Into how many regions do $n$ circles divide the plane, if each pair of circles intersects in two points and no point lies on three circles?

Given a triangle $ABC$ , a circle centered at some point $O$ meets the segments $BC$ , $CA$ , $AB$ in the pairs of points $X$ and $X^{'}$ , $Y$ and $Y^{'}$ , $Z$ and $Z^{'}$ , respectively ,labelled in circular order : $X,X^{'},Y,Y^{'},Z,Z^{'}$. Let $M$ be the Miquel point of the triangle $XYZ$ and let $M^{'}$ be that of the triangle $X^{'}Y^{'}Z^{'}$ . Prove that the segments $OM$ and $OM^{'}$ have equal lehgths.

Let $A,B,C$ be different points on a circle such that $AB = AC$. Point $E$ lies on the segment $BC$, and $D \ne A$ is the intersection point of the circle and line $AE$. Show that the product $AE \cdot AD$ is independent of the choice of $E$.

Suppose that $A,B,C,D$ are points on a circle, $AB$ is the diameter, $CD$ is perpendicular to $AB$ and meets $AB$ and meets $AB$ at $E , AB$ and $CD$ are integers and $AE - EB=\sqrt{3}$. Find $AE$?

A circle with center at the origin and radius $5$ intersects the abscissa in points $A$ and $B$. Let $P$ a point lying on the line $x = 11$, and the point $Q$ is the intersection point of $AP$ with this circle. We know what is the $Q$ point is the midpoint of the $AP$. Find the coordinates of the point $P$.

Two fixed circles $\omega_1$ and $\omega_2$ pass through point $O$. A circle of an arbitrary radius $R$ centered at $O$ meets $\omega_1$ at points $A$ and $B$, and meets $\omega_2$ at points $C$ and $D$. Let $X$ be the common point of lines $AC$ and $BD$. Prove that all the points X are collinear as $R$ changes.

There are $12$ points on a circle. Four checkers, one red, one yellow, one green and one blue sit at neighboring points. In one move any checker can be moved four points to the left or right, onto the fifth point, if it is empty. If after several moves the checkers appear again at the four original points, how might their order have changed?

a) What is the least number of points that can be chosen on a circle of length $1956$, so that for each of these points there is exactly one chosen point at distance $1$, and exactly one chosen point at distance $2$ (distances are measured along the circle)? b) On a circle of length $15$ there are selected $n$ points such that for each of them there is exactly one selected point at distance $1$ from it, and exactly one is selected point at distance $2$ from it. (All distances are measured along the circle.) Prove that $n$ is divisible by $10$.

Three circles $k_1, k_2$, and $k_3$ intersect in point $O$. Let $A, B$, and $C$ be the second intersection points (other than $O$) of $k_2$ and $k_3, k_1$ and $k_3$, and $k_1$ and $k_2$, respectively. Assume that $O$ lies inside of the triangle $ABC$. Let lines $AO,BO$, and $CO$ intersect circles $k_1, k_2$, and $k_3$ for a second time at points $A', B'$, and $C'$, respectively. If $|XY|$ denotes the length of segment $XY$, prove that $\frac{|AO|}{|AA'|}+\frac{|BO|}{|BB'|}+\frac{|CO|}{|CC'|}= 1$

Two circles in a plane intersect. $A$ is one of the points of intersection. Starting simultaneously from $A$ two points move with constant speed, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Prove that there is a fixed point $P$ in the plane such that the two points are always equidistant from $P.$

Let $P$ and $Q$ be any points inside a circle $C$ with center $O$ such that $OP=OQ.$ Determine the location of a point $Z$ on $C$ such that $PZ+QZ$ is minimal.

A circle has center $ (-10,-4) $ and radius $13$. Another circle has center $(3,9) $ and radius $\sqrt{65}$. The line passing through the two points of intersection of the two circles has equation $x+y=c$. What is $c$? $\textbf{(A)} \text{ 3} \qquad \textbf{(B)} \text{ } 3 \sqrt{3} \qquad \textbf{(C)} \text{ } 4\sqrt{2} \qquad \textbf{(D)} \text{ 6} \qquad \textbf{(E)} \text{ }\frac{13}{2}$

The numbers from $1$ through $100$ are written in some order on a circle. We call a pair of numbers on the circle [i]good [/i] if the two numbers are not neighbors on the circle and if at least one of the two arcs they determine on the circle only contains numbers smaller then both of them. What may be the total number of good pairs on the circle.

Given the circle of radius $1$ and several its chords with the sum of lengths $1$. Prove that one can be inscribe a regular hexagon into that circle so that its sides don’t intersect those chords.

Is it possible to divide a circle by three chords, different from diameters, into several equal parts?

