Olympiad problems

2016 USA TSTST, 2

Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Denote by $M$, $N$ the midpoints of $\overline{AH}$, $\overline{BC}$. Suppose the circle $\gamma$ with diameter $\overline{AH}$ meets the circumcircle of $ABC$ at $G \neq A$, and meets line $AN$ at a point $Q \neq A$. The tangent to $\gamma$ at $G$ meets line $OM$ at $P$. Show that the circumcircles of $\triangle GNQ$ and $\triangle MBC$ intersect at a point $T$ on $\overline{PN}$. [i]Proposed by Evan Chen[/i]

2023 Sharygin Geometry Olympiad, 21

Let $ABCD$ be a cyclic quadrilateral; $M_{ac}$ be the midpoint of $AC$; $H_d,H_b$ be the orthocenters of $\triangle ABC,\triangle ADC$ respectively; $P_d,P_b$ be the projections of $H_d$ and $H_b$ to $BM_{ac}$ and $DM_{ac}$ respectively. Define similarly $P_a,P_c$ for the diagonal $BD$. Prove that $P_a,P_b,P_c,P_d$ are concyclic.

projection radical axis humpty point geometry othorcenter cyclic quadrilaterals

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]

euler humpty point inversion power of a point Menelaus' theorem angle chasing barycentric coordinates cyclic quadrilaterals geometric inversion geometry radical axis similar triangles symmedians

2021 Sharygin Geometry Olympiad, 10-11.7

Let $I$ be the incenter of a right-angled triangle $ABC$, and $M$ be the midpoint of hypothenuse $AB$. The tangent to the circumcircle of $ABC$ at $C$ meets the line passing through $I$ and parallel to $AB$ at point $P$. Let $H$ be the orthocenter of triangle $PAB$. Prove that lines $CH$ and $PM$ meet at the incircle of triangle $ABC$.

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2019 IMEO, 1

Let $ABC$ be a scalene triangle with circumcircle $\omega$. The tangent to $\omega$ at $A$ meets $BC$ at $D$. The $A$-median of triangle $ABC$ intersects $BC$ and $\omega$ at $M$ and $N$, respectively. Suppose that $K$ is a point such that $ADMK$ is a parallelogram. Prove that $KA = KN$. [i]Proposed by Alexandru Lopotenco (Moldova)[/i]

humpty point length geometry parallelogram barycentric coordinates cyclic quadrilaterals power of a point spiral similarity symmedians

2014 IMO Shortlist, G6

Let $ABC$ be a fixed acute-angled triangle. Consider some points $E$ and $F$ lying on the sides $AC$ and $AB$, respectively, and let $M$ be the midpoint of $EF$ . Let the perpendicular bisector of $EF$ intersect the line $BC$ at $K$, and let the perpendicular bisector of $MK$ intersect the lines $AC$ and $AB$ at $S$ and $T$ , respectively. We call the pair $(E, F )$ $\textit{interesting}$, if the quadrilateral $KSAT$ is cyclic. Suppose that the pairs $(E_1 , F_1 )$ and $(E_2 , F_2 )$ are interesting. Prove that $\displaystyle\frac{E_1 E_2}{AB}=\frac{F_1 F_2}{AC}$ [i]Proposed by Ali Zamani, Iran[/i]

humpty point geometry angle chasing barycentric coordinates complex numbers construction cyclic quadrilaterals homothety law of cosines law of sines power of a point radical axis similar triangles spiral similarity symmedians