Found problems: 87
2023 Bangladesh Mathematical Olympiad, P9
Let $\Delta ABC$ be an acute angled triangle. $D$ is a point on side $BC$ such that $AD$ bisects angle $\angle BAC$. A line $l$ is tangent to the circumcircles of triangles $ADB$ and $ADC$ at point $K$ and $L$, respectively. Let $M$, $N$ and $P$ be its midpoints of $BD$, $DC$ and $KL$, respectively. Prove that $l$ is tangent to the circumcircle of $\Delta MNP$.
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 $QB = QR$.
Istek Lyceum Math Olympiad 2016, 2
Let $\omega$ be the semicircle with diameter $PQ$. A circle $k$ is tangent internally to $\omega$ and to segment $PQ$ at $C$. Let $AB$ be the tangent to $K$ perpendicular to $PQ$, with $A$ on $\omega$ and $B$ on the segment $CQ$. Show that $AC$ bisects angle $\angle PAB$
2014 Brazil Team Selection Test, 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 $QB = QR$.
2009 USAMO, 5
Trapezoid $ ABCD$, with $ \overline{AB}\parallel{}\overline{CD}$, is inscribed in circle $ \omega$ and point $ G$ lies inside triangle $ BCD$. Rays $ AG$ and $ BG$ meet $ \omega$ again at points $ P$ and $ Q$, respectively. Let the line through $ G$ parallel to $ \overline{AB}$ intersects $ \overline{BD}$ and $ \overline{BC}$ at points $ R$ and $ S$, respectively. Prove that quadrilateral $ PQRS$ is cyclic if and only if $ \overline{BG}$ bisects $ \angle CBD$.
1969 Spain Mathematical Olympiad, 1
Find the locus of the centers of the inversions that transform two points $A, B$ of a given circle $\gamma$ , at diametrically opposite points of the inverse circles of $\gamma$ .
2023 HMIC, P3
Triangle $ABC$ has incircle $\omega$ and $A$-excircle $\omega_A.$ Circle $\gamma_B$ passes through $B$ and is externally tangent to $\omega$ and $\omega_A.$ Circle $\gamma_C$ passes through $C$ and is externally tangent to $\omega$ and $\omega_A.$ If $\gamma_B$ intersects line $BC$ again at $D,$ and $\gamma_C$ intersects line $BC$ again at $E,$ prove that $BD=EC.$
2023 Sharygin Geometry Olympiad, 24
A tetrahedron $ABCD$ is give. A line $\ell$ meets the planes $ABC,BCD,CDA,DAB$ at points $D_0,A_0,B_0,C_0$ respectively. Let $P$ be an arbitrary point not lying on $\ell$ and the planes of the faces, and $A_1,B_1,C_1,D_1$ be the second common points of lines $PA_0,PB_0,PC_0,PD_0$ with the spheres $PBCD,PCDA,PDAB,PABC$ respectively. Prove $P,A_1,B_1,C_1,D_1$ lie on a circle.
2025 Austrian MO National Competition, 2
Let $\triangle{ABC}$ be an acute triangle with $BC > AC$. Let $S$ be the centroid of triangle $ABC$ and let $F$ be the foot of the perpendicular from $C$ to side $AB$. The median $CS$ intersects the circumcircle $\gamma$ of triangle $\triangle{ABC}$ at a second point $P$. Let $M$ be the point where $CS$ intersects $AB$. The line $SF$ intersects the circle $\gamma$ at a point $Q$, such that $F$ lies between $S$ and $Q$. Prove that the points $M,P,Q$ and $F$ lie on a circle.
[i](Karl Czakler)[/i]
2024 Thailand TSTST, 9
Let triangle \( ABC \) be an acute-angled triangle. Square \( AEFB \) and \( ADGC \) lie outside triangle \( ABC \). \( BD \) intersects \( CE \) at point \( H \), and \( BG \) intersects \( CF \) at point \( I \). The circumcircle of triangle \( BFI \) intersects the circumcircle of triangle \( CGI \) again at point \( K \). Prove that line segment \( HK \) bisects \( BC \).
2017 Brazil Team Selection Test, 2
Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.
1969 German National Olympiad, 2
There is a circle $k$ in a plane with center $M$ and radius $r$. The following illustration, through which every point $P \ne M$., is called a “reflection on the circle $k$” from $\varepsilon$ a point $P'$ from $\varepsilon$ is assigned:
(1) $P'$ lies on the ray emanating from$ M$ and passing through $P$.
(2) It is $MP \cdot MP' = r^2$.
a) Construct the mirror point $ P'$ for any given point $P \ne M$ inside $k$.
b) Let another circle $k_1$ be given arbitrarily, but such that $M$ lies outside $k_1$.Construct $k'_1$ , i.e. the set of all mirror points $P'$ of the points $P$ of $k_1$.
