DeuX MO 2020 5p (aops)

geometry problems from Deux Mathematical Olympiad + Shortlist , Mock USAMO & USA(J)MO),  with aops links

it took place inside Aops, details here

                                    2020

DeuX MO 2020 Shortlist G1

Let $\triangle ABC$ with $BC$ being the minimum length, and $D$ as the intersection of the tangents of $(ABC)$ at points $B$ and $C$. Internal angle bisectors of $\angle B, \angle C$ intersects circle with radius $DB$ at $E$ and $F$ respectively. Define $I$ as its incenter. Suppose $M,N$ are the circumcenters of $AIE, AIF$, and $BI, CI$ intersects $(ABC)$ at $X,Y$. Prove that $$\frac{YM}{XN} = \frac{CH - AH}{BH - AH}$$ where $H$ is the orthocenter of $\triangle ABC$.

 by Jonathan Christian, Indonesia

DeuX MO 2020 Shortlist G2 (Level I Problem 2)

Let $ABC$ be an acute triangle such that $D,E,F$ lies on $BC, CA, AB$ respectively and $AD,BE,CF$ be its altitudes. Let $P$ be the common point of $EF$ with the circumcircle of $\triangle ABC$, with $P$ on the minor arc of $AC$. Define $H' \not= B$ as the common point of $BE$ with the circumcircle of $\triangle ABC$. Prove that $\angle ADH' = \angle APF$ if and only if $\angle ABP = \angle CBM$ where $M$ denotes the midpoint of $AC$.

by Orestis Lignos, Greece

DeuX MO  2020 Shortlist G3 (Level II Problem 4)

Given an acute triangle $ABC$. Define $D,E,F$ as the foot of $A,B,C$ altitude respectively. Let $X_A, X_B, X_C$ be the incenters of $\triangle AEF, \triangle BDF, \triangle CDE$. We name a triangle $A$-yemyem if the circle $(AX_B X_C)$ is tangent to $(ABC)$. Define $B$-yemyem and $C$-yemyemsimilarly. Prove that if a triangle is $A$-yemyem and $B$-yemyem, then it is $C$-yemyem as well. 

by Farrel Dwireswara Salim, Indonesia

DeuX MO  2020 Shortlist G4

Let $\triangle ABC$ be a triangle such that $AB \not= AC$ and $\angle A < 60^{\circ}$. Let $B'$ be the reflection of $B$ with respect to $AC$ and $C'$ be the reflection of $C$ with respect to $AB$. Let $\Gamma$ be the circumcircle of $\triangle AB'C'$. Suppose the tangent of $\Gamma$ passing through $A$ intersects $B'C'$ at point $X$. Let $X'$ to be the reflection of $X$ with respect to point $A$. Prove that $X'$ lies on $BC$.

by Yoshua Yonatan, Indonesia

DeuX MO 2020 Shortlist G5

Given a triangle $ABC$ with circumcenter $O$ and orthocenter $H$. Line $OH$ meets $AB, AC$ at $E,F$ respectively.  Define $S$ as the circumcenter of $AEF$. The circumcircle of $AEF$ meets the circumcircle of $ABC$ again at $J$, $J \not= A$. Line $OH$ meets circumcircle of $JSO$ again at $D$, $D \not= O$ and circumcircle of $JSO$ meets circumcircle of $ABC$ again at $K$, $K \not= J$. Define $M$ as the intersection of $JK$ and $OH$ and $DK$ meets circumcircle of $ABC$ at points $K,G$. Prove that circumcircle of $GHM$ and circumcircle of $ABC$ are tangent to each other.

 by 郝敏言, China