Found problems: 821
In triangle $ABC$, the angle bisectors are $BE$ and $CF$ (where $E, F$ are on the sides of the triangle), and their intersection point is $I$. Point $N$ lies on the circumcircle of $AEF$, and the angle $\angle IAN$ is right. The circumcircle of $AEF$ meets the line $NI$ a second time at the point $L$. Show that the circumcenter of $AIL$ lies on line $BC$.
Let $A, B, C$ be centers of three circles that are mutually tangent externally, let $r_A, r_B, r_C$ be the radii of the circles, respectively. Let $r$ be the radius of the incircle of $\vartriangle ABC$. Prove that $$r^2 \le \frac19 (r_A^2 + r_B^2+r_C^2)$$ and identify, with justification, one case where the equality is attained.
Let $ABC$ be a triangle, and let $r$ denote its inradius. Let $R_A$ denote the radius of the circle internally tangent at $A$ to the circle $ABC$ and tangent to the line $BC$; the radii $R_B$ and $R_C$ are defined similarly. Show that $\frac{1}{R_A} + \frac{1}{R_B} + \frac{1}{R_C}\leq\frac{2}{r}$.
Circles $k_1,k_2$ intersect at $B,C$ such that $BC$ is diameter of $k_1$.Tangent of $k_1$ at $C$ touches $k_2$ for the second time at $A$.Line $AB$ intersects $k_1$ at $E$ different from $B$, and line $CE$ intersects $k_2$ at F different from $C$. An arbitrary line through $E$ intersects segment $AF$ at $H$ and $k_1$ for the second time at $G$.If $BG$ and $AC$ intersect at $D$, prove $CH//DF$ .
The figure shows a square and a circle with a common center $O$, with equal areas of striped shapes. Find the value of $\cos a$.
[img]https://2.bp.blogspot.com/-7uwa0H42ELg/XnmsSoPMgcI/AAAAAAAALgk/pHNBqtbsdKgMhcvIRYLm_8JRpOeIYcUeACK4BGAYYCw/s400/96%2Bestonia%2Bopen%2Bs2.4.png[/img]
$1994$ points from the plane are given so that any $100$ of them can be selected $98$ that can be rounded (some points may be at the border of the circle) with a diameter of $1$. Determine the smallest number of circles with radius $1$, sufficient to cover all $1994$
Given a plane with two circles, one with points $A$ and $B$, and the other with points $C$ and $D$ are shown in the figure. The line $AB$ passes through the center of the first circle and touches the second circle while the line $CD$ passes through the center of the second circle and touches the first circle. Prove that the lines $AD$ and $BC$ are parallel.
[img]https://cdn.artofproblemsolving.com/attachments/e/e/92f7b57751e7828a6487a052d4869e27e658b2.png[/img]
Let $D_1$, $D_2$, ..., $D_n$ be closed discs in the plane. (A closed disc is the region limited by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most $2003$ discs $D_i$. Prove that there exists a disc $D_k$ which intersects at most $7\cdot 2003 - 1 = 14020$ other discs $D_i$.
The circle $\omega$ passes through the vertices $B$ and $C$ of triangle $ABC$ and intersects its sides $AC,AB$ at points $A,E$, respectively. On the ray $BD$, a point $K$ such that $BK = AC$ is chosen , and on the ray $CE$, a point $L$ such that $CL = AB$ is chosen. Prove that the center $O$ of the circumscribed circle of the triangle $AKL$ lies on the circle $\omega$.
Let $I$ be the incenter of the $ABC$ triangle. The circumference that passes through $I$ and has center
in $A$ intersects the circumscribed circumference of the $ABC$ triangle at points $M$ and
$N$. Prove that the line $MN$ is tangent to the inscribed circle of the $ABC$ triangle.
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.
Let $ABC$ be an acute triangle. Suppose that circle $\Gamma_1$ has it's center on the side $AC$ and is tangent to the sides $AB$ and $BC$, and circle $\Gamma_2$ has it's center on the side $AB$ and is tangent to the sides $AC$ and $BC$. The circles $\Gamma_1$ and $ \Gamma_2$ intersect at two points $P$ and $Q$. Show that if $A, P, Q$ are collinear, then $AB = AC$.
A circle is divided into $n$ equal arcs by $n$ points. Assume that, no matter how we color the $n$ points in two colors, there always exists an axis of symmetry of the set of points such that any two of the $n$ points which are symmetric with respect to that axis have the same color. Find all possible values of $n$.
A kingdom consists of $12$ cities located on a one-way circular road. A magician comes on the $13$th of every month to cast spells. He starts at the city which was the 5th down the road from the one that he started at during the last month (for example, if the cities are numbered $1–12$ clockwise, and the direction of travel is clockwise, and he started at city #$9$ last month, he will start at city #$2$ this month). At each city that he visits, the magician casts a spell if the city is not already under the spell, and then moves on to the next city. If he arrives at a city which is already under the spell, then he removes the spell from this city, and leaves the kingdom until the next month. Last Thanksgiving the capital city was free of the spell. Prove that it will be free of the spell this Thanksgiving as well.
