Three circles are pair-wise tangent to each other. Prove that the circle passing through the three tangent points is perpendicular to each of the initial three circles.

There are $100$ school students from two clubs $A$ and $B$ standing in circle. Among them $62$ students stand next to at least one student from club $A$, and $54$ students stand next to at least one student from club $B$. 1) How many students stand side-by-side with one friend from club $A$ and one friend from club $B$? 2) What is the number of students from club $A$?

(a) $10$ points dividing a circle into $10$ equal arcs are connected in pairs by $5$ chords. Is it necessary that two of these chords are of equal length? (b) $20$ points dividing a circle into $20$ equal arcs are connected in pairs by $10$ chords. Prove that among these $10$ chords there are two chords of equal length. (VV Proizvolov, Moscow)

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 $P.$

The two intersections of the circles $k_i$ and $k_{i + 1}$ are $P_i$ and $Q_i$ ($1 \le i \le 5, k_6 = k_1$). On the circle $k_1$ lies an arbitrary point $A$. Then the points $B, C, D, E, F, G, H, I, J, K$ lie on the circles $k_2, k_3, k_4, k_5, k_1, k_2, k_3, k_4, k_5, k_1$ respectively, such that $AP_1B, BP_2C, CP_3D, DP_4E, EP_5F, F Q_1G, GQ_2H, HQ_3I, IQ_4J, JQ_5K$ are straight line triplets. Prove that that $K = A$. [img]https://1.bp.blogspot.com/-g6rF1hcPE08/X9j1SEJT7-I/AAAAAAAAMzc/2rWIiWTHZ34zfWVeGujkCxRW1hSCw5oOwCLcBGAsYHQ/s16000/2016%2BDurer%2BC..4.png[/img] [i]Circles can have different radii, and They can be located in different ways from the figure. We assume that during editing none neither of the two points mentioned above coincide.[/i]

Consider an isosceles triangle. let $R$ be the radius of its circumscribed circle and $r$ be the radius of its inscribed circle. Prove that the distance $d$ between the centers of these two circle is \[ d=\sqrt{R(R-2r)} \]

A parallelogram $ABCD$ is given. Two circles with centers at the vertices $A$ and $C$ pass through $B$. The straight line $\ell$ that passes through $B$ and crosses the circles at second time at points $X, Y$ respectively. Prove that $DX = DY$.

Three circles of radius $1$ are inscribed in a square of side length $s$, such that the circles do not overlap or coincide with each other. What is the minimum $s$ where such a configuration is possible?

Three circles $S_A, S_B$, and $S_C$ in the plane with centers in $A, B$, and $C$, respectively, are mutually tangential on the outside. The touchpoint between $S_A$ and $S_B$ we call $C'$, the one $S_A$ between $S_C$ we call $B'$, and the one between $S_B$ and $S_C$ we call $A'$. The common tangent between $S_A$ and $S_C$ (passing through B') we call $\ell_B$, and the common tangent between $S_B$ and $S_C$ (passing through $A'$) we call $\ell_A$. The intersection point of $\ell_A$ and $\ell_B$ is called $X$. The point $Y$ is located so that $\angle XBY$ and $\angle YAX$ are both right angles. Show that the points $X, Y$, and $C'$ lie on a line if and only if $AC = BC$.

Consider two fixed points $B,C$ on a circle $w$. Find the locus of the incenters of all triangles $ABC$ when point $A$ describes $w$.

One hundred points are marked inside a circle, with no three in a line. Prove that it is possible to connect the points in pairs such that all fifty lines intersect one another inside the circle.

Prove that the circle with radius $2$ can be completely covered with $7$ unit circles

Georg and his mother love pizza. They buy a pizza shaped as an equilateral triangle. Georg demands to be allowed to divide the pizza by a straight cut and then make the first choice. The mother accepts this reluctantly, but she wants to choose a point of the pizza through which the cut must pass. Determine the largest fraction of the pizza which the mother is certain to get by this procedure.

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 $Q.$

The circles $ S_1 $ and $ S_2 $ intersect at points $ A $ and $ B $. Circle $ S_3 $ externally touches $ S_1 $ and $ S_2 $ at points $ C $ and $ D $ respectively. Let $ K $ be the midpoint of the chord cut by the line $ AB $ on circles $ S_3 $. Prove that $ \angle CKA = \angle DKA $.

