Found problems: 821
Let: $ C_1, C_2, C_3 $ three circles , intersecting in pairs, such that the secant line common to two of them (any) passes through the center of the third. Prove that the three lines thus defined are concurrent.
Two concentric circles have radii $R$ and $r$ respectively. Determine the greatest possible number of circles that are tangent to both these circles and mutually nonintersecting. Prove that this number lies between $\frac 32 \cdot \frac{\sqrt R +\sqrt r }{\sqrt R -\sqrt r } -1$ and $\frac{63}{20} \cdot \frac{R+r}{R-r}.$
Let $ABCD$ be a square with circumcircle $\Gamma_1$. Let $P$ be a point on the arc $AC$ that also contains $B$. A circle $\Gamma_2$ touches $\Gamma_1$ in $P$ and also touches the diagonal $AC$ in $Q$. Let $R$ be a point on $\Gamma_2$ such that the line $DR$ touches $\Gamma_2$. Proof that $|DR| = |DA|$.
Let $\Omega$ be a circle disk with radius $1$. Determine the minimum $r$ that has the following property:
You can select three points on $\Omega$ so that each circle disk located in $\Omega$ and has a radius greater than $r$ contains at least one of the three points.
Two circles are given in the plane, $\Omega$ and inside it $\omega$. The center of $\omega$ is $I$. $P$ is a point moving on $\Omega$. The second intersection of the tangents from $P$ to $\omega$ and circle $\Omega$ are $Q$ and $R.$ The second intersection of circle $IQR$ and lines $PI$, $PQ$ and $PR$ are $J$, $S$ and $T,$ respectively. The reflection of point $J$ across line $ST$ is $K.$
Prove that lines $PK$ are concurrent.
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 $r( A_1, A_2,...,A_n)=r( A_i, A_j,A_k)$
A square with sides $1$ and four circles of radius $1$ considered each having a vertex of have the square as the center. Find area of the shaded part (see figure).
[img]https://cdn.artofproblemsolving.com/attachments/b/6/6e28d94094d69bac13c2702853ac2c906a80a1.png[/img]
Let $\Gamma_1$ be a circle with centre $A$ and $\Gamma_2$ be a circle with centre $B$, with $A$ lying on $\Gamma_2$. On $\Gamma_2$ there is a (variable) point $P$ not lying on $AB$. A line through $P$ is a tangent of $\Gamma_1$ at $S$, and it intersects $\Gamma_2$ again in $Q$, with $P$ and $Q$ lying on the same side of $AB$. A different line through $Q$ is tangent to $\Gamma_1$ at $T$. Moreover, let $M$ be the foot of the perpendicular to $AB$ through $P$. Let $N$ be the intersection of $AQ$ and $MT$.
Show that $N$ lies on a line independent of the position of $P$ on $\Gamma_2$.
Two circles centers $O$ and $O'$, radii $R$ and $R'$, meet at two points. A variable line $L$ meets the circles at $A, C, B, D$ in that order and $\frac{AC}{AD} = \frac{CB}{BD}$. The perpendiculars from $O$ and $O'$ to $L$ have feet $H$ and $H'$.
Find the locus of $H$ and $H'$.
If $OO'^2 < R^2 + R'^2$, find a point $P$ on $L$ such that $PO + PO'$ has the smallest possible value.
Show that this value does not depend on the position of $L$.
Comment on the case $OO'^2 > R^2 + R'^2$.
On the plane there are two non-intersecting circles with equal radii and with centres $O_1$ and $O_2$, line $s$ going through these centres, and their common tangent $t$. The third circle is tangent to these two circles in points $K$ and $L$ respectively, line $s$ in point $M$ and line $t$ in point $P$. The point of tangency of line $t$ and the first circle is $N$.
a) Find the length of the segment $O_1O_2$.
b) Prove that the points $M, K$ and $N$ lie on the same line
The medians $AM$ and $BN$ of a triangle $ABC$ are the diameters of the circles $\omega_1$ and $\omega_2$. If $\omega_1$ touches the altitude $CH$, prove that $\omega_2$ also touches $CH$.
I. Gorodnin
Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$.
(1) Prove that there exists an equilateral triangle whose vertices lie in different discs.
(2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$.
[i]Radu Gologan, Romania[/i]
[hide="Remark"]
The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url].
