Found problems: 3882
2004 Croatia Team Selection Test, 3
A line intersects a semicircle with diameter $AB$ and center $O$ at $C$ and $D$, and the line $AB$ at $M$, where $MB < MA$ and $MD < MC.$ If the circumcircles of the triangles $AOC$ and $DOB$ meet again at $K,$ prove that $\angle MKO$ is right.
2001 India Regional Mathematical Olympiad, 5
In a triangle $ABC$, $D$ is a point on $BC$ such that $AD$ is the internal bisector of $\angle A$. Suppose $\angle B = 2 \angle C$ and $CD =AB$. prove that $\angle A = 72^{\circ}$.
1998 National High School Mathematics League, 1
Circumcenter and incentre of $\triangle ABC$ are $O,I$. $AD$ is the height on side $BC$. If $I$ is on line $OC$, prove that the radius of circumcircle and escribed circle (in \angle BAC) are equal.
2005 District Olympiad, 3
Let $O$ be a point equally distanced from the vertices of the tetrahedron $ABCD$. If the distances from $O$ to the planes $(BCD)$, $(ACD)$, $(ABD)$ and $(ABC)$ are equal, prove that the sum of the distances from a point $M \in \textrm{int}[ABCD]$, to the four planes, is constant.
2021 Oral Moscow Geometry Olympiad, 4
On the diagonal $AC$ of cyclic quadrilateral $ABCD$ a point $E$ is chosen such that $\angle ABE = \angle CBD$. Points $O,O_1,O_2$ are the circumcircles of triangles $ABC, ABE$ and $CBE$ respectively. Prove that lines $DO,AO_{1}$ and $CO_{2}$ are concurrent.
2009 Korea Junior Math Olympiad, 5
Acute triangle $\triangle ABC$ satis es $AB < AC$. Let the circumcircle of this triangle be $O$, and the midpoint of $BC,CA,AB$ be $D,E,F$. Let $P$ be the intersection of the circle with $AB$ as its diameter and line $DF$, which is in the same side of $C$ with respect to $AB$. Let $Q$ be the intersection of the circle with $AC$ as its diameter and the line $DE$, which is in the same side of $B$ with respect to $AC$. Let $PQ \cap BC = R$, and let the line passing through $R$ and perpendicular to $BC$ meet $AO$ at $X$. Prove that $AX = XR$.
Swiss NMO - geometry, 2008.1
Let $ABC$ be a triangle with $\angle BAC \ne 45^o$ and $\angle ABC \ne 135^o$. Let $P$ be the point on the line $AB$ with $\angle CPB = 45^o$. Let $O_1$ and $O_2$ be the centers of the circumcircles of the triangles $ACP$ and $BCP$ respectively. Show that the area of the square $CO_1P O_2$ is equal to the area of the triangle $ABC$.
2012 Online Math Open Problems, 40
Suppose $x,y,z$, and $w$ are positive reals such that
\[ x^2 + y^2 - \frac{xy}{2} = w^2 + z^2 + \frac{wz}{2} = 36 \] \[ xz + yw = 30. \] Find the largest possible value of $(xy + wz)^2$.
[i]Author: Alex Zhu[/i]
1970 IMO Longlists, 15
Given $\triangle ABC$, let $R$ be its circumradius and $q$ be the perimeter of its excentral triangle. Prove that $q\le 6\sqrt{3} R$.
Typesetter's Note: the excentral triangle has vertices which are the excenters of the original triangle.
2015 Taiwan TST Round 2, 1
Let $ABC$ be a triangle with incircle $\omega$, incenter $I$ and circumcircle $\Gamma$. Let $D$ be the tangency point of $\omega$ with $BC$, let $M$ be the midpoint of $ID$, and let $A'$ be the diametral opposite of $A$ with respect to $\Gamma$. If we denote $X=A'M\cap \Gamma$, then prove that the circumcircle of triangle $AXD$ is tangent to $BC$.
