This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 8

2009 Poland - Second Round, 3
Tags: menelaus , projective geometry , geometry , perpendicular bisector
Disjoint circles $ o_1, o_2$, with centers $ I_1, I_2$ respectively, are tangent to the line $ k$ at $ A_1, A_2$ respectively and they lie on the same side of this line. Point $ C$ lies on segment $ I_1I_2$ and $ \angle A_1CA_2 \equal{} 90^{\circ}$. Let $ B_1$ be the second intersection of $ A_1C$ with $ o_1$, and let $ B_2$ be the second intersection of $ A_2C$ with $ o_2$. Prove that $ B_1B_2$ is tangent to the circles $ o_1, o_2$.

2021 Bulgaria EGMO TST, 1
Tags: geometry , length , menelaus , area
On the side $AB$ of a triangle $ABC$ is chosen a point $P$. Let $Q$ be the midpoint of $BC$ and let $CP$ and $AQ$ intersect at $R$. If $AB + AP = CP$, prove that $CR = AB$.

2004 Germany Team Selection Test, 2
Tags: menelaus , angle bisector , geometry
In a triangle $ABC$, let $D$ be the midpoint of the side $BC$, and let $E$ be a point on the side $AC$. The lines $BE$ and $AD$ meet at a point $F$. Prove: If $\frac{BF}{FE}=\frac{BC}{AB}+1$, then the line $BE$ bisects the angle $ABC$.

2004 Germany Team Selection Test, 2
Tags: angle bisector , geometry , menelaus
In a triangle $ABC$, let $D$ be the midpoint of the side $BC$, and let $E$ be a point on the side $AC$. The lines $BE$ and $AD$ meet at a point $F$. Prove: If $\frac{BF}{FE}=\frac{BC}{AB}+1$, then the line $BE$ bisects the angle $ABC$.

2024 Bulgaria National Olympiad, 6
Tags: menelaus , tangent circle , geometry
Given is a triangle $ABC$ and a circle $\omega$ with center $I$ that touches $AB, AC$ and meets $BC$ at $X, Y$. The line through $I$ perpendicular to $BC$ meets the line through $A$ parallel to $BC$ at $Z$. Show that the circumcircles of $\triangle XYZ$ and $\triangle ABC$ are tangent to each other.

2023 Romania National Olympiad, 2
Tags: menelaus , similar triangles , geometry
In the parallelogram $ABCD$, $AC \cap BD = { O }$, and $M$ is the midpoint of $AB$. Let $P \in (OC)$ and $MP \cap BC = { Q }$. We draw a line parallel to $MP$ from $O$, which intersects line $CD$ at point $N$. Show that $A,N,Q$ are collinear if and only if $P$ is the midpoint of $OC$.

2011 IFYM, Sozopol, 7
Tags: menelaus , geometry
The inscribed circle of $\Delta ABC$ $(AC<BC)$ is tangent to $AC$ and $BC$ in points $X$ and $Y$ respectively. A line is constructed through the middle point $M$ of $AB$, parallel to $XY$, which intersects $BC$ in $N$. Let $L\in BC$ be such that $NL=AC$ and $L$ is between $C$ and $N$. The lines $ML$ and $AC$ intersect in point $K$. Prove that $BN=CK$.

2019 Iran MO (3rd Round), 1
Tags: reflection , geometric transformation , geometry , incenter , circumcircle , law of sines , menelaus
Consider a triangle $ABC$ with incenter $I$. Let $D$ be the intersection of $BI,AC$ and $CI$ intersects the circumcircle of $ABC$ at $M$. Point $K$ lies on the line $MD$ and $\angle KIA=90^\circ$. Let $F$ be the reflection of $B$ about $C$. Prove that $BIKF$ is cyclic.