Let $\Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4$ be distinct circles such that $\Gamma_1,\Gamma_3$ are externally tangent at $P$, and $\Gamma_2,\Gamma_4$ are externally tangent at the same point $P$. Suppose that $\Gamma_1$ and $\Gamma_2$; $\Gamma_2$ and $\Gamma_3$; $\Gamma_3$ and $\Gamma_4$; $\Gamma_4$ and $\Gamma_1$ meet at $A,B,C,D,$ respectively, and that all of these points are different from $P$. Prove that
$$\frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}$$
Given are $n$ different concentric circles on the plane.Inside the disk with the smallest radius (strictly inside it),we consider two distinct points $A,B$.We consider $k$ distinct lines passing through $A$ and $m$ distinct lines passing through $B$.There is no line passing through both $A$ and $B$ and all the lines passing through $k$ intersect with all the lines passing through $B$.The intersections do not lie on some of the circles.Determine the maximum and the minimum number of regions formed by the lines and the circles and are inside the circles.
Let $\Gamma_1$ and $\Gamma_2$ be the circles with diameters $AP$ and $AQ$. Let $T$ be another point of intersection of the circles $\Gamma_1$ and $\Gamma_2$. Let $Q_1$ be another point of intersection of the circle $\Gamma_1$ and the line $AQ$, and $P_1$ the other point of intersection of the circle $\Gamma_2$ and the line $AP$. The circle $\Gamma_3$ passes through the points $T$, $P$ and $P_1$ and the circle $\Gamma_4$ passes through the points $T$, $Q$ and $Q_1$. Prove that the line containing the common chord of the circles $\Gamma_3$ and $\Gamma_4$ passes through$A$.
The two circles $\Gamma_1$ and $\Gamma_2$ intersect at $P$ and $Q$. The common tangent that's on the same side as $P$, intersects the circles at $A$ and $B$,respectively. Let $C$ be the second intersection with $\Gamma_2$ of the tangent to $\Gamma_1$ at $P$, and let $D$ be the second intersection with $\Gamma_1$ of the tangent to $\Gamma_2$ at $Q$. Let $E$ be the intersection of $AP$ and $BC$, and let $F$ be the intersection of $BP$ and $AD$. Let $M$ be the image of $P$ under point reflection with respect to the midpoint of $AB$. Prove that $AMBEQF$ is a cyclic hexagon.
Each member of a set of circles in the $xy$-plane is tangent to the $x$-axis and no two of the circles intersect. Show that
(a) the points of tangency can include all rational points on the axis.
(b) the points of tangency cannot include all the irrational points.
Let $A$ and $B$ be points on the circle $k$ with center $O$, so that $AB> AO$. Let $C$ be the intersection of the bisectors of $\angle OAB$ and $k$, different from $A$. Let $D$ be the intersection of the straight line $AB$ with the circumcircle of the triangle $OBC$, different from $B$. Show that $AD = AO$ .
Let $A$ be one of the two points where the circles whose centers are the points $M$ and $N$ intersect. The tangents in $A$ to such circles intersect them again in $B$ and $C$, respectively. Let $P$ be a point such that the quadrilateral $AMPN$ is a parallelogram. Show that $P$ is the circumcenter of triangle $ABC$.
Let $ABC$ be an arbitrary scalene triangle. Define $\sum$ to be the set of all circles $y$ that have the following properties:
[b](i)[/b] $y$ meets each side of $ABC$ in two (possibly coincident) points;
[b](ii)[/b] if the points of intersection of $y$ with the sides of the triangle are labeled by $P, Q, R, S, T , U$, with the points occurring on the sides in orders $\mathcal B(B,P,Q,C), \mathcal B(C, R, S,A), \mathcal B(A, T,U,B)$, then the following relations of parallelism hold: $TS \parallel BC; PU\parallel CA; RQ\parallel AB$. (In the limiting cases, some of the conditions of parallelism will hold vacuously; e.g., if $A$ lies on the circle $y$, then $T$ , $S$ both coincide with $A$ and the relation $TS \parallel BC$ holds vacuously.)
[i](a)[/i] Under what circumstances is $\sum$ nonempty?
[i](b)[/i] Assuming that Σ is nonempty, show how to construct the locus of centers of the circles in the set $\sum$.
[i](c)[/i] Given that the set $\sum$has just one element, deduce the size of the largest angle of $ABC.$
[i](d)[/i] Show how to construct the circles in $\sum$ that have, respectively, the largest and the smallest radii.
Square $ABCD$ and circle $O$ intersect in eight points, forming four curvilinear triangles, $AEF , BGH , CIJ$ and $DKL$ ($EF , GH, IJ$ and $KL$ are arcs of the circle) . Prove that
(a) The sum of lengths of $EF$ and $IJ$ equals the sum of the lengths of $GH$ and $KL$.
(b) The sum of the perimeters of curvilinear triangles $AEF$ and $CIJ$ equals the sum of the perimeters of the curvilinear triangles $BGH$ and $DKL$.
( V . V . Proizvolov , Moscow)
In the figure, the chord $[CD]$ is perpendicular to the diameter $[AB]$ and intersects it at $H$. Length of $AB$ is a two-digit natural number. Changing the order of these two digits gives length of $CD$. Knowing that distance from $H$ to the center $O$ is a positive rational number, calculate $AB$.
[img]https://cdn.artofproblemsolving.com/attachments/5/f/eb9c61579a38118b4f753bbc19a9a50e0732dc.png[/img]
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 $BF$.