Nadia bought a compass and after opening its package realized that the length of the needle leg is $10$ centimeters whereas the length of the pencil leg is $16$ centimeters! Assume that in order to draw a circle with this compass, the angle between the pencil leg and the paper must be at least $30$ degrees but the needle leg could be positioned at any angle with respect to the paper. Let $n$ be the difference between the radii of the largest and the smallest circles that Nadia can draw with this compass in centimeters. Which of the following options is closest to $n$? $\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 20$

A regular hexagon of side $1$ is covered by six unit disks. Prove that none of the vertices of the hexagon is covered by two (or more) discs.

Let $P$ be an arbitrary point in the plane of triangle $ABC$. Lines $PA, PB, PC$ meets the perpendicular bisectors of $BC,CA,AB$ at $O_a,O_b,O_c$ respectively. Let $(O_a)$ be the circle with center $O_a$ passing through two points $B,C$, two circles $(O_b), (O_c)$ are defined in the same manner. Two circles $(O_b), (O_c)$ meets at $A_1$, distinct from $A$. Points $B_1,C_1$ are defined in the same manner. Let $Q$ be an arbitrary point in the plane of $ABC$ and $QB,QC$ meets $(O_c)$ and $(O_b)$ at $A_2,A_3$ distinct from $B,C$. Similarly, we have points $B_2,B_3,C_2,C_3$. Let $(K_a), (K_b), (K_c)$ be the circumcircles of triangles $A_1A_2A_3, B_1B_2B_3, C_1C_2C_3$. Prove that (a) three circles $(K_a), (K_b), (K_c)$ have a common point. (b) two triangles $K_aK_bK_c, ABC$ are similar. Trần Quang Hùng

Let $\omega_1$ and $\omega_2$ be two circles , with respective centres $O_1$ and $O_2$ , that intersect each other in $A$ and $B$. The line $O_1A$ intersects $\omega_2$ in $A$ and $C$ and the line $O_2A$ inetersects $\omega_1$ in $A$ and $D$. The line through $B$ parallel to $AD$ intersects $\omega_1$ in $B$ and $E$. Suppose that $O_1A$ is parallel to $DE$. Show that $CD$ is perpendicular to $O_2C$.

Given a circle $k$ and two points $A$ and $B$ outside the circle. Specify how to can construct a circle with a compass and ruler, so that $A$ and $B$ lie on that circle and that circle is tangent to $k$.

Let \(\Omega\) be a circle with a chord \(AB\) which is not a diameter. \(\Gamma_{1}\) be a circle on one side of \(AB\) such that it is tangent to \(AB\) at \(C\) and internally tangent to \(\Omega\) at \(D\). Likewise, let \(\Gamma_{2}\) be a circle on the other side of \(AB\) such that it is tangent to \(AB\) at \(E\) and internally tangent to \(\Omega\) at \(F\). Suppose the line \(DC\) intersects \(\Omega\) at \(X \neq D\) and the line \(FE\) intersects \(\Omega\) at \(Y \neq F\). Prove that \(XY\) is a diameter of \(\Omega\) .

Points $A$ and $B$ are fixed on a circle $c_1$. Circle $c_2$, whose centre lies on $c_1$, touches line $AB$ at $B$. Another line through $A$ intersects $c_2$ at points $D$ and $E$, where $D$ lies between $A$ and $E$. Line $BD$ intersects $c_1$ again at $F$. Prove that line $EB$ is tangent to $c_1$ if and only if $D$ is the midpoint of the segment $BF$.

We are given a circle $K$ with center $S$ and radius $1$ and a square $Q$ with center $M$ and side $2$. Let $XY$ be the hypotenuse of an isosceles right triangle $XY Z$. Describe the locus of points $Z$ as $X$ varies along $K$ and $Y$ varies along the boundary of $Q.$

Circles $S_1$ and $S_2$ intersect at points $M$ and $N$. Through point $A$ of circle $S_1$, draw straight lines $AM$ and $AN$ intersecting $S_2$ at points $B$ and $C$, and through point $D$ of circle $S_2$, draw straight lines $DM$ and $DN$ intersecting $S_1$ at points $E$ and $F$, and $A$, $E$, $F$ lie along one side of line $MN$, and $D$, $B$, $C$ lie on the other side (see figure). Prove that if $AB = DE$, then points $A$, $F$, $C$ and $D$ lie on the same circle, the position of the center of which does not depend on choosing points $A$ and $D$. [img]https://cdn.artofproblemsolving.com/attachments/7/0/d1f9c2f39352e2b39e55bd2538677073618ef9.png[/img]

Let $ABC$ be a triangle with $CA=CB$ and $\angle{ACB}=120^\circ$, and let $M$ be the midpoint of $AB$. Let $P$ be a variable point of the circumcircle of $ABC$, and let $Q$ be the point on the segment $CP$ such that $QP = 2QC$. It is given that the line through $P$ and perpendicular to $AB$ intersects the line $MQ$ at a unique point $N$. Prove that there exists a fixed circle such that $N$ lies on this circle for all possible positions of $P$.