Found problems: 287
1998 Switzerland Team Selection Test, 5
Points $A$ and $B$ are chosen on a circle $k$. Let AP and $BQ$ be segments of the same length tangent to $k$, drawn on different sides of line $AB$. Prove that the line $AB$ bisects the segment $PQ$.
1966 IMO Longlists, 1
Given $n>3$ points in the plane such that no three of the points are collinear. Does there exist a circle passing through (at least) $3$ of the given points and not containing any other of the $n$ points in its interior ?
1995 Portugal MO, 4
The diameter $[AC]$ of a circle is divided into four equal segments by points $P, M$ and $Q$. Consider a segment $[BD]$ that passes through $P$ and cuts the circle at $B$ and $D$, such that $PD =\frac{3}{2} AP$. Knowing that the area of the triangle $[ABP]$ has measure $1$ cm$^2$ , calculate the area of $[ABCD]$?
[img]https://1.bp.blogspot.com/-ibre0taeRo8/X4KiWWSROEI/AAAAAAAAMl4/xFNfpQBxmMMVLngp5OWOXRLMuaxf3nolQCLcBGAsYHQ/s154/1995%2Bportugal%2Bp5.png[/img]
1953 Putnam, B6
Let $P$ and $Q$ be any points inside a circle $C$ with center $O$ such that $OP=OQ.$ Determine the location of a point $Z$ on $C$ such that $PZ+QZ$ is minimal.
2015 Singapore Senior Math Olympiad, 2
There are $n=1681$ children, $a_1,a_2,...,a_{n}$ seated clockwise in a circle on the floor. The teacher walks behind the children in the clockwise direction with a box of $1000$ candies. She drops a candy behind the first child $a_1$. She then skips one child and drops a candy behind the third child, $a_3$. Now she skips two children and drops a candy behind the next child, $a_6$. She continues this way, at each stage skipping one child more than at the preceding stage before dropping a candy behind the next child. How many children will never receive a candy? Justify your answer.
1978 Chisinau City MO, 166
It is known that at least one coordinate of the center $(x_0, y_0)$ of the circle $(x -x_0)^2+ (y -y_0)^2 = R^2$ is irrational. Prove that on the circle itself there are at most two points with rational coordinates.
Estonia Open Junior - geometry, 2012.1.5
A hiking club wants to hike around a lake along an exactly circular route. On the shoreline they determine two points, which are the most distant from each other, and start to walk along the circle, which has these two points as the endpoints of its diameter. Can they be sure that, independent of the shape of the lake, they do not have to swim across the lake on any part of their route?
1986 Tournament Of Towns, (131) 7
On the circumference of a circle are $21$ points. Prove that among the arcs which join any two of these points, at least $100$ of them must subtend an angle at the centre of the circle not exceeding $120^o$ .
( A . F . Sidorenko)
2010 Czech And Slovak Olympiad III A, 2
A circular target with a radius of $12$ cm was hit by $19$ shots. Prove that the distance between two hits is less than $7$ cm.
2018 Azerbaijan Junior NMO, 4
A circle $\omega$ and a point $T$ outside the circle is given. Let a tangent from $T$ to $\omega$ touch $\omega$ at $A$, and take points $B,C$ lying on $\omega$ such that $T,B,C$ are colinear. The bisector of $\angle ATC$ intersects $AB$ and $AC$ at $P$ and $Q$,respectively. Prove that $PA=\sqrt{PB\cdot QC}$
2020 Iran MO (3rd Round), 2
For each $n$ find the number of ways one can put the numbers $\{1,2,3,...,n\}$ numbers on the circle, such that if for any $4$ numbers $a,b,c,d$ where $n|a+b-c-d$. The segments joining $a,b$ and $c,d$ do not meet inside the circle. (Two ways are said to be identical , if one can be obtained from rotaiting the other)
1959 AMC 12/AHSME, 43
The sides of a triangle are $25,39,$ and $40$. The diameter of the circumscribed circle is:
$ \textbf{(A)}\ \frac{133}{3}\qquad\textbf{(B)}\ \frac{125}{3}\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 41\qquad\textbf{(E)}\ 40 $
1997 Estonia National Olympiad, 3
The points $A, B, M$ and $N$ are on a circle with center $O$ such that the radii $OA$ and $OB$ are perpendicular to each other, and $MN$ is parallel to $AB$ and intersects the radius $OA$ at $P$. Find the radius of the circle if $|MP|= 12$ and $|P N| = 2 \sqrt{14}$
2007 Dutch Mathematical Olympiad, 1
Consider the equilateral triangle $ABC$ with $|BC| = |CA| = |AB| = 1$.
On the extension of side $BC$, we define points $A_1$ (on the same side as B) and $A_2$ (on the same side as C) such that $|A_1B| = |BC| = |CA_2| = 1$. Similarly, we define $B_1$ and $B_2$ on the extension of side $CA$ such that $|B_1C| = |CA| =|AB_2| = 1$, and $C_1$ and $C_2$ on the extension of side $AB$ such that $|C_1A| = |AB| = |BC_2| = 1$. Now the circumcentre of 4ABC is also the centre of the circle that passes through the points $A_1,B_2,C_1,A_2,B_1$ and $C_2$.
Calculate the radius of the circle through $A_1,B_2,C_1,A_2,B_1$ and $C_2$.
