geometry problems from International Mathematical Excellence Olympiad (IMEO)
with aops links in the names
it did not take place in 2018
Let $O$ be the circumcenter of a triangle$ ABC$. Let $M$ be the midpoint of $AO$. The $BO$ and $CO$ intersect the altitude $AD$ at points $E$ and $F$,respectively. Let $O_1$ and $O_2$ be the circumcenters of the triangle $ABE$ and $ACF$, respectively. Prove that $M$ lies on $O_1O_2$.
it did not take place in 2018
2019 IMEO p1
Let $ABC$ be a scalene triangle with circumcircle $\omega$. The tangent to $\omega$ at $A$ meets $BC$ at $D$. The $A$-median of triangle $ABC$ intersects $BC$ and $\omega$ at $M$ and $N$, respectively. Suppose that $K$ is a point such that $ADMK$ is a parallelogram. Prove that $KA = KN$.
Let $ABC$ be a scalene triangle with incenter $I$ and circumcircle $\omega$. The internal and external bisectors of angle $\angle BAC$ intersect $BC$ at $D$ and $E$, respectively. Let $M$ be the point on segment $AC$ such that $MC = MB$. The tangent to $\omega$ at $B$ meets $MD$ at $S$. The circumcircles of triangles $ADE$ and $BIC$ intersect each other at $P$ and $Q$. If $AS$ meets $\omega$ at a point $K$ other than $A$, prove that $K$ lies on $PQ$.
with aops links in the names
it did not take place in 2018
2017 - 2020
2017 IMEO p2Let $O$ be the circumcenter of a triangle$ ABC$. Let $M$ be the midpoint of $AO$. The $BO$ and $CO$ intersect the altitude $AD$ at points $E$ and $F$,respectively. Let $O_1$ and $O_2$ be the circumcenters of the triangle $ABE$ and $ACF$, respectively. Prove that $M$ lies on $O_1O_2$.
2019 IMEO p1
Let $ABC$ be a scalene triangle with circumcircle $\omega$. The tangent to $\omega$ at $A$ meets $BC$ at $D$. The $A$-median of triangle $ABC$ intersects $BC$ and $\omega$ at $M$ and $N$, respectively. Suppose that $K$ is a point such that $ADMK$ is a parallelogram. Prove that $KA = KN$.
Alexandru Lopotenco (Moldova)
2019 IMEO p6Let $ABC$ be a scalene triangle with incenter $I$ and circumcircle $\omega$. The internal and external bisectors of angle $\angle BAC$ intersect $BC$ at $D$ and $E$, respectively. Let $M$ be the point on segment $AC$ such that $MC = MB$. The tangent to $\omega$ at $B$ meets $MD$ at $S$. The circumcircles of triangles $ADE$ and $BIC$ intersect each other at $P$ and $Q$. If $AS$ meets $\omega$ at a point $K$ other than $A$, prove that $K$ lies on $PQ$.
Alexandru Lopotenco (Moldova)
Pitchayut Saengrungkongka
Let $ABC$ be a triangle and $A'$ be the reflection of $A$ about $BC$. Let $P$ and $Q$ be points on $AB$ and $AC$, respectively, such that $PA'=PC$ and $QA'=QB$. Prove that the perpendicular from $A'$ to $PQ$ passes through the circumcenter of $\triangle ABC$.
Fedir Yudin
Let $O$, $I$, and $\omega$ be the circumcenter, the incenter, and the incircle of nonequilateral $\triangle ABC$. Let $\omega_A$ be the unique circle tangent to $AB$ and $AC$, such that the common chord of $\omega_A$ and $\omega$ passes through the center of $\omega_A$ . Let $O_A$ be the center of $\omega_A$. Define $\omega_B, O_B, \omega_C, O_C$ similarly. If $\omega$ touches $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively, prove that the perpendiculars from $D$, $E$, $F$ to $O_BO_C , O_CO_A , O_AO_B$ are concurrent on the line $OI$.
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