Found problems: 821
Two circles $c$ and $c'$ with centers $O$ and $O'$ lie completely outside each other. Points $A, B$, and $C$ lie on the circle $c$ and points $A', B'$, and $C$ lie on the circle $c'$ so that segment $AB\parallel A'B'$, $BC \parallel B'C'$, and $\angle ABC = \angle A'B'C'$. The lines $AA', BB$', and $CC'$ are all different and intersect in one point $P$, which does not coincide with any of the vertices of the triangles $ABC$ or $A'B'C'$. Prove that $\angle AOB = \angle A'O'B'$.
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]
Circle of radius $R_1$ is inscribed in an acute angle $\alpha$. Second circle with radius $R_2$ touches one of the sides forming the angle $\alpha$ in same point as first circle and intersects the second side in points $A$ and $B$, such that centers of both circles lie inside angle $\alpha$. Prove that $$AB=4\cos{\frac{\alpha}{2}}\sqrt{(R_2-R_1)\left(R_1 \cos^2 \frac{\alpha}{2}+R_2 \sin^2 \frac{\alpha}{2}\right)}$$
Circle $C$ center $O$ touches externally circle $C'$ center $O'$. A line touches $C$ at $A$ and $C'$ at $B$. $P$ is the midpoint of $AB$. Show that $\angle OPO' = 90^o$.
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 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 $\frac{|AO|}{|AA'|}+\frac{|BO|}{|BB'|}+\frac{|CO|}{|CC'|}= 1$
Circles $C_1$ and $C_2$ intersect at different points $A$ and $B$. The straight lines tangents to $C_1$ that pass through $A$ and $B$ intersect at $T$. Let $M$ be a point on $C_1$ that is out of $C_2$. The $MT$ line intersects $C_1$ at $C$ again, the $MA$ line intersects again to $C_2$ in $K$ and the line $AC$ intersects again to the circumference $C_2$ in $L$. Prove that the $MC$ line passes through the midpoint of the $KL$ segment.
There are given $1000$ integers $a_1,... , a_{1000}$. Their squares $a^2_1, . . . , a^2_{1000}$ are written in a circle. It so happened that the sum of any $41$ consecutive numbers on this circle is a multiple of $41^2$. Is it necessarily true that every integer $a_1,... , a_{1000}$ is a multiple of $41$?
(Boris Frenkin)
On the extension of chord $AB$ of a circle centroid at $O$ a point $X$ is taken and tangents $XC$ and $XD$ to the circle are drawn from it with $C$ and $D$ lying on the circle, let $E$ be the midpoint of the line segment $CD$. If $\angle OEB = 140^o$ then determine with proof the magnitude of $\angle AOB$.
Given two intersecting circles with centers $O_1, O_2$. Construct the circle touching one of them externally and the second one internally such that the distance from its center to $O_1O_2$ is maximal.
(M.Volchkevich)
[i]A. Akopyan, A. Akopyan, A. Akopyan, I. Bogdanov[/i]
A conjurer Arutyun and his assistant Amayak are going to show following super-trick. A circle is drawn on the board in the room. Spectators mark $2007$ points on this circle, after that Amayak
removes one of them. Then Arutyun comes to the room and shows a semicircle, to which the removed point belonged. Explain, how Arutyun and Amayak may show this super-trick.
Given two points $M$ and $K$ on the circumference with radius $r_1$ and centre $O_1$. The circumference with radius $r_2$ and centre $O_2$ is inscribed in $\angle MO_1K$ . Find the area of quadrangle $MO_1KO_2$ .
Let $\Gamma$ be a circle and $P$ be a point outside, $PA$ and $PB$ be tangents to $\Gamma$ , $A, B \in \Gamma$ . Point $K$ is an arbitrary point on the segment $AB$. The circumscirbed circle of $\vartriangle PKB$ intersects $\Gamma$ for the second time at point $T$, point $P'$ is symmetric to point $P$ wrt point $A$. Prove that $\angle PBT = \angle P'KA$.
On the circle $(O)$ of radius $15$ cm are given $2$ points $A, B$. The altitude $OH$ of the triangle $OAB$ intersect $(O)$ at $C$. What is $AC$ if $AB = 16$ cm?
$k_1$ denotes one of the arcs formed by intersection of the circumference $k$ and the chord $AB$. $C$ is the middle point of $k_1$. On the half line (ray) $PC$ is drawn the segment $PM$. Find the locus formed from the point $M$ when $P$ is moving on $k_1$.
[i]G. Ganchev[/i]
Segment $AB$ intersects two equal circles, is parallel to the line joining their centres, and all the points of intersection of the segment and the circles lie between $A$ and $B$. From the point $A$ tangents to the circle nearest to $A$ are drawn, and from the point $B$ tangents to the circle nearest to $B$ are also drawn. It turns out that the quadrilateral formed by the four tangents extended contains both circles. Prove that a circle can be drawn so that it touches all four sides of the quadrilateral.
(P Kozhevnikov)
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]
Suppose there are $m$ Martians and $n$ Earthlings at an intergalactic peace conference. To ensure the Martians stay peaceful at the conference, we must make sure that no two Martians sit together, such that between any two Martians there is always at least one Earthling.
(a) Suppose all $m + n$ Martians and Earthlings are seated in a line. How many ways can the Earthlings and Martians be seated in a line?
(b) Suppose now that the $m+n$ Martians and Earthlings are seated around a circular round-table. How many ways can the Earthlings and Martians be seated around the round-table?
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.
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.
Circles $\omega$ and $\Omega$ are inscribed into the same angle. Line $\ell$ meets the sides of angles, $\omega$ and $\Omega$ in points $A$ and $F, B$ and $C, D$ and $E$ respectively (the order of points on the line is $A,B,C,D,E, F$). It is known that$ BC = DE$. Prove that $AB = EF$.
(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)
Fixed two circles $w_1$ and $w_2$, $\ell$ one of their external tangent and $m$ one of their internal tangent . On the line $m$, a point $X$ is chosen, and on the line $\ell$, points $Y$ and $Z$ are constructed so that $XY$ and $XZ$ touch $w_1$ and $w_2$, respectively, and the triangle $XYZ$ contains circles $w_1$ and $w_2$. Prove that the centers of the circles inscribed in triangles $XYZ$ lie on one line.
(P. Kozhevnikov)
In $\Delta ABC$, $AB=6$, $BC=7$ and $CA=8$. Let $D$ be the mid-point of minor arc $AB$ on the circumcircle of $\Delta ABC$. Find $AD^2$
A circle $C$ with radius $3$ has an equilateral triangle inscribed in it. Let $D$ be a circle lying outside the equilateral triangle, tangent to $C$, and tangent to the equilateral triangle at the midpoint of one of its sides. The radius of $D$ can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.