Olympiad problems

2017 Estonia Team Selection Test, 3

Tags: geometry , incenter , reflection , IMO Shortlist , Inversion , Cyclic Quadrilaterals

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 $BDE$.

2020 Indonesia MO, 6

Tags: geometry , Cyclic Quadrilaterals , parallel

Given a cyclic quadrilateral $ABCD$. Let $X$ be a point on segment $BC$ ($X \not= BC$) such that line $AX$ is perpendicular to the angle bisector of $\angle CBD$, and $Y$ be a point on segment $AD$ ($Y \not= D)$ such that $BY$ is perpendicular to the angle bisector of $\angle CAD$. Prove that $XY$ is parallel to $CD$.

2017 Taiwan TST Round 1, 2

Tags: geometry , incenter , reflection , IMO Shortlist , Inversion , Cyclic Quadrilaterals

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 $BDE$.

2016 IMO Shortlist, G4

Tags: geometry , incenter , reflection , IMO Shortlist , Inversion , Cyclic Quadrilaterals

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 $BDE$.

Cono Sur Shortlist - geometry, 2021.G1.2

Tags: geometry , Cyclic Quadrilaterals , triangle -incenter , circumcircle

Let $ABC$ be a triangle and $I$ its incenter. The lines $BI$ and $CI$ intersect the circumcircle of $ABC$ again at $M$ and $N$, respectively. Let $C_1$ and $C_2$ be the circumferences of diameters $NI$ and $MI$, respectively. The circle $C_1$ intersects $AB$ at $P$ and $Q$, and the circle $C_2$ intersects $AC$ at $R$ and $S$. Show that $P$, $Q$, $R$ and $S$ are concyclic.

2017 Azerbaijan Team Selection Test, 2

Tags: geometry , incenter , reflection , IMO Shortlist , Inversion , Cyclic Quadrilaterals

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 Cono Sur Olympiad, 2

Tags: geometry , Cyclic Quadrilaterals , triangle -incenter , circumcircle

Let $ABC$ be a triangle and $I$ its incenter. The lines $BI$ and $CI$ intersect the circumcircle of $ABC$ again at $M$ and $N$, respectively. Let $C_1$ and $C_2$ be the circumferences of diameters $NI$ and $MI$, respectively. The circle $C_1$ intersects $AB$ at $P$ and $Q$, and the circle $C_2$ intersects $AC$ at $R$ and $S$. Show that $P$, $Q$, $R$ and $S$ are concyclic.

2017 India IMO Training Camp, 2

Tags: geometry , incenter , reflection , IMO Shortlist , Inversion , Cyclic Quadrilaterals

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 $BDE$.

2004 National Olympiad First Round, 29

Tags: geometry , circumcircle , trapezoid , trigonometry , cyclic quadrilateral , Cyclic Quadrilaterals , Law of cosine

Let $M$ be the intersection of the diagonals $AC$ and $BD$ of cyclic quadrilateral $ABCD$. If $|AB|=5$, $|CD|=3$, and $m(\widehat{AMB}) = 60^\circ$, what is the circumradius of the quadrilateral? $ \textbf{(A)}\ 5\sqrt 3 \qquad\textbf{(B)}\ \dfrac {7\sqrt 3}{3} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \sqrt{34} $

Russian TST 2017, P1

Tags: geometry , incenter , reflection , IMO Shortlist , Inversion , Cyclic Quadrilaterals

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 Germany Team Selection Test, 3

Tags: geometry , incenter , reflection , IMO Shortlist , Inversion , Cyclic Quadrilaterals

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 German National Olympiad, 4

Tags: geometry , geometry proposed , cyclic quadrilateral , Cyclic Quadrilaterals , tangent circles , Germany

Let $ABCD$ be a cyclic quadrilateral. The point $P$ is chosen on the line $AB$ such that the circle passing through $C,D$ and $P$ touches the line $AB$. Similarly, the point $Q$ is chosen on the line $CD$ such that the circle passing through $A,B$ and $Q$ touches the line $CD$. Prove that the distance between $P$ and the line $CD$ equals the distance between $Q$ and $AB$.

2021 Nigerian MO Round 3, Problem 2

Tags: geometry , Cyclic Quadrilaterals

Let $B, C, D, E$ be four pairwise distinct collinear points and let $A$ be a point not on ine $BC$. Now, let the circumcircle of $\triangle ABC$ meet $AD$ and $AE$ respectively again at $F$ and $G$. Show that $DEFG$ is cyclic if and only if $AB=AC$.

2017 Brazil Team Selection Test, 2

Tags: geometry , incenter , reflection , IMO Shortlist , Inversion , Cyclic Quadrilaterals

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 India IMO Training Camp, 2

Tags: geometry , incenter , reflection , IMO Shortlist , Inversion , Cyclic Quadrilaterals

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

Tags: geometry , incenter , reflection , IMO Shortlist , Inversion , Cyclic Quadrilaterals

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 Germany Team Selection Test, 3

Tags: geometry , incenter , reflection , IMO Shortlist , Inversion , Cyclic Quadrilaterals

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 $BDE$.