MO Discord Server 2020 (MODS) (uc)

geometry problems from Mathematics Olympiads Discord Server (MODS)
with aops links in the names

collected inside aops here
4 problems each paper

2020 under construction

2020 MO Discord Server January Beginner p
2020 MO Discord Server January Intermediate p
2020 MO Discord Server January Advanced p
2020 MO Discord Server February Beginner p

2020 MO Discord Server February Intermediate p4
A square $ABCD$ has side length 1 and centre $O$. A point $P$, distinct from $O$, is chosen in the interior of $ABCD$. Let $P_a$ lie on the ray $AP$ such that $AP_a \cdot AP = 1$. Define $P_b$, $P_c$, and $P_d$ similarly. Suppose that $P_aPP_c$ and $P_bPP_d$ are non-degenerate triangles. Prove that their circumcircles intersect on line $OP$ at a point other than $P$.

2020 MO Discord Server February Advanced p
2020 MO Discord Server March Beginner p
2020 MO Discord Server March Intermediate p

2020 MO Discord Server March Advanced p2
An acute triangle $ABC$ has circumcircle $\Gamma$ and circumcentre $O$. The incentres of $AOB$ and $AOC$ are $I_b$ and $I_c$ respectively. Let $M$ be the the point on $\Gamma$ such that $MB = MC$ and $M$ lies on the same side of $BC$ as $A$. Prove that the points $M$, $A$, $I_b$, and $I_c$ are concyclic.

2020 MO Discord Server April Beginner p2
Let $ABC$ be a triangle with $\angle BAC$ not a right angle. Let $X$ be the point on ray $BC$ such that $BA = BX$, and $Y$ be the point on ray $CB$ such that $CA = CY$. Let $P \neq A$ be the point on $AX$ such that $CA = CP$, and $Q \neq A$ be the point on $AY$ such that $BA = BQ$. Show that $CP$, $BQ$ and the perpendicular bisector of $BC$ are concurrent.

2020 MO Discord Server April Intermediate p
2020 MO Discord Server April Advanced p
2020 MO Discord Server May Beginner p

2020 MO Discord Server May Intermediate p3
Let $ABC$ be an isosceles triangle with $AB = AC$ and $\angle BAC < 60^{\circ}$. Let $M$ be the midpoint of $AB$ and $\Gamma$ be the circumcircle of $\triangle MBC$. Let $D$ be a point on $\Gamma$. Suppose that the circle centered at $D$ passing through $A$ intersects $\Gamma$ at distinct points $X$ and $Y$. Let $P$ and $Q$ lie on $XY$ such that $PB$ and $QC$ are tangent to $\Gamma$, and denote by $R$ the intersection of $PB$ and $QC$. Prove that regardless of the choice of $D$, the triangle $PQR$ has constant perimeter.


2020 MO Discord Server May Advanced p
2020 MO Discord Server June Beginner p
2020 MO Discord Server June Intermediate p


2020 MO Discord Server June Advanced p3
Let a lattice tetrahedron denote a tetrahedron whose vertices have integer coordinates. Given a lattice tetrahedron, a move consists of picking some vertex and moving it parallel to one of the three edges of the face opposite the vertex so that it lands on a different point with integer coordinates. Prove that any two lattice tetrahedra with the same volume can be transformed into each other by a series of moves

2020 MO Discord Server July  Beginner p
2020 MO Discord Server July  Intermediate p
2020 MO Discord Server July  Advanced p



source: https://github.com/Mathematical-Olympiads-Discord-Server/files/tree/master/Contests