passing through $A_1, B_1$, in point $D$. Segments $AD$ and $BB_1$ meet in point $M$, segments $BD$ and $AA_1$ meet in point $N$. Prove that the circumcircles of triangles $B_1DM$ and $A_1DN$ touch.

Given are three circles $(O_1), (O_2), (O_3)$, pairwise intersecting each other, that is, every single circle meets the other two circles at two distinct points. Let $(X_1)$ be the circle externally tangent to $(O_1)$ and internally tangent to the circles $(O_2), (O_3),$ circles $(X_2), (X_3)$ are defined in the same manner. Let $(Y_1)$ be the circle internally tangent to $(O_1)$ and externally tangent to the circles $(O_2), (O_3)$, the circles $(Y_2), (Y_3)$ are defined in the same way. Let $(Z_1), (Z_2)$ be two circles internally tangent to all three circles $(O_1), (O_2), (O_3)$. Prove that the four lines $X_1Y_1, X_2Y_2, X_3Y_3, Z_1Z_2$ are concurrent. Nguyễn Văn Linh