Found problems: 53
1914 Eotvos Mathematical Competition, 1
Let $A$ and $B$ be points on a circle $k$. Suppose that an arc $k'$ of another circle, $\ell$, connects $A$ with $B$ and divides the area inside the circle $k$ into two equal parts. Prove that arc $k'$ is longer than the diameter of $k$.
2016 Estonia Team Selection Test, 6
A circle is divided into arcs of equal size by $n$ points ($n \ge 1$). For any positive integer $x$, let $P_n(x)$ denote the number of possibilities for colouring all those points, using colours from $x$ given colours, so that any rotation of the colouring by $ i \cdot \frac{360^o}{n}$ , where i is a positive integer less than $n$, gives a colouring that differs from the original in at least one point. Prove that the function $P_n(x)$ is a polynomial with respect to $x$.
1984 Spain Mathematical Olympiad, 5
Let $A$ and $A' $ be fixed points on two equal circles in the plane and let $AB$ and $A' B'$ be arcs of these circles of the same length $x$. Find the locus of the midpoint of segment $BB'$ when $x$ varies:
(a) if the arcs have the same direction,
(b) if the arcs have opposite directions.
1982 Tournament Of Towns, (026) 4
(a) $10$ points dividing a circle into $10$ equal arcs are connected in pairs by $5$ chords.
Is it necessary that two of these chords are of equal length?
(b) $20$ points dividing a circle into $20$ equal arcs are connected in pairs by $10$ chords.
Prove that among these $10$ chords there are two chords of equal length.
(VV Proizvolov, Moscow)
2016 NZMOC Camp Selection Problems, 3
Points $A, B, C$ are vertices of an equilateral triangle inscribed in a circle. Point $D$ lies on the shorter arc $\overarc {AB}$ . Prove that $AD + BD = DC$.
1987 Tournament Of Towns, (153) 4
We are given a figure bounded by arc $AC$ of a circle, and a broken line $ABC$, with the arc and broken line being on opposite sides of the chord $AC$. Construct a line passing through the mid-point of arc $AC$ and dividing the area of the figure into two regions of equal area.
2012 IMAC Arhimede, 5
On the circumference of a circle, there are $3n$ colored points that divide the circle on $3n$ arches, $n$ of which have lenght $1$, $n$ of which have length $2$ and the rest of them have length $3$ . Prove that there are two colored points on the same diameter of the circle.
2019 Poland - Second Round, 1
A cyclic quadrilateral $ABCD$ is given. Point $K_1, K_2$ lie on the segment $AB$, points $L_1, L_2$ on the segment $BC$, points $M_1, M_2$ on the segment $CD$ and points $N_1, N_2$ on the segment $DA$. Moreover, points $K_1, K_2, L_1, L_2, M_1, M_2, N_1, N_2$ lie on a circle $\omega$ in that order. Denote by $a, b, c, d$ the lengths of the arcs $N_2K_1, K_2L_1, L_2M_1, M
_2N_1$ of the circle $\omega$ not containing points $K_2, L_2, M_2, N_2$, respectively. Prove that
\begin{align*}
a+c=b+d.
\end{align*}
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)
1987 Austrian-Polish Competition, 8
A circle of perimeter $1$ has been dissected into four equal arcs $B_1, B_2, B_3, B_4$. A closed smooth non-selfintersecting curve $C$ has been composed of translates of these arcs (each $B_j$ possibly occurring several times). Prove that the length of $C$ is an integer.
2002 Moldova Team Selection Test, 4
Let $C$ be the circle with center $O(0,0)$ and radius $1$, and $A(1,0), B(0,1)$ be points on the circle. Distinct points $A_1,A_2, ....,A_{n-1}$ on $C$ divide the smaller arc $AB$ into $n$ equal parts ($n \ge 2$). If $P_i$ is the orthogonal projection of $A_i$ on $OA$ ($i =1, ... ,n-1$), find all values of $n$ such that $P_1A^{2p}_1 +P_2A^{2p}_2 +...+P_{n-1}A^{2p}_{n-1}$ is an integer for every positive integer $p$.
1990 Bulgaria National Olympiad, Problem 5
Given a circular arc, find a triangle of the smallest possible area which covers the arc so that the endpoints of the arc lie on the same side of the triangle.
2012 Czech-Polish-Slovak Junior Match, 2
On the circle $k$, the points $A,B$ are given, while $AB$ is not the diameter of the circle $k$. Point $C$ moves along the long arc $AB$ of circle $k$ so that the triangle $ABC$ is acute. Let $D,E$ be the feet of the altitudes from $A, B$ respectively. Let $F$ be the projection of point $D$ on line $AC$ and $G$ be the projection of point $E$ on line $BC$.
(a) Prove that the lines $AB$ and $FG$ are parallel.
(b) Determine the set of midpoints $S$ of segment $FG$ while along all allowable positions of point $C$.
2023 Yasinsky Geometry Olympiad, 5
Point $O$ is the center of the circumscribed circle of triangle $ABC$. Ray $AO$ intersects the side $BC$ at point $T$. With $AT$ as a diameter, a circle is constructed. At the intersection with the sides of the triangle $ABC$, three arcs were formed outside it. Prove that the larger of these arcs is equal to the sum of the other two.
