Discord Geometry Olympiad 2021 (DGO) 10p

 geometry problems from Discord Geometry Olympiad (DGO),  with aops links in the names

collected inside aops here

2021

2021 Discord Geometry Olympiad Individual 1.1

Let $ABC$ be a triangle and let $M$ be a point on side $BC$ such that $BM < MC$. Let $B', C'$ be points on sides $AC, AB$ such that $AM, BB', CC'$ are concurrent. Determine if it is possible for $B'C'$ to be parallel to $BC$.

by wassupevery1

2021 Discord Geometry Olympiad Individual 1.2

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. $OH$ and $OA$ intersect circumcircle of triangle $BOC$ again at $P, K$. Suppose that $P, O$ lie on the same side with respect to $BC$. Prove that $P B + PC = PK$.

by wassupevery1

2021 Discord Geometry Olympiad Individual and Team 1.3

Let triangle $ABC$ be a triangle with incenter $I$ and circumcircle $\Omega$ with circumcenter $O$. The incircle touches $CA, AB$ at $E, F$ respectively. $R$ is another intersection point of external bisector of $\angle BAC$ with $\Omega$, and $T$ is $\text{A-mixtillinear}$ incircle touch point to $\Omega$. Let $W, X, Z$ be points lie on $\Omega$. $RX$ intersect $AI$ at $Y$ . Assume that $R \ne X$. Suppose that $E, F, X, Y$ and $W, Z, E, F$ are concyclic, and $AZ, EF, RX$ are concurrent.

Prove that

$\bullet$ $AZ, RW, OI$ are concurrent.

$\bullet$ $\text{A-symmedian}$, tangent line to $\Omega$ at $T$ and $WZ$ are concurrent.

by wassupevery1 and k12byda5h

2021 Discord Geometry Olympiad Individual 2.1

Let $M$ be an interior point inside a triangle $ABC$ such that $\angle MAB < \angle MAC$. Is it always true that

$\frac{MB}{MC}<\frac{AB}{AC}$ ?

by wassupevery1

2021 Discord Geometry Olympiad Individual 2.2

For each triangle $XYZ$, we say that a triangle $ABC$ is relatively good triangle of triangle $XYZ$ if and only if $A, B, C$ lie on side $YZ, XZ, XY$ respectively, and $XA, YB, ZC$ are concurrent. Given a triangle $ABC$, let $A_1B_1C_1$ to be a relatively good triangle of triangle $ABC$. Define triangle $A_nB_nC_n$ to be relatively good of triangle $A_{n-1}B_{n-1}C_{n-1}$ for every positve integer $n$ greater than $1$. Determine all integers $n\geq  2$ such that $AA_n, BB_n, CC_n$ are always concurrent.

by wassupevery1

2021 Discord Geometry Olympiad Individual and Team 2.3

Let $O$ be the circumcenter of triangle $ABC$. The altitudes from $A, B, C$ of triangle $ABC$ intersects the circumcircle of the triangle $ABC$ at $A_1, B_1, C_1$ respectively. $AO, BO, CO$ meets $BC, CA, AB$ at $A_2, B_2, C_2$ respectively. Prove that the circumcircles of triangles $AA_1A_2, BB_1B_2, CC_1C_2$ share two common points.

by wassupevery1

2021 Discord Geometry Olympiad Team 1.1

Given triangle $ABC$ with orthocenter $H$ and $AD,BE,CF$ be the altitudes. Let $K$ be the symmedian point of triangle $ABC$. $N$ is the circumcenter of triangle $DEF$. $L$ is the isogonal conjugate point of $K$ with respect to triangle $DEF$. Prove that $H,N,L$ are collinear.

2021 Discord Geometry Olympiad Team 1.2

Let triangle $ABC$ be the triangle with circumcenter $O$. $P$ be the point on circumcircle of triangle $BOC$. $AP,AO$ intersect the circle $BOC$ again at $X,Y$. Suppose that $PY,OX,BC$ are concurrent. Suppose that $P,O$ lie on the same side with respect to $BC$. Prove that $PB+PC = PY$.

by wassupevery1 and k12byda5h

2021 Discord Geometry Olympiad Team and Individual 1.3

Let triangle $ABC$ be a triangle with incenter $I$ and circumcircle $\Omega$ with circumcenter $O$. The incircle touches $CA, AB$ at $E, F$ respectively. $R$ is another intersection point of external bisector of $\angle BAC$ with $\Omega$, and $T$ is $\text{A-mixtillinear}$ incircle touch point to $\Omega$. Let $W, X, Z$ be points lie on $\Omega$. $RX$ intersect $AI$ at $Y$ . Assume that $R \ne X$. Suppose that $E, F, X, Y$ and $W, Z, E, F$ are concyclic, and $AZ, EF, RX$ are concurrent.

Prove that

$\bullet$ $AZ, RW, OI$ are concurrent.

$\bullet$ $\text{A-symmedian}$, tangent line to $\Omega$ at $T$ and $WZ$ are concurrent.

by wassupevery1 and k12byda5h

2021 Discord Geometry Olympiad Team 2.1

Let triangle $ABC$ be triangle with orthocenter $H$ and circumcircle $O$. A point $X$ lies on line $BC$. $AH$ intersects the circumcircle of triangle $ABC$ again at $H'$. $AX$ intersects circumcircle of triangle $H'HX$ again at $Y$ and intersects circumcircle of triangle $ABC$ again at $Z$. Let $G$ be the intersection of $BC$ with $H'O$. Let $P$ lies on $AB$ such that $PH'A = 90^\circ - \angle BAC$. Prove that

1. the ratio and the angle between $YH$ and $ZG$ do not depend on the choices of $X$.

2. $\angle PYH = \angle BZG$.

by k12byda5h

2021 Discord Geometry Olympiad Team 2.2

Given a right triangle $ABC$ ($\angle CAB = 90^\circ$) with $AC > AB$. Points $X,Y$ lie on the line $AB$ such that $X,A,B,Y$ lie in this order. Point $Z,W$ lie on the line $AC$ such that $A,C,Z,W$ lie in this order. Suppose that $BX = BY = AC,CZ=AX,CW = AY$. Prove that the quadrilateral formed by the lines $CX,CY,BW,BZ$ is cyclic.

2021 Discord Geometry Olympiad Team and Individual 2.3

Let $O$ be the circumcenter of triangle $ABC$. The altitudes from $A, B, C$ of triangle $ABC$ intersects the circumcircle of the triangle $ABC$ at $A_1, B_1, C_1$ respectively. $AO, BO, CO$ meets $BC, CA, AB$ at $A_2, B_2, C_2$ respectively. Prove that the circumcircles of triangles $AA_1A_2, BB_1B_2, CC_1C_2$ share two common points.

by wassupevery1