Quarantine MO 2020 (Argentinian, Mexican, Global) 7p

geometry problems from Quarantine Mathematical Olympiads with aops links in the names,
so far below is Argentinian, Mexican and Global

Global Quarantine Mathematical Olympiad 2020

 CGMO 2020 EN in pdf with solutions and marking scheme

collected inside aops here

2020 Global Quarantine MO Beginner Exam p3
Let $A$ and $B$ be two distinct points in the plane. Let $M$ be the midpoint of the segment $AB$, and let $\omega$ be a circle that goes through $A$ and $M$. Let $T$ be a point on $\omega$ such that the line $BT$ is tangent to $\omega$. Let $X$ be a point (other than $B$) on the line $AB$ such that $TB = TX$, and let $Y$ be the foot of the perpendicular from $A$ onto the line $BT$.
Prove that the lines $AT$ and $XY$ are parallel.

by Navneel Singhal, India

2020 Global Quarantine MO Advanced Exam p1
Let $ABC$ be a triangle with incentre $I$. The incircle of the triangle $ABC$ touches the sides $AC$ and $AB$ at points $E$ and $F$ respectively. Let $\ell_B$ and $\ell_C$ be the tangents to the circumcircle of $BIC$ at $B$ and $C$ respectively. Show that there is a circle tangent to $EF, \ell_B$ and $\ell_C$ with centre on the line $BC$.

by Navneel Singhal, India

2020 Global Quarantine MO Advanced Exam p8
Let $ABC$ be an acute scalene triangle, with the feet of $A,B,C$ onto $BC,CA,AB$ being $D,E,F$ respectively. Let $W$ be a point inside $ABC$ whose reflections over $BC,CA,AB$ are $W_a,W_b,W_c$ respectively. Finally, let $N$ and $I$ be the circumcenter and the incenter of $W_aW_bW_c$ respectively. Prove that, if $N$ coincides with the nine-point center of $DEF$, the line $WI$ is parallel to the Euler line of $ABC$.

by Navneel Singhal, India 

and Massimiliano Foschi, Italy

Mexican Quarantine Mathematical Olympiad 2020

collected inside aops here

2020 Mexican Quarantine MO p2

Let $\Gamma_1$ and $\Gamma_2$ be circles intersecting at points $A$ and $B$. A line through $A$ intersects $\Gamma_1$ and $\Gamma_2$ at $C$ and $D$ respectively. Let $P$ be the intersection of the lines tangent to $\Gamma_1$ at $A$ and $C$, and let $Q$ be the intersection of the lines tangent to $\Gamma_2$ at $A$ and $D$. Let $X$ be the second intersection point of the circumcircles of $BCP$ and $BDQ$, and let $Y$ be the intersection of lines $AB$ and $PQ$. Prove that $C$, $D$, $X$ and $Y$ are concyclic. 

2020 Mexican Quarantine MO p4
Let $ABC$ be an acute triangle with orthocenter $H$. Let $A_1$, $B_1$ and $C_1$ be the feet of the altitudes of triangle $ABC$ opposite to vertices $A$, $B$, and $C$ respectively. Let $B_2$ and $C_2$ be the midpoints of $BB_1$ and $CC_1$, respectively. Let $O$ be the intersection of lines $BC_2$ and $CB_2$. Prove that $O$ is the circumcenter of triangle $ABC$ if and only if $H$ is the midpoint of $AA_1$.

by Dorlir Ahmeti

Quarantine - Argentinian Olympic Revenge

collected inside aops here

2020 Quarantine - Argentinian Olympic Revenge p4

Let $ABC$ be an acute triangle with $\angle A <45$. Let $D$ be a point in triangle $ABC$ such that $BD = CD$ and $\angle  BDC = 4\angle  A$. Point $E$ is the reflection of $C$ wrt $AB$ and point $F$ is the reflection of $B$ wrt $AC$. Prove that $AD$ and $EF$ are perpendicular.

2020 Quarantine - Argentinian Olympic Revenge p5

Find all positive reals $c$ such that for every acute triangle $ABC$ with area $T$ and for every point $P$ in the plane

$$(AP + BP + CP)^2 \ge cT$$

source:  https://www.omaforos.com.ar/archivo.php?id=5496