(a)Two circles $S_1$ and $S_2$ are placed so that they do not overlap each other, neither completely nor partially. They have centres in $O_1$ and $O_2$, respectively. Further, $L_1$ and $M_1$ are different points on $S_1$ so that $O_2L_1$ and $O_2M_1$ are tangent to $S_1$, and similarly $L_2$ and $M_2$ are different points on $S_2$ so that $O_1L_2$ and $O_1M_2$ are tangent to $S_2$. Show that there exists a unique circle which is tangent to the four line segments $O_2L_1, O_2M_1, O_1L_2$, and $O_1M_2$. (b) Four circles $S_1, S_2, S_3$ and $S_4$ are placed so that none of them overlap each other, neither completely nor partially. They have centres in $O_1, O_2, O_3$, and $O_4$, respectively. For each pair $(S_i, S_j )$ of circles, with $1 \le i < j \le 4$, we find a circle $S_{ij}$ as in part [b]a[/b]. The circle $S_{ij}$ has radius $R_{ij}$ . Show that $\frac{1}{R_{12}} + \frac{1}{R_{23}}+\frac{1}{R_{34}}+\frac{1}{R_{14}}= 2 \left(\frac{1}{R_{13}} +\frac{1}{R_{24}}\right)$

Let $ C_1, C_2 $ be two circles of equal radius, disjoint, of centers $ O_1, O_2 $, such that $ C_1 $ is to the left of $ C_2 $. Let $ l $ be a line parallel to the line $ O_1O_2 $, secant to both circles. Let $ P_1 $ be a point of $ l $, to the left of $ C_1 $ and $ P_2 $ a point of $ l $, to the right of $ C_2 $ such that the tangents of $ P_1 $ to $ C_1 $ and of $ P_2 $ a $ C_2 $ form a quadrilateral. Show that there is a circle tangent to the four sides of said quadrilateral.

A convex quadrilateral $ABCD$ is inscribed into a circle $\omega$ . Suppose that there is a point $X$ on the segment $AC$ such that the $XB$ and $XD$ tangents to the circle $\omega$ . Tangent of $\omega$ at $C$, intersect $XD$ at $Q$. Let $E$ ($E\ne A$) be the intersection of the line $AQ$ with $\omega$ . Prove that $AD, BE$, and $CQ$ are concurrent.

Quadrilateral $ABCD$ is circumscribed. Its incircle touches sides $AB, BC, CD, DA$ in points $K, L, M, N$ respectively. Points $A', B', C', D'$ are the midpoints of segments $LM, MN, NK, KL$. Prove that the quadrilateral formed by lines $AA', BB', CC', DD'$ is cyclic.

Let $ABCD$ be a quadrilateral and let $\Gamma$ be a circle of center $O$ that is internally tangent to its four sides. If $M$ is the midpoint of $AC$ and $N$ is the midpoint of $BD$, prove that $M,O, N$ are collinear.

On sides $BC,CA,AB$ of a triangle $ABC$ points $D,E,F$ respectively are chosen so that $AD,BE,CF$ have a common point, say $G$. Suppose that one can inscribe circles in the quadrilaterals $AFGE,BDGF,CEGD$ so that each two of them have a common point. Prove that triangle $ABC$ is equilateral.

In the convex quadrilateral $ ABCD $, the corresponding points $ M $ and $ N $ are chosen on the adjacent sides $ \overline{AB} $ and $ \overline{BC} $ and the intersection point of the segments $ AN $ and $ GM $ is marked by 0. Prove that if circles can be inscribed in the quadrilaterals $ AOCD $ and $ BMON $, then a circle can also be inscribed in the quadrilateral $ ABCD $.

A cyclic quadrilateral $ABCD$ is given. Let $I{}$ and $J{}$ be the centers of circles inscribed in the triangles $ABC$ and $ADC$. It turns out that the points $B, I, J, D$ lie on the same circle. Prove that the quadrilateral $ABCD$ is tangential.

On the sides $AB, BC$ and $CA$ of triangle $ABC$, points $L, M$ and $N$ are chosen, respectively, such that the lines $CL, AM$ and $BN$ intersect at a common point O inside the triangle and the quadrilaterals $ALON, BMOL$ and $CNOM$ have incircles. Prove that $$\frac{1}{AL\cdot BM} +\frac{1}{BM\cdot CN} +\frac{1}{CN \cdot AL} =\frac{1}{AN\cdot BL} +\frac{1}{BL\cdot CM} +\frac{1}{CM\cdot AN} $$