2005 Taiwan TST Round 3, 1
Let $P$ be a point in the interior of $\triangle ABC$. The lengths of the sides of $\triangle ABC$ is $a,b,c$, and the distance from $P$ to the sides of $\triangle ABC$ is $p,q,r$. Show that the circumradius $R$ of $\triangle ABC$ satisfies \[\displaystyle R\le \frac{a^2+b^2+c^2}{18\sqrt[3]{pqr}}.\] When does equality hold?
2007 Canada National Olympiad, 5
Let the incircle of triangle $ ABC$ touch sides $ BC,\, CA$ and $ AB$ at $ D,\, E$ and $ F,$ respectively. Let $ \omega,\,\omega_{1},\,\omega_{2}$ and $ \omega_{3}$ denote the circumcircles of triangle $ ABC,\, AEF,\, BDF$ and $ CDE$ respectively.
Let $ \omega$ and $ \omega_{1}$ intersect at $ A$ and $ P,\,\omega$ and $ \omega_{2}$ intersect at $ B$ and $ Q,\,\omega$ and $ \omega_{3}$ intersect at $ C$ and $ R.$
$ a.$ Prove that $ \omega_{1},\,\omega_{2}$ and $ \omega_{3}$ intersect in a common point.
$ b.$ Show that $ PD,\, QE$ and $ RF$ are concurrent.
2018 Harvard-MIT Mathematics Tournament, 8
Equilateral triangle $ABC$ has circumcircle $\Omega$. Points $D$ and $E$ are chosen on minor arcs $AB$ and $AC$ of $\Omega$ respectively such that $BC=DE$. Given that triangle $ABE$ has area $3$ and triangle $ACD$ has area $4$, find the area of triangle $ABC$.
2011 Sharygin Geometry Olympiad, 22
Let $CX, CY$ be the tangents from vertex $C$ of triangle $ABC$ to the circle passing through the midpoints of its sides. Prove that lines $XY , AB$ and the tangent to the circumcircle of $ABC$ at point $C$ concur.
1956 AMC 12/AHSME, 26
Which one of the following combinations of given parts does not determine the indicated triangle?
$ \textbf{(A)}\ \text{base angle and vertex angle; isosceles triangle}$
$ \textbf{(B)}\ \text{vertex angle and the base; isosceles triangle}$
$ \textbf{(C)}\ \text{the radius of the circumscribed circle; equilateral triangle}$
$ \textbf{(D)}\ \text{one arm and the radius of the inscribed circle; right triangle}$
$ \textbf{(E)}\ \text{two angles and a side opposite one of them; scalene triangle}$
2011 Sharygin Geometry Olympiad, 2
In triangle $ABC, \angle B = 2\angle C$. Points $P$ and $Q$ on the medial perpendicular to $CB$ are such that $\angle CAP = \angle PAQ = \angle QAB = \frac{\angle A}{3}$ . Prove that $Q$ is the circumcenter of triangle $CPB$.
2023 Turkey Team Selection Test, 1
Let $ABCD$ be a trapezoid with $AB \parallel CD$. A point $T$ which is inside the trapezoid satisfies $ \angle ATD = \angle CTB$. Let line $AT$ intersects circumcircle of $ACD$ at $K$ and line $BT$ intersects circumcircle of $BCD$ at $L$.($K \neq A$ , $L \neq B$) Prove that $KL \parallel AB$.
2013 All-Russian Olympiad, 3
The incircle of triangle $ ABC $ has centre $I$ and touches the sides $ BC $, $ CA $, $ AB $ at points $ A_1 $, $ B_1 $, $ C_1 $, respectively. Let $ I_a $, $ I_b $, $ I_c $ be excentres of triangle $ ABC $, touching the sides $ BC $, $ CA $, $ AB $ respectively. The segments $ I_aB_1 $ and $ I_bA_1 $ intersect at $ C_2 $. Similarly, segments $ I_bC_1 $ and $ I_cB_1 $ intersect at $ A_2 $, and the segments $ I_cA_1 $ and $ I_aC_1 $ at $ B_2 $. Prove that $ I $ is the center of the circumcircle of the triangle $ A_2B_2C_2 $.
