India IITB Mathathon 2019, 2021 5p

  geometry problems from India IITB Mathathon,  with aops links in the names

2021 R2,R3 collected inside aops here

2019, 2021

2018 IITB Mathathon Round 4 Sample  (2013 Rioplatense Olympiad, Level 3, p2)

Let ABCD be a square, and let E and F be points in AB and BC respectively such that BE=BF. In the triangle EBC, let N be the foot of the altitude relative to EC. Let G be the intersection between AD and the extension of the previously mentioned altitude. FG and EC intersect at point P, and the lines NF and DC intersect at point T. Suppose DP meets BC at X. Prove that X is orthocentre of triangle BTD.

2019 IITB Mathathon Round 1 p8 (2016 HMMT Team  p10)

Let $ABC$ be a triangle with incenter $I$ whose incircle is tangent to $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D$, $E$, $F$. Point $P$ lies on $\overline{EF}$ such that $\overline{DP} \perp \overline{EF}$. Ray $BP$ meets $\overline{AC}$ at $Y$ and ray $CP$ meets $\overline{AB}$ at $Z$. Point $Q$ is selected on the circumcircle of $\triangle AYZ$ so that $\overline{AQ} \perp \overline{BC}$. Prove that $P$, $I$, $Q$ are collinear.

2019 IITB Mathathon Round 4 p2 (USA TSTST 2018 p5)

Let $ABC$ be an acute triangle with circumcircle $\omega$, and let $H$ be the foot of the altitude from $A$ to $\overline{BC}$. Let $P$ and $Q$ be the points on $\omega$ with $PA = PH$ and $QA = QH$. The tangent to $\omega$ at $P$ intersects lines $AC$ and $AB$ at $E_1$ and $F_1$ respectively; the tangent to $\omega$ at $Q$ intersects lines $AC$ and $AB$ at $E_2$ and $F_2$ respectively. Show that the circumcircles of $\triangle AE_1F_1$ and $\triangle AE_2F_2$ are congruent, and the line through their centers is parallel to the tangent to $\omega$ at $A$.

by Ankan Bhattacharya and Evan Chen

2021 IITB Mathathon Round 1 p10

In $\Delta ABC , AB = 12 , AC = 13$ and $BC = 14$. Let $I$ be the incenter. The circle with diameter $AI$ meets $(ABC)$ again at $D$. Let external angle bisector of $\angle BDC$ intersect $BC$ at $E$. Find the length of $EB$.

2021 IITB Mathathon Round 2 p1

 A triangle $\triangle ABC$ has its incenter $I$. The lines $AI,BI,CI$ intersect the circumcircle of $\triangle ABC$ in $D,E,F$ resp. Let $M,N,P$ be the circumcenters of triangles $IEF$, $IDF$ and $IDE$ respectively. Prove that the lines $DM$, $EN$ and $FP$ are concurrent.

2021 IITB Mathathon Round 3 p2

In $\triangle ABC$, $AC>AB$. Let $D$ be the point on segment $AC$ such that $CD=AB$. Let the internal angle bisector of $\angle BAC$ intersect $\odot(ABC)$ at $M$. Let the perpendicular bisector of segment $AD$ intersect $AM$ at $E$, and let $F$ be the midpoint of segment $EM$. Prove that $FD=FB$.