Found problems: 3882
2013 Kazakhstan National Olympiad, 3
Let $ABCD$ be cyclic quadrilateral. Let $AC$ and $BD$ intersect at $R$, and let $AB$ and $CD$ intersect at $K$. Let $M$ and $N$ are points on $AB$ and $CD$ such that $\frac{AM}{MB}=\frac{CN}{ND}$. Let $P$ and $Q$ be the intersections of $MN$ with the diagonals of $ABCD$. Prove that circumcircles of triangles $KMN$ and $PQR$ are tangent at a fixed point.
2024 Bosnia and Herzegovina Junior BMO TST, 3.
Let $ABC$ be a right-angled triangle where ∠$ACB$=90°.Let $CD$ be an altitude of that triangle and points $M$ and $N$ be the midpoints of $CD$ and $BC$, respectively.If $S$ is the circumcenter of the triangle $AMN$, prove that $AS$ and $BC$ are paralel.
OMMC POTM, 2022 12
Let $\triangle ABC$ be such that the midpoint of $BC$ is $D$. Let $E$ be the point on the opposite side of $AC$ as $B$ on the circumcircle of $\triangle ABC$ such that $\angle DEA = \angle DEC$ and let $\omega$ be the circumcircle of $\triangle CED$. If $\omega$ intersects $AE$ at $X$ and the tangent to $\omega$ at $D$ intersects $AB$ at $Y$, show that $XY$ is parallel to $BC$.
[i]Proposed by Taco12[/i]
2011 ISI B.Stat Entrance Exam, 10
Show that the triangle whose angles satisfy the equality
\[\frac{\sin^2A+\sin^2B+\sin^2C}{\cos^2A+\cos^2B+\cos^2C} = 2\]
is right angled.
1978 IMO Longlists, 32
Let $\mathcal{C}$ be the circumcircle of the square with vertices $(0, 0), (0, 1978), (1978, 0), (1978, 1978)$ in the Cartesian plane. Prove that $\mathcal{C}$ contains no other point for which both coordinates are integers.
2014 India IMO Training Camp, 3
Let $ABC$ be a triangle with $\angle B > \angle C$. Let $P$ and $Q$ be two different points on line $AC$ such that $\angle PBA = \angle QBA = \angle ACB $ and $A$ is located between $P$ and $C$. Suppose that there exists an interior point $D$ of segment $BQ$ for which $PD=PB$. Let the ray $AD$ intersect the circle $ABC$ at $R \neq A$. Prove that $QB = QR$.
2008 Brazil Team Selection Test, 3
Denote by $ M$ midpoint of side $ BC$ in an isosceles triangle $ \triangle ABC$ with $ AC = AB$. Take a point $ X$ on a smaller arc $ \overarc{MA}$ of circumcircle of triangle $ \triangle ABM$. Denote by $ T$ point inside of angle $ BMA$ such that $ \angle TMX = 90$ and $ TX = BX$.
Prove that $ \angle MTB - \angle CTM$ does not depend on choice of $ X$.
[i]Author: Farzan Barekat, Canada[/i]
2012 Sharygin Geometry Olympiad, 22
A circle $\omega$ with center $I$ is inscribed into a segment of the disk, formed by an arc and a chord $AB$. Point $M$ is the midpoint of this arc $AB$, and point $N$ is the midpoint of the complementary arc. The tangents from $N$ touch $\omega$ in points $C$ and $D$. The opposite sidelines $AC$ and $BD$ of quadrilateral $ABCD$ meet in point $X$, and the diagonals of $ABCD$ meet in point $Y$. Prove that points $X, Y, I$ and $M$ are collinear.
2014 Czech-Polish-Slovak Junior Match, 2
Let $ABCD$ be a parallelogram with $\angle BAD<90^o$ and $AB> BC$ . The angle bisector of $BAD$ intersects line $CD$ at point $P$ and line $BC$ at point $Q$. Prove that the center of the circle circumscirbed around the triangle $CPQ$ is equidistant from points $B$ and $D$.
2025 Philippine MO, P7
In acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$, let $D$ be an arbitrary point on the circumcircle of triangle $ABC$ such that $D$ does not lie on line $OB$ and that line $OD$ is not parallel to line $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$ such that $DE$ is perpendicular to $BC$, and let $F$ be the point on line $AC$ such that $FA = FE$. Let $P$ and $R$ be the points on the circumcircle of triangle $ABC$ such that $PE$ is a diameter, and $BH$ and $DR$ are parallel. Let $M$ be the midpoint of $DH$.
(a) Show that $AP$ and $BR$ are perpendicular. \\
(b) Show that $FM$ and $BM$ are perpendicular.
2012 USAJMO, 1
Given a triangle $ABC$, let $P$ and $Q$ be points on segments $\overline{AB}$ and $\overline{AC}$, respectively, such that $AP=AQ$. Let $S$ and $R$ be distinct points on segment $\overline{BC}$ such that $S$ lies between $B$ and $R$, $\angle BPS=\angle PRS$, and $\angle CQR=\angle QSR$. Prove that $P,Q,R,S$ are concyclic (in other words, these four points lie on a circle).
