US Ersatz Math Olympiad 2020-21 (USEMO) 6p

geometry problems from US Ersatz Math Olympiad (USEMO)

with aops links in the names

It started in 2020

2020 - 2021

May 2020

2020 May USEMO p1
Let $ABCD$ be a cyclic quadrilateral. A circle centered at $O$ passes through $B$ and $D$ and meets lines $BA$ and $BC$ again at points $E$ and $F$ (distinct from $A,B,C$). Let $H$ denote the orthocenter of triangle $DEF.$ Prove that if lines $AC,$ $DO,$ $EF$ are concurrent, then triangle $ABC$ and $EHF$ are similar.

Robin Son

2020 May USEMO p6
Let $ABC$ be an acute scalene triangle with circumcenter $O$ and altitudes $\overline{AD}$, $\overline{BE}$, $\overline{CF}$. Let $X$, $Y$, $Z$ be the midpoints of $\overline{AD}$, $\overline{BE}$, $\overline{CF}$. Lines $AD$ and $YZ$ intersect at $P$, lines $BE$ and $ZX$ intersect at $Q$, and lines $CF$ and $XY$ intersect at $R$. Suppose that lines $YZ$ and $BC$ intersect at $A'$, and lines $QR$ and $EF$ intersect at $D'$. Prove that the perpendiculars from $A$, $B$, $C$, $O$, to the lines $QR$, $RP$, $PQ$, $A'D'$, respectively, are concurrent.

Ankan Bhattacharya

October 2020

2020 October USEMO p3

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Let $\Gamma$ denote the circumcircle of triangle $ABC$, and $N$ the midpoint of $OH$. The tangents to $\Gamma$ at $B$ and $C$, and the line through $H$ perpendicular to line $AN$, determine a triangle whose circumcircle we denote by $\omega_A$. Define $\omega_B$ and $\omega_C$ similarly. Prove that the common chords of $\omega_A$,$\omega_B$ and $\omega_C$ are concurrent on line $OH$.

 Anant Mudgal

2020 October USEMO p5

The sides of a convex $200$-gon $A_1 A_2 \dots A_{200}$ are colored red and blue in an alternating fashion. Suppose the extensions of the red sides determine a regular $100$-gon, as do the extensions of the blue sides. Prove that the $50$ diagonals $\overline{A_1A_{101}},\ \overline{A_3A_{103}},\ \dots,\ \overline{A_{99}A_{199}}$ are concurrent.

 Ankan Bhattacharya

October 2021

2021 May USEMO p1

Let $A_1C_2B_1A_2C_1B_2$ be an equilateral hexagon. Let $O_1$ and $H_1$ denote the circumcenter and orthocenter of $\triangle A_1B_1C_1$, and let $O_2$ and $H_2$ denote the circumcenter and orthocenter of $\triangle A_2B_2C_2$. Suppose that $O_1 \ne O_2$ and $H_1 \ne H_2$. Prove that the lines $O_1O_2$ and $H_1H_2$ are either parallel or coincide.

Ankan Bhattacharya

2021 May USEMO p4

Let $ABC$ be a triangle with circumcircle $\omega$, and let $X$ be the reflection of $A$ in $B$. Line $CX$ meets $\omega$ again at $D$. Lines $BD$ and $AC$ meet at $E$, and lines $AD$ and $BC$ meet at $F$. Let $M$ and $N$ denote the midpoints of $AB$ and $AC$.

Can line $EF$ share a point with the circumcircle of triangle $AMN?$

Sayandeep Shee

source: web.evanchen.cc/usemo.html