[asy]
unitsize(1.5 cm);
pair[] A, B, C;
A[0] = (0,0);
B[0] = (1,0);
C[0] = dir(60);
A[1] = B[0] + dir(-60);
A[2] = C[0] + dir(120);
B[1] = C[0] + dir(60);
B[2] = A[0] + dir(240);
C[1] = A[0] + (-1,0);
C[2] = B[0] + (1,0);
draw(A[1]--A[2]);
draw(B[1]--B[2]);
draw(C[1]--C[2]);
draw(circumcircle(A[2],B[1],C[2]));
dot("$A$", A[0], SE);
dot("$A_1$", A[1], SE);
dot("$A_2$", A[2], NW);
dot("$B$", B[0], SW);
dot("$B_1$", B[1], NE);
dot("$B_2$", B[2], SW);
dot("$C$", C[0], N);
dot("$C_1$", C[1], W);
dot("$C_2$", C[2], E);
[/asy]
2000 IMO Shortlist, 2
Two circles $ G_1$ and $ G_2$ intersect at two points $ M$ and $ N$. Let $ AB$ be the line tangent to these circles at $ A$ and $ B$, respectively, so that $ M$ lies closer to $ AB$ than $ N$. Let $ CD$ be the line parallel to $ AB$ and passing through the point $ M$, with $ C$ on $ G_1$ and $ D$ on $ G_2$. Lines $ AC$ and $ BD$ meet at $ E$; lines $ AN$ and $ CD$ meet at $ P$; lines $ BN$ and $ CD$ meet at $ Q$. Show that $ EP \equal{} EQ$.
1966 IMO Shortlist, 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.
1966 IMO Longlists, 15
Given four points $A,$ $B,$ $C,$ $D$ on a circle such that $AB$ is a diameter and $CD$ is not a diameter. Show that the line joining the point of intersection of the tangents to the circle at the points $C$ and $D$ with the point of intersection of the lines $AC$ and $BD$ is perpendicular to the line $AB.$
1986 All Soviet Union Mathematical Olympiad, 438
A triangle and a square are circumscribed around the unit circle. Prove that the intersection area is more than $3.4$.
Is it possible to assert that it is more than $3.5$?
2021 Sharygin Geometry Olympiad, 16
Let circles $\Omega$ and $\omega$ touch internally at point $A$. A chord $BC$ of $\Omega$ touches $\omega$ at point $K$. Let $O$ be the center of $\omega$. Prove that the circle $BOC$ bisects segment $AK$.
1995 Chile National Olympiad, 7
In a semicircle of radius $4$ three circles are inscribed, as indicated in the figure. Larger circles have radii $ R_1 $ and $ R_2 $, and the larger circle has radius $ r $.
a) Prove that $ \dfrac {1} {\sqrt{r}} = \dfrac {1} {\sqrt{R_1}} + \dfrac {1} {\sqrt{R_2}} $
b) Prove that $ R_1 + R_2 \le 8 (\sqrt{2} -1) $
c) Prove that $ r \le \sqrt{2} -1 $
[img]https://cdn.artofproblemsolving.com/attachments/0/9/aaaa65d1f4da4883973751e1363df804b9944c.jpg[/img]
2005 Sharygin Geometry Olympiad, 10.6
Let $H$ be the orthocenter of triangle $ABC$, $X$ be an arbitrary point. A circle with a diameter of $XH$ intersects lines $AH, BH, CH$ at points $A_1, B_1, C_1$ for the second time, and lines $AX BX, CX$ at points $A_2, B_2, C_2$. Prove that lines A$_1A_2, B_1B_2, C_1C_2$ intersect at one point.
1976 Bundeswettbewerb Mathematik, 3
A circle is divided by $2n$ points into $2n$ equal arcs. Let $P_1, P_2, \ldots, P_{2n}$ be an arbitrary permutation of the $2n$ division points. Prove that the polygonal line $P_1 P_2 \cdots P_{2n} P_1$ contains at least two parallel segments.
2016 Saudi Arabia IMO TST, 2
Let $ABC$ be a triangle inscribed in the circle $(O)$ and $P$ is a point inside the triangle $ABC$. Let $D$ be a point on $(O)$ such that $AD \perp AP$. The line $CD$ cuts the perpendicular bisector of $BC$ at $M$. The line $AD$ cuts the line passing through $B$ and is perpendicular to $BP$ at $Q$. Let $N$ be the reflection of $Q$ through $M$. Prove that $CN \perp CP$.
2014 Finnish National High School Mathematics, 4
The radius $r$ of a circle with center at the origin is an odd integer.
There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers.
Determine $r$.
1985 Tournament Of Towns, (094) 2
The radius $OM$ of a circle rotates uniformly at a rate of $360/n$ degrees per second , where $n$ is a positive integer . The initial radius is $OM_0$. After $1$ second the radius is $OM_1$ , after two more seconds (i.e. after three seconds altogether) the radius is $OM_2$ , after $3$ more seconds (after $6$ seconds altogether) the radius is $OM_3$, ..., after $n - 1$ more seconds its position is $OM_{n-1}$. For which values of $n$ do the points $M_0, M_1 , ..., M_{n-1}$ divide the circle into $n$ equal arcs?
(a) Is it true that the powers of $2$ are such values?
(b) Does there exist such a value which is not a power of $2$?
(V. V. Proizvolov , Moscow)