Found problems: 287
2021 Malaysia IMONST 2, 1
Given a circle with center $O$. Points $A$ and $B$ lie on the circle such that triangle $OBA$ is equilateral. Let $C$ be a point outside the circle with $\angle ACB = 45^{\circ}$. Line $CA$ intersects the circle at point $D$, and the line $CB$ intersects the circle at point $E$. Find $\angle DBE$.
1996 Singapore Senior Math Olympiad, 1
$PQ, CD$ are parallel chords of a circle. The tangent at $D$ cuts $PQ$ at $T$ and $B$ is the point of contact of the other tangent from $T$ (Fig. ). Prove that $BC$ bisects $PQ$.
[img]https://cdn.artofproblemsolving.com/attachments/2/f/22f69c03601fbb8e388e319cd93567246b705c.png[/img]
2017 India PRMO, 27
Let $\Omega_1$ be a circle with centre $O$ and let $AB$ be diameter of $\Omega_1$. Let $P$ be a point on the segment $OB$ different from $O$. Suppose another circle $\Omega_2$ with centre $P$ lies in the interior of $\Omega_1$. Tangents are drawn from $A$ and $B$ to the circle $\Omega_2$ intersecting $\Omega_1$ again at $A_1$ and B1 respectively such that $A_1$ and $B_1$ are on the opposite sides of $AB$. Given that $A_1 B = 5, AB_1 = 15$ and $OP = 10$, find the radius of $\Omega_1$.
2014 IFYM, Sozopol, 6
Is it true that for each natural number $n$ there exist a circle, which contains exactly $n$ points with integer coordinates?
1992 IMO Longlists, 17
In the plane let $\,C\,$ be a circle, $\,L\,$ a line tangent to the circle $\,C,\,$ and $\,M\,$ a point on $\,L$. Find the locus of all points $\,P\,$ with the following property: there exists two points $\,Q,R\,$ on $\,L\,$ such that $\,M\,$ is the midpoint of $\,QR\,$ and $\,C\,$ is the inscribed circle of triangle $\,PQR$.
2020 Adygea Teachers' Geometry Olympiad, 4
A circle is inscribed in an angle with vertex $O$, touching its sides at points $M$ and $N$. On an arc $MN$ nearest to point $O$, an arbitrary point $P$ is selected. At point $P$, a tangent is drawn to the circle $P$, intersecting the sides of the angle at points $A$ and $B$. Prove that that the length of the segment $AB$ is the smallest when $P$ is its midpoint.
2016 Bosnia And Herzegovina - Regional Olympiad, 3
Circle of radius $R_1$ is inscribed in an acute angle $\alpha$. Second circle with radius $R_2$ touches one of the sides forming the angle $\alpha$ in same point as first circle and intersects the second side in points $A$ and $B$, such that centers of both circles lie inside angle $\alpha$. Prove that $$AB=4\cos{\frac{\alpha}{2}}\sqrt{(R_2-R_1)\left(R_1 \cos^2 \frac{\alpha}{2}+R_2 \sin^2 \frac{\alpha}{2}\right)}$$
2018 Hanoi Open Mathematics Competitions, 10
There are $100$ school students from two clubs $A$ and $B$ standing in circle. Among them $62$ students stand next to at least one student from club $A$, and $54$ students stand next to at least one student from club $B$.
1) How many students stand side-by-side with one friend from club $A$ and one friend from club $B$?
2) What is the number of students from club $A$?
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 $1994$
2016 Bangladesh Mathematical Olympiad, 5
Suppose there are $m$ Martians and $n$ Earthlings at an intergalactic peace conference. To ensure the Martians stay peaceful at the conference, we must make sure that no two Martians sit together, such that between any two Martians there is always at least one Earthling.
(a) Suppose all $m + n$ Martians and Earthlings are seated in a line. How many ways can the Earthlings and Martians be seated in a line?
(b) Suppose now that the $m+n$ Martians and Earthlings are seated around a circular round-table. How many ways can the Earthlings and Martians be seated around the round-table?
1992 IMO Shortlist, 20
In the plane let $\,C\,$ be a circle, $\,L\,$ a line tangent to the circle $\,C,\,$ and $\,M\,$ a point on $\,L$. Find the locus of all points $\,P\,$ with the following property: there exists two points $\,Q,R\,$ on $\,L\,$ such that $\,M\,$ is the midpoint of $\,QR\,$ and $\,C\,$ is the inscribed circle of triangle $\,PQR$.
1997 Greece Junior Math Olympiad, 4
Consider ten concentric circles and ten rays as in the following figure.
At the points where the inner circle is intersected by the rays write successively, in direction clockwise, the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$. In the next circle we write the numbers $11, 12, 13, 14, 15, 16, 17, 18, 19,20$ successively, and so on successively until the last round were we write the numbers $91, 92, 93, 94, 95, 96, 97, 98, 99, 100$ successively. In this orde, the numbers $1, 11, 21, 31, 41, 51, 61, 71, 81, 91$ are in the same ray, and similarly for the other rays. In front of $50$ of those $100$ numbers, we use the sign ''$-$'' such as:
a) in each of the ten rays, exist exactly $5$ signs ''$-$'' , and also
b) in each of the ten concentric circles, to be exactly $5$ signs ''$-$''.
Prove that the sum of the $100$ signed numbers that occur, equals zero.
[img]https://cdn.artofproblemsolving.com/attachments/9/d/ffee6518fcd1b996c31cf06d0ce484a821b4ae.gif[/img]