geometry problems from EMMO [Every Mathematician Must Outperform]
with aops links in the names
2016 EMMO Junior
Let $ABC$ be a triangle, and $D$ a point on the line $BC$. Let $O$ denote the circumcenter of $ABC$. Suppose the line passing through $D$ and perpendicular to $AC$ cuts line $AO$ at $E$,and $AB$ at $F$ . The lines $AD$ and $BE$ meet at $G$. Prove that the circumcircles of $AEG,GDB$ and the circle with diameter $FG$ are coaxial.
2016 EMMO Junior
Two circles $\omega _1$ and $\omega _2$ with centres $O_1$ and $O_2$ respectively intersect each other at $A$ and $B$. The line $O_1B$ cuts $\omega _2$ again at $X$, the line $O_2 B$ cuts $\omega _1$ again at $Y$, such that $A$ and $B$ are on the same side of line $XY$. Let $P$ and $Q$ be the feet of perpendiculars from $B$ to $AY$ and $AX$, and $M$ be the midpoint of $XY$. The lines $O_1B$ and $O_2B$ cut $PQ$ at $E$ and $F$ respectively. Show that the line $MF$ bisects $AX$ and $ME$ bisects $AY$.
2016 EMMO Junior 5
2016 EMMO Senior
In $\triangle ABC$, let point $D$ be the tangency point of incircle $\omega$ with side $BC$. Analogously define $E$ and $F$ for sides $AC$ and $AB$ respectively. Let $P$ be the feet of the perpendicular from $D$ to $EF$. Let $N$ be the midpoint of arc $ ABC$ of circumcircle $\Gamma$ of $\triangle ABC$. Prove that the lines $AP$ and $ND$ concur on $\Gamma$ .
Let $P$ be the orthocenter of the intouch triangle of $\triangle ABC$. Let the reflection of $P$ on the perpendicular bisectors of $BC,CA,AB$ be $X,Y,Z$. Let $A_1,B_1,C_1$, be the midpoints of $YZ,ZX,XY$. Let $D,E,F$ be the midpoints of $BC,CA,AB$ and $I_A,I_B,I_C$ be the excenters of $\triangle ABC$. Prove that $DA_1,EB_1,FC_1$ are concurrent at the radical center of the nine-point circles of triangles $I_ABC,I_BCA,I_CAB$.
with aops links in the names
It is an ELMO style contest for students at IMOTC in India
it lasted only one year, 2016
EMMO 2016
2016 EMMO Junior
Let $ABC$ be a triangle, and $D$ a point on the line $BC$. Let $O$ denote the circumcenter of $ABC$. Suppose the line passing through $D$ and perpendicular to $AC$ cuts line $AO$ at $E$,and $AB$ at $F$ . The lines $AD$ and $BE$ meet at $G$. Prove that the circumcircles of $AEG,GDB$ and the circle with diameter $FG$ are coaxial.
Sutanay Bhattacharya
Two circles $\omega _1$ and $\omega _2$ with centres $O_1$ and $O_2$ respectively intersect each other at $A$ and $B$. The line $O_1B$ cuts $\omega _2$ again at $X$, the line $O_2 B$ cuts $\omega _1$ again at $Y$, such that $A$ and $B$ are on the same side of line $XY$. Let $P$ and $Q$ be the feet of perpendiculars from $B$ to $AY$ and $AX$, and $M$ be the midpoint of $XY$. The lines $O_1B$ and $O_2B$ cut $PQ$ at $E$ and $F$ respectively. Show that the line $MF$ bisects $AX$ and $ME$ bisects $AY$.
Sutanay Bhattacharya and Atharva Tekawade
Let $\triangle ABC$ be a triangle with circumcenter $O$ and circumcircle $\Gamma.$ The point $X$ lies on $\Gamma$ such that $AX$ is the $A$- symmedian of triangle $\triangle ABC.$ The line through $X$ perpendicular to $AX$ intersects $AB,AC$ in $F,E,$ respectively. Denote by $\gamma$ the nine-point circle of triangle $\triangle AEF,$ and let $\Gamma$ and $\gamma$ intersect again in $P \neq X.$ Further, let the tangent to $\Gamma$ at $A$ meet the line $BC$ in $Y,$ and let $Z$ be the antipode of $A$ with respect to circle $\Gamma.$ Prove that the points $Y,P,Z$ are collinear.
Notes:
1. The $A$-symmedian of triangle $\triangle ABC$ is the reflection of the $A$-median in the $A$-angle bisector.
2. The antipode of a point with respect to a circle is the point on the circle diametrically opposite to it.
Adithya Bhaskar
[based largely on a problem of Tran Quang Hung (buratinogigle)
and its solution by Telv.]
2016 EMMO Senior
In $\triangle ABC$, let point $D$ be the tangency point of incircle $\omega$ with side $BC$. Analogously define $E$ and $F$ for sides $AC$ and $AB$ respectively. Let $P$ be the feet of the perpendicular from $D$ to $EF$. Let $N$ be the midpoint of arc $ ABC$ of circumcircle $\Gamma$ of $\triangle ABC$. Prove that the lines $AP$ and $ND$ concur on $\Gamma$ .
Anant Mudgal
2016 EMMO Senior
In $\triangle ABC$, points $A_1,B_1,C_1$ are feet of alititudes opposite to vertices $A,B,C$ respectively. Let points $A_b,A_c$ be on the lines $AB,AC$ such that $\angle BA_bB_1=\angle ABC$ and $\angle CA_cC_1=\angle ACB$. Let $A_bA_c\cap B_1C_1=X_A$. Analogously define $X_B,X_C$ are defined. Prove that lines $AX_A,BX_B,CX_C$ concur on circumcircle of $\triangle ABC$.
Anant Mudgal
2016 EMMO Senior Let $P$ be the orthocenter of the intouch triangle of $\triangle ABC$. Let the reflection of $P$ on the perpendicular bisectors of $BC,CA,AB$ be $X,Y,Z$. Let $A_1,B_1,C_1$, be the midpoints of $YZ,ZX,XY$. Let $D,E,F$ be the midpoints of $BC,CA,AB$ and $I_A,I_B,I_C$ be the excenters of $\triangle ABC$. Prove that $DA_1,EB_1,FC_1$ are concurrent at the radical center of the nine-point circles of triangles $I_ABC,I_BCA,I_CAB$.
Sagnik Majumder
WizardMath
WizardMath
bonus:
LMAO 2017 shortlist removed
(LMAO was shortof an EMMO replacement, started in 2017)
You are given a triangle $ABC$. Let $I$ be the incenter of $ABC$. $AI$ meets $BC$ at $X$. Let the midpoint of $AX$ be $D$. Define $E$ and $F$ similarly. Prove that the orthocenter $\mathbb{H}$ of $\triangle DEF$ lies on the Euler line $\mathcal{L}_\mathrm{E}$ of $\triangle ABC$.
Let the cevian triangle of the isotomic conjugate of the circumcenter of $\triangle ABC$ be $\triangle XYZ$ and let the orthocenter of $\triangle ABC$ be $H$. Then prove that the isogonal conjugate of $H$ wrt $\triangle XYZ$ lies on the Euler line $\mathcal{L}_\mathrm{E}$ of $\triangle ABC$.
No comments:
Post a Comment