Olympiad problems

2020 Israel Olympic Revenge, G

Let $ABC$ be an acute triangle with $AB\neq AC$. The angle bisector of $\angle BAC$ intersects with $BC$ at a point $D$. $BE,CF$ are the altitudes of the triangle and $Ap_1,Ap_2$ are the isodynamic points of triangle $ABC$.Let the $A$-median of $ABC$ intersect $EF$ at $T$. Show that the line connecting $T$ with the nine-point center of $ABC$ is perpendicular to $BC$ if and only if $\angle Ap_1DAp_2=90^\circ$.

2011 Greece Team Selection Test, 4

Tags: nine point center , poncelet point , circumcircle , geometry , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral and let $K,L,M,N,S,T$ the midpoints of $AB, BC, CD, AD, AC, BD$ respectively. Prove that the circumcenters of $KLS, LMT, MNS, NKT$ form a cyclic quadrilateral which is similar to $ABCD$.

2011 Greece Team Selection Test, 4

Tags: cyclic quadrilateral , nine point center , poncelet point , geometry , circumcircle

Let $ABCD$ be a cyclic quadrilateral and let $K,L,M,N,S,T$ the midpoints of $AB, BC, CD, AD, AC, BD$ respectively. Prove that the circumcenters of $KLS, LMT, MNS, NKT$ form a cyclic quadrilateral which is similar to $ABCD$.

2019 Belarus Team Selection Test, 2.2

Tags: circumcircle , perpendicular bisector , nine point center , euler line , geometry

Let $O$ be the circumcenter and $H$ be the orthocenter of an acute-angled triangle $ABC$. Point $T$ is the midpoint of the segment $AO$. The perpendicular bisector of $AO$ intersects the line $BC$ at point $S$. Prove that the circumcircle of the triangle $AST$ bisects the segment $OH$. [i](M. Berindeanu, RMC 2018 book)[/i]

Russian TST 2017, P1

Tags: geometry , nine point center

The diagonals of a convex quadrilateral divide it into four triangles. Prove that the nine point centers of these four triangles either lie on one straight line, or are the vertices of a parallelogram.