Found problems: 821
2012 Denmark MO - Mohr Contest, 1
Inside a circle with radius $6$ lie four smaller circles with centres $A,B,C$ and $D$. The circles touch each other as shown. The point where the circles with centres $A$ and $C$ touch each other is the centre of the big circle. Calculate the area of quadrilateral $ABCD$.
[img]https://1.bp.blogspot.com/-FFsiOOdcjao/XzT_oJYuQAI/AAAAAAAAMVk/PpyUNpDBeEIESMsiElbexKOFMoCXRVaZwCLcBGAsYHQ/s0/2012%2BMohr%2Bp1.png[/img]
2020 BMT Fall, 9
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 $m + n$.
2019 Tuymaada Olympiad, 2
A triangle $ABC$ with $AB < AC$ is inscribed in a circle $\omega$. Circles $\gamma_1$ and $\gamma_2$ touch the lines $AB$ and $AC$, and their centres lie on the circumference of $\omega$. Prove that $C$ lies on a common external tangent to $\gamma_1$ and $\gamma_2$.
1999 Estonia National Olympiad, 5
The numbers $0, 1, 2, . . . , 9$ are written (in some order) on the circumference. Prove that
a) there are three consecutive numbers with the sum being at least $15$,
b) it is not necessarily the case that there exist three consecutive numbers with the sum more than $15$.
2010 All-Russian Olympiad Regional Round, 10.6
The tangent lines to the circle $\omega$ at points $B$ and $D$ intersect at point $P$. The line passing through $P$ cuts out from circle chord $AC$. Through an arbitrary point on the segment $AC$ a straight line parallel to $BD$ is drawn. Prove that it divides the lengths of polygonal $ABC$ and $ADC$ in the same ratio.
[hide=last sentence was in Russian: ]Докажите, что она делит длины ломаных ABC и ADC в одинаковых отношениях. [/hide]
1985 Tournament Of Towns, (099) 3
A teacher gives each student in the class the following task in their exercise book .
"Take two concentric circles of radius $1$ and $10$ . To the smaller circle produce three tangents whose intersections $A, B$ and $C$ lie in the larger circle . Measure the area $S$ of triangle $ABC$, and areas $S_1 , S_2$ and $S_3$ , the three sector-like regions with vertices at $A, B$ and $C$ (as in the diagram). Find the value of $S_1 +S_2 +S_3 -S$."
Prove that each student would obtain the same result .
[img]https://1.bp.blogspot.com/-K3kHWWWgxgU/XWHRQ8WqqPI/AAAAAAAAKjE/0iO4-Yz6p9AcM2mklprX_M18xTyg9O81gCK4BGAYYCw/s200/TOT%2B1985%2BAutumn%2BJ3.png[/img]
( A . K . Tolpygo, Kiev)
1999 Argentina National Olympiad, 2
Let $C_1$ and $C_2$ be the outer circumferences of centers $O_1$ and $O_2$, respectively. The two tangents to the circumference $C_2$ are drawn by $O_1$, intersecting $C_1$ at $P$ and $P'$. The two tangents to the circumference $C_1$ are drawn by $O_2$, intersecting $C_2$ at $Q$ and $Q'$. Prove that the segment $PP'$ is equal to the segment $QQ'$.
2017 Turkey Team Selection Test, 3
At the $ABC$ triangle the midpoints of $BC, AC, AB$ are respectively $D, E, F$ and the triangle tangent to the incircle at $G$, $H$ and $I$ in the same order.The midpoint of $AD$ is $J$. $BJ$ and $AG$ intersect at point $K$. The $C-$centered circle passing through $A$ cuts the $[CB$ ray at point $X$. The line passing through $K$ and parallel to the $BC$ and $AX$ meet at $U$. $IU$ and $BC$ intersect at the $P$ point. There is $Y$ point chosen at incircle. $PY$ is tangent to incircle at point $Y$. Prove that $D, E, F, Y$ are cyclic.
Indonesia MO Shortlist - geometry, g2
Two circles that are not equal are tangent externally at point $R$. Suppose point $P$ is the intersection of the external common tangents of the two circles. Let $A$ and $B$ are two points on different circles so that $RA$ is perpendicular to $RB$. Show that the line $AB$ passes through $P$.
2019 Yasinsky Geometry Olympiad, p3
Two circles $\omega_1$ and $\omega_2$ are tangent externally at the point $P$. Through the point $A$ of the circle $\omega_1$ is drawn a tangent to this circle, which intersects the circle $\omega_2$ at points $B$ and $C$ (see figure). Line $CP$ intersects again the circle $\omega_1$ to $D$. Prove that the $PA$ is a bisector of the angle $DPB$.