(Oleksii Karliuchenko)
2004 Junior Tuymaada Olympiad, 3
Point $ O $ is the center of the circumscribed circle of an acute triangle $ Abc $. A certain circle passes through the points $ B $ and $ C $ and intersects sides $ AB $ and $ AC $ of a triangle. On its arc lying inside the triangle, points $ D $ and $ E $ are chosen so that the segments $ BD $ and $ CE $ pass through the point $ O $. Perpendicular $ DD_1 $ to $ AB $ side and perpendicular $ EE_1 $ to $ AC $ side intersect at $ M $. Prove that the points $ A $, $ M $ and $ O $ lie on the same straight line.
2013 Sharygin Geometry Olympiad, 8
Let P be an arbitrary point on the arc $AC$ of the circumcircle of a fixed triangle $ABC$, not containing $B$. The bisector of angle $APB$ meets the bisector of angle $BAC$ at point $P_a$ the bisector of angle $CPB$ meets the bisector of angle $BCA$ at point $P_c$. Prove that for all points $P$, the circumcenters of triangles $PP_aP_c$ are collinear.
by I. Dmitriev
2002 Moldova Team Selection Test, 3
A triangle $ABC$ is inscribed in a circle $G$. Points $M$ and $N$ are the midpoints of the arcs $BC$ and $AC$ respectively, and $D$ is an arbitrary point on the arc $AB$ (not containing $C$). Points $I_1$ and $I_2$ are the incenters of the triangles $ADC$ and $BDC$, respectively. If the circumcircle of triangle $DI_1I_2$ meets $G$ again at $P$, prove that triangles $PNI_1$ and $PMI_2$ are similar.
2015 NZMOC Camp Selection Problems, 3
Let $ABC$ be an acute angled triangle. The arc between $A$ and $B$ of the circumcircle of $ABC$ is reflected through the line $AB$, and the arc between $A$ and $C$ of the circumcircle of $ABC$ is reflected over the line $AC$. Obviously these two reflected arcs intersect at the point $A$. Prove that they also intersect at another point inside the triangle $ABC$.
Estonia Open Junior - geometry, 2015.1.5
Let $ABC$ be an acute triangle. The arcs $AB$ and $AC$ of the circumcircle of the triangle are reflected over the lines AB and $AC$, respectively. Prove that the two arcs obtained intersect in another point besides $A$.
1996 Chile National Olympiad, 6
Two circles, $C$ and $K$, are secant at $A$ and $B$. Let $P$ be a point on the arc $AB$ of $C$. Lines $PA$ and $PB$ intersect $K$ again at $R$ and $S$ respectively. Let $P'$ be another point at same arc as $P$, so that lines $P'A$ and $P'B$ again intersect $K$ at $R'$ and $S'$, respectively. Prove that the arcs $RS$ and $R'S'$ have equal measures.
[img]https://cdn.artofproblemsolving.com/attachments/2/4/88693c36159179fb2b098b671a2f8281b37aae.png[/img]
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)
2017 Yasinsky Geometry Olympiad, 3
Given circle arc, whose center is an inaccessible point. $A$ is a point on this arc (see fig.). How to construct using compass and ruler without divisions, a tangent to given circle arc at point $A$ ?
[img]https://1.bp.blogspot.com/-7oQBNJGLsVw/W6dYm4Xw7bI/AAAAAAAAJH8/sJ-rgAQZkW0kvlPOPwYiGjnOXGQZuDnRgCK4BGAYYCw/s1600/Yasinsky%2B2017%2BVIII-IX%2Bp3.png[/img]
2025 JBMO TST - Turkey, 8
Pairwise distinct points $P_1,\dots,P_{1024}$, which lie on a circle, are marked by distinct reals $a_1,\dots,a_{1024}$. Let $P_i$ be $Q-$good for a $Q$ on the circle different than $P_1,\dots,P_{1024}$, if and only if $a_i$ is the greatest number on at least one of the two arcs $P_iQ$. Let the score of $Q$ be the number of $Q-$good points on the circle. Determine the greatest $k$ such that regardless of the values of $a_1,\dots,a_{1024}$, there exists a point $Q$ with score at least $k$.
2021 JBMO Shortlist, G2
Let $P$ be an interior point of the isosceles triangle $ABC$ with $\hat{A} = 90^{\circ}$. If
$$\widehat{PAB} + \widehat{PBC} + \widehat{PCA} = 90^{\circ},$$
prove that $AP \perp BC$.
Proposed by [i]Mehmet Akif Yıldız, Turkey[/i]
2017 Denmark MO - Mohr Contest, 3
The figure shows an arc $\ell$ on the unit circle and two regions $A$ and $B$.
Prove that the area of $A$ plus the area of $B$ equals the length of $\ell$.
[img]https://1.bp.blogspot.com/-SYoSrFowZ30/XzRz0ygiOVI/AAAAAAAAMUs/0FCduUoxKGwq0gSR-b3dtb3SvDjZ89x_ACLcBGAsYHQ/s0/2017%2BMohr%2Bp3.png[/img]