Olympiad problems

2021 Sharygin Geometry Olympiad, 13

Tags: geometry , excircle , incenter , mixtilinear , concurrency , Sharygin Geometry Olympiad , anant mudgal geo

In triangle $ABC$ with circumcircle $\Omega$ and incenter $I$, point $M$ bisects arc $BAC$ and line $\overline{AI}$ meets $\Omega$ at $N\ne A$. The excircle opposite to $A$ touches $\overline{BC}$ at point $E$. Point $Q\ne I$ on the circumcircle of $\triangle MIN$ is such that $\overline{QI}\parallel\overline{BC}$. Prove that the lines $\overline{AE}$ and $\overline{QN}$ meet on $\Omega$.

2023 India National Olympiad, 6

Tags: geometry , INMO , INMO 2023 , anant mudgal geo

Euclid has a tool called [i]cyclos[/i] which allows him to do the following: [list] [*] Given three non-collinear marked points, draw the circle passing through them. [*] Given two marked points, draw the circle with them as endpoints of a diameter. [*] Mark any intersection points of two drawn circles or mark a new point on a drawn circle. [/list] Show that given two marked points, Euclid can draw a circle centered at one of them and passing through the other, using only the cyclos. [i]Proposed by Rohan Goyal, Anant Mudgal, and Daniel Hu[/i]

2021 Sharygin Geometry Olympiad, 15

Tags: geometry , anant mudgal geo

Let $APBCQ$ be a cyclic pentagon. A point $M$ inside triangle $ABC$ is such that $\angle MAB = \angle MCA$, $\angle MAC = \angle MBA$ and $\angle PMB = \angle QMC = 90^\circ$. Prove that $AM$, $BP$, and $CQ$ concur. [i]Anant Mudgal and Navilarekallu Tejaswi[/i]

2023 India IMO Training Camp, 2

Tags: india , geometry , TST , anant mudgal geo

In triangle $ABC$, let $D$ be the foot of the perpendicular from $A$ to line $BC$. Point $K$ lies inside triangle $ABC$ such that $\angle KAB = \angle KCA$ and $\angle KAC = \angle KBA$. The line through $K$ perpendicular to like $DK$ meets the circle with diameter $BC$ at points $X,Y$. Prove that $AX \cdot DY = DX \cdot AY$

2020 Sharygin Geometry Olympiad, 12

Tags: geometry , Sharygin 2020 , anant mudgal geo

Let $H$ be the orthocenter of a nonisosceles triangle $ABC$. The bisector of angle $BHC$ meets $AB$ and $AC$ at points $P$ and $Q$ respectively. The perpendiculars to $AB$ and $AC$ from $P$ and $Q$ meet at $K$. Prove that $KH$ bisects the segment $BC$.

2020 USEMO, 3

Tags: radical axis , Euler , geometry , USEMO , USEMO 2020 , anant mudgal geo

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Let $\Gamma$ denote the circumcircle of triangle $ABC$, and $N$ the midpoint of $OH$. The tangents to $\Gamma$ at $B$ and $C$, and the line through $H$ perpendicular to line $AN$, determine a triangle whose circumcircle we denote by $\omega_A$. Define $\omega_B$ and $\omega_C$ similarly. Prove that the common chords of $\omega_A$,$\omega_B$ and $\omega_C$ are concurrent on line $OH$. Proposed by Anant Mudgal

2023 Sharygin Geometry Olympiad, 22

Tags: geometry , tangent , incircle , projective geometry , Sharygin Geometry Olympiad , Sharygin 2023 , anant mudgal geo

Let $ABC$ be a scalene triangle, $M$ be the midpoint of $BC,P$ be the common point of $AM$ and the incircle of $ABC$ closest to $A$, and $Q$ be the common point of the ray $AM$ and the excircle farthest from $A$. The tangent to the incircle at $P$ meets $BC$ at point $X$, and the tangent to the excircle at $Q$ meets $BC$ at $Y$. Prove that $MX=MY$.

STEMS 2021 Math Cat A, Q3

Tags: geometry , anant mudgal geo , Angle Chasing

An acute scalene triangle $\triangle{ABC}$ with altitudes $\overline{AD}, \overline{BE},$ and $\overline{CF}$ is inscribed in circle $\Gamma$. Medians from $B$ and $C$ meet $\Gamma$ again at $K$ and $L$ respectively. Prove that the circumcircles of $\triangle{BFK}, \triangle{CEL}$ and $\triangle{DEF}$ concur.