2012 China Team Selection Test, 1
Given two circles ${\omega _1},{\omega _2}$, $S$ denotes all $\Delta ABC$ satisfies that ${\omega _1}$ is the circumcircle of $\Delta ABC$, ${\omega _2}$ is the $A$- excircle of $\Delta ABC$ , ${\omega _2}$ touches $BC,CA,AB$ at $D,E,F$.
$S$ is not empty, prove that the centroid of $\Delta DEF$ is a fixed point.
2016 IMO Shortlist, G4
Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.
2017 Estonia Team Selection Test, 3
Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.
2021 Romanian Master of Mathematics Shortlist, G2
Let $ABC$ be a triangle with incenter $I$. The line through $I$, perpendicular to $AI$, intersects the circumcircle of $ABC$ at points $P$ and $Q$. It turns out there exists a point $T$ on the side $BC$ such that $AB + BT = AC + CT$ and $AT^2 = AB \cdot AC$. Determine all possible values of the ratio $IP/IQ$.
2024 Thailand October Camp, 3
Let triangle \( ABC \) be an acute-angled triangle. Square \( AEFB \) and \( ADGC \) lie outside triangle \( ABC \). \( BD \) intersects \( CE \) at point \( H \), and \( BG \) intersects \( CF \) at point \( I \). The circumcircle of triangle \( BFI \) intersects the circumcircle of triangle \( CGI \) again at point \( K \). Prove that line segment \( HK \) bisects \( BC \).
2014 EGMO, 2
Let $D$ and $E$ be points in the interiors of sides $AB$ and $AC$, respectively, of a triangle $ABC$, such that $DB = BC = CE$. Let the lines $CD$ and $BE$ meet at $F$. Prove that the incentre $I$ of triangle $ABC$, the orthocentre $H$ of triangle $DEF$ and the midpoint $M$ of the arc $BAC$ of the circumcircle of triangle $ABC$ are collinear.
2003 IMO Shortlist, 4
Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $\Gamma_4$ be distinct circles such that $\Gamma_1$, $\Gamma_3$ are externally tangent at $P$, and $\Gamma_2$, $\Gamma_4$ are externally tangent at the same point $P$. Suppose that $\Gamma_1$ and $\Gamma_2$; $\Gamma_2$ and $\Gamma_3$; $\Gamma_3$ and $\Gamma_4$; $\Gamma_4$ and $\Gamma_1$ meet at $A$, $B$, $C$, $D$, respectively, and that all these points are different from $P$. Prove that
\[
\frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}.
\]
2024 Bangladesh Mathematical Olympiad, P9
Let $ABC$ be a triangle and $M$ be the midpoint of side $BC$. The perpendicular bisector of $BC$ intersects the circumcircle of $\triangle ABC$ at points $K$ and $L$ ($K$ and $A$ lie on the opposite sides of $BC$). A circle passing through $L$ and $M$ intersects $AK$ at points $P$ and $Q$ ($P$ lies on the line segment $AQ$). $LQ$ intersects the circumcircle of $\triangle KMQ$ again at $R$. Prove that $BPCR$ is a cyclic quadrilateral.
2017 Estonia Team Selection Test, 3
Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.
2018 Serbia Team Selection Test, 5
Let $H $ be the orthocenter of $ABC $ ,$AB\neq AC $ ,and let $F $ be a point on circumcircle of $ABC $ such that $\angle AFH=90^{\circ} $.$K $ is the symmetric point of $H $ wrt $B $.Let $P $ be a point such that $\angle PHB=\angle PBC=90^{\circ} $,and $Q $ is the foot of $B $ to $CP $.Prove that $HQ $ is tangent to tge circumcircle of $FHK $.
2014 Taiwan TST Round 3, 4
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 $QB = QR$.
2015 USA TSTST, 2
Let ABC be a scalene triangle. Let $K_a$, $L_a$ and $M_a$ be the respective intersections with BC of the internal angle bisector, external angle bisector, and the median from A. The circumcircle of $AK_aL_a$ intersects $AM_a$ a second time at point $X_a$ different from A. Define $X_b$ and $X_c$ analogously. Prove that the circumcenter of $X_aX_bX_c$ lies on the Euler line of ABC.
(The Euler line of ABC is the line passing through the circumcenter, centroid, and orthocenter of ABC.)
[i]Proposed by Ivan Borsenco[/i]
1967 Spain Mathematical Olympiad, 6
An equilateral triangle $ABC$ with center $O$ and radius $OA = R$ is given, and consider the seven regions that the lines of the sides determine on the plane. It is asked to draw and describe the region of the plane transformed from the two shaded regions in the attached figure, by the inversion of center $O$ and power $R^2$.
[img]https://cdn.artofproblemsolving.com/attachments/e/c/bf1cb12c961467d216d54885f3387b328ce744.png[/img]