[i]L. Emelyanov, A. Polyansky[/i]
2005 Canada National Olympiad, 3
Let $S$ be a set of $n\ge 3$ points in the interior of a circle.
$a)$ Show that there are three distinct points $a,b,c\in S$ and three distinct points $A,B,C$ on the circle such that $a$ is (strictly) closer to $A$ than any other point in $S$, $b$ is closer to $B$ than any other point in $S$ and $c$ is closer to $C$ than any other point in $S$.
$b)$ Show that for no value of $n$ can four such points in $S$ (and corresponding points on the circle) be guaranteed.
Geometry Mathley 2011-12, 3.1
$AB,AC$ are tangent to a circle $(O)$, $B,C$ are the points of tangency. $Q$ is a point iside the angle $BAC$, on the ray $AQ$, take a point $P$ suc that $OP$ is perpendicular to $AQ$. The line $OP$ meets the circumcircles triangles $BPQ$ and $CPQ$ at $I, J$. Prove that $OI = OJ$.
Hồ Quang Vinh
2015 Azerbaijan IMO TST, 1
Let $\omega$ be the circumcircle of an acute-angled triangle $ABC$. The lines tangent to $\omega$ at the points $A$ and $B$ meet at $K$. The line passing through $K$ and parallel to $BC$ intersects the side $AC$ at $S$. Prove that $BS=CS$
2018 Mediterranean Mathematics OIympiad, 2
Let $ABC$ be acute triangle. Let $E$ and $F$ be points on $BC$, such that angles $BAE$ and $FAC$ are equal. Lines $AE$ and $AF$ intersect cirumcircle of $ABC$ at points $M$ and $N$. On rays $AB$ and $AC$ we have points $P$ and $R$, such that angle $PEA$ is equal to angle $B$ and angle $AER$ is equal to angle $C$. Let $L$ be intersection of $AE$ and $PR$ and $D$ be intersection of $BC$ and $LN$. Prove that
$$\frac{1}{|MN|}+\frac{1}{|EF|}=\frac{1}{|ED|}.$$
2019 Iran MO (3rd Round), 3
Consider a triangle $ABC$ with circumcenter $O$ and incenter $I$. Incircle touches sides $BC,CA$ and $AB$ at $D, E$ and $F$. $K$ is a point such that $KF$ is tangent to circumcircle of $BFD$ and $KE$ is tangent to circumcircle of $CED$. Prove that $BC,OI$ and $AK$ are concurrent.
Kyiv City MO Seniors Round2 2010+ geometry, 2020.11.2
A point $P$ was chosen on the smaller arc $BC$ of the circumcircle of the acute-angled triangle $ABC$. Points $R$ and $S$ on the sides$ AB$ and $AC$ are respectively selected so that $CPRS$ is a parallelogram. Point $T$ on the arc $AC$ of the circumscribed circle of $\vartriangle ABC$ such that $BT \parallel CP$. Prove that $\angle TSC = \angle BAC$.
(Anton Trygub)
2013 ELMO Shortlist, 13
In $\triangle ABC$, $AB<AC$. $D$ and $P$ are the feet of the internal and external angle bisectors of $\angle BAC$, respectively. $M$ is the midpoint of segment $BC$, and $\omega$ is the circumcircle of $\triangle APD$. Suppose $Q$ is on the minor arc $AD$ of $\omega$ such that $MQ$ is tangent to $\omega$. $QB$ meets $\omega$ again at $R$, and the line through $R$ perpendicular to $BC$ meets $PQ$ at $S$. Prove $SD$ is tangent to the circumcircle of $\triangle QDM$.
[i]Proposed by Ray Li[/i]