2010 Greece JBMO TST, 3
Given an acute and scalene triangle $ABC$ with $AB<AC$ and random line $(e)$ that passes throuh the center of the circumscribed circles $c(O,R)$. Line $(e)$, intersects sides $BC,AC,AB$ at points $A_1,B_1,C_1$ respectively (point $C_1$ lies on the extension of $AB$ towards $B$). Perpendicular from $A$ on line $(e)$ and $AA_1$ intersect circumscribed circle $c(O,R)$ at points $M$ and $A_2$ respectively. Prove that
a) points $O,A_1,A_2, M$ are consyclic
b) if $(c_2)$ is the circumcircle of triangle $(OBC_1)$ and $(c_3)$ is the circumcircle of triangle $(OCB_1)$, then circles $(c_1),(c_2)$ and $(c_3)$ have a common chord
2021 Oral Moscow Geometry Olympiad, 5
The trapezoid is inscribed in a circle. Prove that the sum of distances from any point of the circle to the midpoints of the lateral sides are not less than the diagonal of the trapezoid.
2007 Pre-Preparation Course Examination, 3
$ABC$ is an arbitrary triangle. $A',B',C'$ are midpoints of arcs $BC, AC, AB$. Sides of triangle $ABC$, intersect sides of triangle $A'B'C'$ at points $P,Q,R,S,T,F$. Prove that \[\frac{S_{PQRSTF}}{S_{ABC}}=1-\frac{ab+ac+bc}{(a+b+c)^{2}}\]
2005 Korea National Olympiad, 2
For triangle $ABC$, $P$ and $Q$ satisfy $\angle BPA + \angle AQC=90^{\circ}$. It is provided that the vertices of the triangle $BAP$ and $ACQ$ are ordered counterclockwise(or clockwise). Let the intersection of the circumcircles of the two triangles be $N$ ($A \neq N$, however if $A$ is the only intersection $A=N$), and the midpoint of segment $BC$ be $M$. Show that the length of $MN$ does not depend on $P$ and $Q$.
2016 Sharygin Geometry Olympiad, 2
Let $H$ be the orthocenter of an acute-angled triangle $ABC$. Point $X_A$ lying on the tangent at $H$ to the circumcircle of triangle $BHC$ is such that $AH=AX_A$ and $X_A \not= H$. Points $X_B,X_C$ are defined similarly. Prove that the triangle $X_AX_BX_C$ and the orthotriangle of $ABC$ are similar.
2018 JBMO TST-Turkey, 6
A point $E$ is located inside a parallelogram $ABCD$ such that $\angle BAE = \angle BCE$. The centers of the circumcircles of the triangles $ABE,ECB, CDE$ and $DAE$ are concyclic.
2006 MOP Homework, 2
Let $ABC$ be an acute triangle. Determine the locus of points $M$ in the interior of the triangle such that $AB-FG=\frac{MF \cdot AG+MG \cdot BF}{CM}$, where $F$ and $G$ are the feet of the perpendiculars from $M$ to lines $BC$ and $AC$, respectively.
2023 Germany Team Selection Test, 1
Let $ABC$ be an acute triangle and let $\omega$ be its circumcircle. Let the tangents to $\omega$ through $B,C$ meet each other at point $P$. Prove that the perpendicular bisector of $AB$ and the parallel to $AB$ through $P$ meet at line $AC$.
2005 France Team Selection Test, 2
Two right angled triangles are given, such that the incircle of the first one is equal to the circumcircle of the second one. Let $S$ (respectively $S'$) be the area of the first triangle (respectively of the second triangle).
Prove that $\frac{S}{S'}\geq 3+2\sqrt{2}$.
2020 CCA Math Bonanza, T3
Five unit squares are arranged in a plus shape as shown below:
[asy]
size(3cm);
real s=0.1;
draw(s*(0,1)--s*(0,2));
draw(s*(1,0)--s*(1,3));
draw(s*(2,0)--s*(2,3));
draw(s*(3,1)--s*(3,2));
draw(s*(1,0)--s*(2,0));
draw(s*(0,1)--s*(3,1));
draw(s*(0,2)--s*(3,2));
draw(s*(1,3)--s*(2,3));
[/asy]
What is the area of the smallest circle containing the interior and boundary of the plus shape?
[i]2020 CCA Math Bonanza Team Round #3[/i]
2010 Contests, 3
In an acute-angled triangle $ABC$, $CF$ is an altitude, with $F$ on $AB$, and $BM$ is a median, with $M$ on $CA$. Given that $BM=CF$ and $\angle MBC=\angle FCA$, prove that triangle $ABC$ is equilateral.
1998 Belarus Team Selection Test, 4
The altitudes through the vertices $ A,B,C$ of an acute-angled triangle $ ABC$ meet the opposite sides at $ D,E, F,$ respectively. The line through $ D$ parallel to $ EF$ meets the lines $ AC$ and $ AB$ at $ Q$ and $ R,$ respectively. The line $ EF$ meets $ BC$ at $ P.$ Prove that the circumcircle of the triangle $ PQR$ passes through the midpoint of $ BC.$
2018 Brazil National Olympiad, 3
Let $ABC$ be an acute-angled triangle with circumcenter $O$ and orthocenter $H$. The circle with center $X_a$ passes in the points $A$ and $H$ and is tangent to the circumcircle of $ABC$. Define $X_b, X_c$ analogously, let $O_a, O_b, O_c$ the symmetric of $O$ to the sides $BC, AC$ and $AB$, respectively. Prove that the lines $O_aX_a, O_bX_b, O_cX_c$ are concurrents.
2016 Poland - Second Round, 2
In acute triangle $ABC$ bisector of angle $BAC$ intersects side $BC$ in point $D$. Bisector of line segment $AD$ intersects circumcircle of triangle $ABC$ in points $E$ and $F$. Show that circumcircle of triangle $DEF$ is tangent to line $BC$.