Found problems: 821
Swiss NMO - geometry, 2017.1
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 $AD = AO$ .
2016 Regional Olympiad of Mexico Southeast, 1
In a circumference there are $99$ natural numbers. If $a$ and $b$ are two consecutive numbers in the circle, then they must satisfies one of the following conditions: $a-b=1, a-b=2$ or $\frac{a}{b}=2$. Prove that, in the circle exists a number multiple of $3$.
2014 Sharygin Geometry Olympiad, 7
Nine circles are drawn around an arbitrary triangle as in the figure. All circles tangent to the same side of the triangle have equal radii. Three lines are drawn, each one connecting one of the triangle’s vertices to the center of one of the circles touching the opposite side, as in the figure. Show that the three lines are concurrent.
(N. Beluhov)
1984 Polish MO Finals, 3
Let $W$ be a regular octahedron and $O$ be its center. In a plane $P$ containing $O$ circles $k_1(O,r_1)$ and $k_2(O,r_2)$ are chosen so that $k_1 \subset P\cap W \subset k_2$. Prove that $\frac{r_1}{r_2}\le \frac{\sqrt3}{2}$
2003 IMO Shortlist, 2
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 $D_i$.
2008 Estonia Team Selection Test, 5
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 $BF$.
1945 Moscow Mathematical Olympiad, 105
A circle rolls along a side of an equilateral triangle. The radius of the circle is equal to the height of the triangle. Prove that the measure of the arc intercepted by the sides of the triangle on this circle is equal to $60^o$ at all times.
1979 IMO Shortlist, 24
A circle $C$ with center $O$ on base $BC$ of an isosceles triangle $ABC$ is tangent to the equal sides $AB,AC$. If point $P$ on $AB$ and point $Q$ on $AC$ are selected such that $PB \times CQ = (\frac{BC}{2})^2$, prove that line segment $PQ$ is tangent to circle $C$, and prove the converse.
Indonesia MO Shortlist - geometry, g3
Suppose $L_1$ is a circle with center $O$, and $L_2$ is a circle with center $O'$. The circles intersect at $ A$ and $ B$ such that $\angle OAO' = 90^o$. Suppose that point $X$ lies on the circumcircle of triangle $OAB$, but lies inside $L_2$. Let the extension of $OX$ intersect $L_1$ at $Y$ and $Z$. Let the extension of $O'X$ intersect $L_2$ at $W$ and $V$ . Prove that $\vartriangle XWZ$ is congruent with $\vartriangle XYV$.
Denmark (Mohr) - geometry, 2007.4
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]
2013 Saudi Arabia GMO TST, 4
In acute triangle $ABC$, points $D$ and $E$ are the feet of the perpendiculars from $A$ to $BC$ and $B$ to $CA$, respectively. Segment $AD$ is a diameter of circle $\omega$. Circle $\omega$ intersects sides $AC$ and $AB$ at $F$ and $G$ (other than $A$), respectively. Segment $BE$ intersects segments $GD$ and $GF$ at $X$ and $Y$ respectively. Ray $DY$ intersects side $AB$ at $Z$. Prove that lines $XZ$ and $BC$ are perpendicular
2021 Iran RMM TST, 1
Suppose that two circles $\alpha, \beta$ with centers $P,Q$, respectively , intersect orthogonally at $A$,$B$. Let $CD$ be a diameter of $\beta$ that is exterior to $\alpha$. Let $E,F$ be points on $\alpha$ such that $CE,DF$ are tangent to $\alpha$ , with $C,E$ on one side of $PQ$ and $D,F$ on the other side of $PQ$. Let $S$ be the intersection of $CF,AQ$ and $T$ be the intersection of $DE,QB$. Prove that $ST$ is parallel to $CD$ and is tangent to $\alpha$
2006 Estonia National Olympiad, 2
Prove that the circle with radius $2$ can be completely covered with $7$ unit circles
1966 IMO Shortlist, 14
What is the maximal number of regions a circle can be divided in by segments joining $n$ points on the boundary of the circle ?
[i]Posted already on the board I think...[/i]
1985 IMO Longlists, 5
If possible, construct an equilateral triangle whose three vertices are on three given circles.
1966 IMO Longlists, 28
In the plane, consider a circle with center $S$ and radius $1.$ Let $ABC$ be an arbitrary triangle having this circle as its incircle, and assume that $SA\leq SB\leq SC.$ Find the locus of
[b]a.)[/b] all vertices $A$ of such triangles;
[b]b.)[/b] all vertices $B$ of such triangles;
[b]c.)[/b] all vertices $C$ of such triangles.
2001 239 Open Mathematical Olympiad, 3
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 $ PQ $ be a chord cut by the line $ AB $ on circle $ S_3 $, and $ K $ be the midpoint of $ CD $. Prove that $ \angle PKC = \angle QKC $.
2008 Tournament Of Towns, 7
Each of three lines cuts chords of equal lengths in two given circles. The points of intersection of these lines form a triangle. Prove that its circumcircle passes through the midpoint of the segment joining the centres of the circles.
2005 Federal Math Competition of S&M, Problem 4
Inside a circle $k$ of radius $R$ some round spots are made. The area of each spot is $1$. Every radius of circle $k$, as well as every circle concentric with $k$, meets in no more than one spot. Prove that the total area of all the spots is less than
$$\pi\sqrt R+\frac12R\sqrt R.$$
1995 Abels Math Contest (Norwegian MO), 2b
Two circles of the same radii intersect in two distinct points $P$ and $Q$. A line passing through $P$, not touching any of the circles, intersects the circles again at $A$ and $B$. Prove that $Q$ lies on the perpendicular bisector of $AB$.
1997 Chile National Olympiad, 5
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.
1966 IMO Longlists, 38
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 $\frac{63}{20} \cdot \frac{R+r}{R-r}.$
1995 Tuymaada Olympiad, 6
Given a circle of radius $r= 1995$. Show that around it you can describe exactly $16$ primitive Pythagorean triangles. The primitive Pythagorean triangle is a right-angled triangle, the lengths of the sides of which are expressed by coprime integers.
Geometry Mathley 2011-12, 4.1
Five points $K_i, i = 1, 2, 3, 4$ and $P$ are chosen arbitrarily on the same circle. Denote by $P(i, j)$ the distance from $P$ to the line passing through $K_i$ and $K_j$ . Prove that $$P(1, 2)P(3, 4) = P(1, 4)P(2, 3) = P(1, 3)P(2, 4)$$
Bùi Quang Tuấn
2013 IMAR Test, 3
The closure (interior and boundary) of a convex quadrangle is covered by four closed discs centered at each vertex of the quadrangle each. Show that three of these discs cover the closure of the triangle determined by their centers.