[img]https://1.bp.blogspot.com/-nmKZGdBXfao/XOd51gRFuyI/AAAAAAAAKO0/EYo2SCW0eGcJsF64-Avo6w73ugkIIQ30ACK4BGAYYCw/s1600/Yasinsky%2B2019%2Bp2.png[/img]
2006 Sharygin Geometry Olympiad, 9
$L(a)$ is the line connecting the points of the unit circle corresponding to the angles $a$ and $\pi - 2a$.
Prove that if $a + b + c = 2\pi$, then the lines $L (a), L (b)$ and $L (c)$ intersect at one point.
2000 Croatia National Olympiad, Problem 2
Let $ABC$ be a triangle with $AB = AC$. With center in a point of the side $BC$, the circle $S$ is constructed that is tangent to the sides $AB$ and $AC$. Let $P$ and $Q$ be any points on the sides $AB$ and $AC$ respectively, such that $PQ$ is tangent to $S$. Show that $PB \cdot CQ = \left(\frac{BC}{2}\right)^2$
2006 Korea Junior Math Olympiad, 7
A line through point $P$ outside of circle $O$ meets the said circle at $B,C$ ($PB < PC$). Let $PO$ meet circle $O$ at $Q,D$ (with $PQ < PD$). Let the line passing $Q$ and perpendicular to $BC$ meet circle $O$ at $A$. If $BD^2 = AD\cdot CP$, prove that $PA$ is a tangent to $O$.
1984 All Soviet Union Mathematical Olympiad, 393
Given three circles $c_1,c_2,c_3$ with $r_1,r_2,r_3$ radiuses, $r_1 > r_2, r_1 > r_3$. Each lies outside of two others. The A point -- an intersection of the outer common tangents to $c_1$ and $c_2$ -- is outside $c_3$. The $B$ point -- an intersection of the outer common tangents to $c_1$ and $c_3$ -- is outside $c_2$. Two pairs of tangents -- from $A$ to $c_3$ and from $B$ to $c_2$ -- are drawn. Prove that the quadrangle, they make, is circumscribed around some circle and find its radius.
1962 IMO, 6
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 \[ d=\sqrt{R(R-2r)} \]
2001 239 Open Mathematical Olympiad, 5
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 $.
2022 China Team Selection Test, 1
Given two circles $\omega_1$ and $\omega_2$ where $\omega_2$ is inside $\omega_1$. Show that there exists a point $P$ such that for any line $\ell$ not passing through $P$, if $\ell$ intersects circle $\omega_1$ at $A,B$ and $\ell$ intersects circle $\omega_2$ at $C,D$, where $A,C,D,B$ lie on $\ell$ in this order, then $\angle APC=\angle BPD$.
Estonia Open Senior - geometry, 1996.2.4
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]
2015 Peru IMO TST, 8
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.
2003 Cuba MO, 2
Let $A$ be a point outside the circle $\omega$ . The tangents from $A$ touch the circle at $B$ and $C$. Let $P$ be an arbitrary point on extension of $AC$ towards $C$, $Q$ the projection of $C$ onto $PB$ and $E$ the second intersection point of the circumcircle of $ABP$ with the circle $\omega$ . Prove that $\angle PEQ = 2\angle APB$
1984 All Soviet Union Mathematical Olympiad, 384
The centre of the coin with radius $r$ is moved along some polygon with the perimeter $P$, that is circumscribed around the circle with radius $R$ ($R>r$). Find the coin trace area (a sort of polygon ring).
1991 Denmark MO - Mohr Contest, 5
Show that no matter how $15$ points are plotted within a circle of radius $2$ (circle border included), there will be a circle with radius $1$ (circle border including) which contains at least three of the $15$ points.
2009 Greece JBMO TST, 2
Given convex quadrilateral $ABCD$ inscribed in circle $(O,R)$ (with center $O$ and radius $R$). With centers the vertices of the quadrilateral and radii $R$, we consider the circles $C_A(A,R), C_B(B,R), C_C(C,R), C_D(D,R)$. Circles $C_A$ and $C_B$ intersect at point $K$, circles $C_B$ and $C_C$ intersect at point $L$, circles $C_C$ and $C_D$ intersect at point $M$ and circles $C_D$ and $C_A$ intersect at point $N$ (points $K,L,M,N$ are the second common points of the circles given they all pass through point $O$). Prove that quadrilateral $KLMN$ is a parallelogram.
1994 North Macedonia National Olympiad, 4
$1994$ 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 $1994$
2002 Abels Math Contest (Norwegian MO), 3a
A circle with center in $O$ is given. Two parallel tangents tangent to the circle at points $M$ and $N$. Another tangent intersects the first two tangents at points $K$ and $L$. Show that the circle having the line segment $KL$ as diameter passes through $O$.