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 $k.$

Consider a triangle $ABC$ such that $\angle A = 90^o, \angle C =60^o$ and $|AC|= 6$. Three circles with centers $A, B$ and $C$ are pairwise tangent in points on the three sides of the triangle. Determine the area of the region enclosed by the three circles (the grey area in the figure). [asy] unitsize(0.2 cm); pair A, B, C; real[] r; A = (6,0); B = (6,6*sqrt(3)); C = (0,0); r[1] = 3*sqrt(3) - 3; r[2] = 3*sqrt(3) + 3; r[3] = 9 - 3*sqrt(3); fill(arc(A,r[1],180,90)--arc(B,r[2],270,240)--arc(C,r[3],60,0)--cycle, gray(0.7)); draw(A--B--C--cycle); draw(Circle(A,r[1])); draw(Circle(B,r[2])); draw(Circle(C,r[3])); dot("$A$", A, SE); dot("$B$", B, NE); dot("$C$", C, SW); [/asy]

Three disks touch pairwise from outside in the points $X,Y,Z$. Then the radiuses of the disks were expanded by $2/\sqrt3$ times, and the centres were reserved. Prove that the triangle $XYZ$ is completely covered by the expanded disks.

In the right triangle $ABC$, $AC=12$, $BC=5$, and angle $C$ is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle? [asy] draw((0,0)--(12,0)--(12,5)--(0,0)); draw(arc((8.67,0),(12,0),(5.33,0))); label("$A$", (0,0), W); label("$C$", (12,0), E); label("$B$", (12,5), NE); label("$12$", (6, 0), S); label("$5$", (12, 2.5), E);[/asy] $\textbf{(A) }\frac{7}{6}\qquad\textbf{(B) }\frac{13}{5}\qquad\textbf{(C) }\frac{59}{18}\qquad\textbf{(D) }\frac{10}{3}\qquad\textbf{(E) }\frac{60}{13}$

For any set of points $A_1, A_2,...,A_n$ on the plane, one defines $r( A_1, A_2,...,A_n)$ as the radius of the smallest circle that contains all of these points. Prove that if $n \ge 3$, there are indices $i,j,k$ such that $r( A_1, A_2,...,A_n)=r( A_i, A_j,A_k)$

The two circles $\Gamma_1$ and $\Gamma_2$ with the midpoints $O_1$ resp. $O_2$ intersect in the two distinct points $A$ and $B$. A line through $A$ meets $\Gamma_1$ in $C \neq A$ and $\Gamma_2$ in $D \neq A$. The lines $CO_1$ and $DO_2$ intersect in $X$. Prove that the four points $O_1,O_2,B$ and $X$ are concyclic.

Let $A$ and $B$ be two points inside a circle $C$. Show that there exists a circle that contains $A$ and $B$ and lies completely inside $C$.

The figure shows a large circle with radius $2$ m and four small circles with radii $1$ m. It is to be painted using the three shown colours. What is the cost of painting the figure? [img]https://1.bp.blogspot.com/-oWnh8uhyTIo/XzP30gZueKI/AAAAAAAAMUY/GlC3puNU_6g6YRf6hPpbQW8IE8IqMP3ugCLcBGAsYHQ/s0/2018%2BMohr%2Bp2.png[/img]

In a circumference $\Gamma$ a chord $PQ$ is considered such that the segment that joins the midpoint of the smallest arc $PQ$ and the midpoint of the segment $PQ$ measures $1$. Let $\Gamma_1, \Gamma_2$ and $\Gamma_3$ be three tangent circumferences to the chord $PQ$ that are in the same half plane than the center of $\Gamma$ with respect to the line $PQ$. Furthermore, $\Gamma_1$ and $\Gamma_3$ are internally tangent to $\Gamma$ and externally tangent to$ \Gamma_2$, and the centers of $\Gamma_1$ and $\Gamma_3$ are on different halfplanes with respect to the line determined by the centers of $\Gamma$ and $\Gamma_2$. If the sum of the radii of $\Gamma_1, \Gamma_2$ and $\Gamma_3$ is equal to the radius of $\Gamma$, calculate the radius of $\Gamma_2$.

Two circles $\gamma_1, \gamma_2$ are given, with centers at points $O_1, O_2$ respectively. Select a point $K$ on circle $\gamma_2$ and construct two circles, one $\gamma_3$ that touches circle $\gamma_2$ at point $K$ and circle $\gamma_1$ at a point $A$, and the other $\gamma_4$ that touches circle $\gamma_2$ at point $K$ and circle $\gamma_1$ at a point $B$. Prove that, regardless of the choice of point K on circle $\gamma_2$, all lines $AB$ pass through a fixed point of the plane.