geometry problems from Mathematics Olympiads Discord Server (MODS)
with aops links in the names
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
with aops links in the names
2020 under construction
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
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