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Circles $\Gamma_1$ and $\Gamma_2$ meet at points $X$ and $Y$. A circle $S_1$ touches internally $\Gamma_1$ at $A$ and $\Gamma_2$ externally at $B$. A circle $S_2$ touches $\Gamma_2$ internally at $C$ and $\Gamma_1$ externally at $D$. Prove that the points $A, B, C, D$ are either collinear or concyclic.
(A. Voidelevich)
Let $\omega$ be a circle with center $O$ and let $A$ be a point outside $\omega$. The tangents from $A$ touch $\omega$ at points $B$, and $C$. Let $D$ be the point at which the line $AO$ intersects the circle such that $O$ is between $A$ and $D$. Denote by $X$ the orthogonal projection of $B$ onto $CD$, by $Y$ the midpoint of the segment $BX$ and by $Z$ the second point of intersection of the line $DY$ with $\omega$. Prove that $ZA$ and $ZC$ are perpendicular to each other.
Suppose $28$ objects are placed along a circle at equal distances. In how many ways can $3$ objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?
On the unit circle $k$ with center $O$, points $A$ and $B$ with $AB=1$ are chosen and unit circles $k_1$ and $k_2$ with centers $A$ and $B$ are drawn. A sequence of circles $(l_n)$ is defined as follows: circle $l_1$ is tangent to $k$ internally at $D_1$ and to $k_1,k_2$ externally, and for $n>1$ circle $l_n$ is tangent to $k_1$ and $k_2$ and to $l_{n-1}$ at $D_n$. For each $n$, compute $d_n=OD_n$ and the radius $r_n$ of $l_n$.
In the figure shown, the small circles have radius $1$. Calculate the area of the gray part of the figure.
[img]https://1.bp.blogspot.com/-oy-WirJ6u9o/XzcFc3roVDI/AAAAAAAAMX8/qxNy5I_0RWUOxl-ZE52fnrwo0v0T7If9QCLcBGAsYHQ/s0/1998%2BMohr%2Bp1.png[/img]
In a coordinate system, a circle with radius $7$ and center is on the y-axis placed inside the parabola with equation $y = x^2$ , so that it just touches the parabola in two points. Determine the coordinate set for the center of the circle.
Circles with centers $P, Q$ and $R$, having radii $1, 2$ and $3$, respectively, lie on the same side of line $l$ and are tangent to $l$ at $P', Q'$ and $R'$, respectively, with $Q'$ between $P'$ and $R'$. The circle with center $Q$ is externally tangent to each of the other two circles. What is the area of triangle $PQR$?
$\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}$
A circle $\Gamma$ and a line $\ell$ which does not intersect $\Gamma$ are given. Suppose $P, Q,R, S$ are variable points on circle $\Gamma$ such that the points $A = PQ\cap RS$ and $B = PS \cap QR$ lie on $\ell$. Prove that the circle on $AB$ as a diameter passes through two fixed points.
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 $C$.
Let $\triangle ABC$ be a triangle and let $X$ be a point in the interior of the triangle. The second intersection points of the lines $XA,XB$ and $XC$ with the circumcircle of $\triangle ABC$ are $P,Q$ and $R$. Let $U$ be a point on the ray $XP$ (these are the points on the line $XP$ such that $P$ and $U$ lie on the same side of $X$). The line through $U$ parallel to $AB$ intersects $BQ$ in $V$ . The line through $U$ parallel to $AC$ intersects $CR$ in $W$. Prove that $Q, R, V$ , and $W$ lie on a circle.
$ABC$ is an acute triangle with $AB> AC$. $\Gamma_B$ is a circle that passes through $A,B$ and is tangent to $AC$ on $A$. Define similar for $ \Gamma_C$. Let $D$ be the intersection $\Gamma_B$ and $\Gamma_C$ and $M$ be the midpoint of $BC$. $AM$ cuts $\Gamma_C$ at $E$. Let $O$ be the center of the circumscibed circle of the triangle $ABC$. Prove that the circumscibed circle of the triangle $ODE$ is tangent to $\Gamma_B$.
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.
We are given three points $A,B,C$ on a semicircle. The tangent lines at $A$ and $B$ to the semicircle meet the extension of the diameter at points $M,N$ respectively. The line passing through $A$ that is perpendicular to the diameter meets $NC$ at $R$, and the line passing through $B$ that is perpendicular to the diameter meets $MC$ at $S$. If the line $RS$ meets the extension of the diameter at $Z$, prove that $ZC$ is tangent to the semicircle.