### Olympic Revenge 2009-19 (Brazilian) 14p

geometry problems from (Brazilian) Olympic Revenge
with aops links in the names

It is a competition for teachers,  written by students.

2009 - 2019

2009 Olympic Revenge P1
Given a scalene triangle $ABC$ with circuncenter $O$ and circumscribed circle $\Gamma$. Let $D, E ,F$ the midpoints of $BC, AC, AB$. Let $M=OE \cap AD$, $N=OF \cap AD$ and $P=CM \cap BN$. Let $X=AO \cap PE$, $Y=AP \cap OF$. Let $r$ the tangent of $\Gamma$ through $A$. Prove that $r, EF, XY$ are concurrent.

2009 Olympic Revenge P3
Let $ABC$ to be a triangle with incenter $I$. $\omega_{A}$, $\omega_{B}$ and $\omega_{C}$ are the incircles of the triangles $BIC$, $CIA$ and $AIB$, repectively. After all, $T$ is the tangent point between $\omega_{A}$ and $BC$. Prove that the other internal common tangent to $\omega_{B}$ and $\omega_{C}$ passes through the point $T$.

2010 Olympic Revenge P6
Let $ABC$ to be a triangle and $\Gamma$ its circumcircle. Also, let $D, F, G$ and $E$, in this order, on the arc $BC$ which does not contain $A$ satisfying $\angle BAD = \angle CAE$ and $\angle BAF = \angle CAG$. Let $D, F, G$ and $E$ to be the intersections of $AD, AF, AG$ and $AE$ with $BC$, respectively. Moreover, $X$ is the intersection of $DF$ with $EG$, $Y$ is the intersection of $DF$ with $EG$, $Z$ is the intersection of $DG$ with $EF$ and $W$ is the intersection of $EF$ with $DG$. Prove that $X, Y$ and $A$ are collinear, such as $W, Z$ and $A$. Moreover, prove that $\angle BAX = \angle CAZ$.

2011 Olympic Revenge P4
Let $ABCD$ to be a quadrilateral inscribed in a circle $\Gamma$. Let $r$ and $s$ to be the tangents to $\Gamma$ through $B$ and $C$, respectively, $M$ the intersection between the lines $r$ and $AD$ and $N$ the intersection between the lines $s$ and $AD$. After all, let $E$ to be the intersection between the lines $BN$ and $CM$, $F$ the intersection between the lines $AE$ and $BC$ and $L$ the midpoint of $BC$. Prove that the circuncircle of the triangle $DLF$ is tangent to $\Gamma$.

2012 Olympic Revenge P6
Let $ABC$ be an scalene triangle and $I$ and $H$ its incenter, ortocenter respectively. The incircle touchs $BC$, $CA$ and $AB$ at $D,E$ an $F$. $DF$ and $AC$ intersects at $K$ while $EF$ and $BC$ intersets at $M$. Shows that $KM$ cannot be paralel to $IH$.

2013 Olympic Revenge P2
Let $ABC$ to be an acute triangle. Also, let $K$ and $L$ to be the two intersections of the perpendicular from $B$ with respect to side $AC$ with the circle of diameter  $AC$, with $K$ closer to $B$ than $L$. Analogously, $X$ and $Y$ are the two intersections of the perpendicular from $C$ with respect to side $AB$ with the circle of diamter $AB$, with $X$ closer to $C$ than $Y$. Prove that the intersection of $XL$ and $KY$ lies on $BC$.

2014 Olympic Revenge P1
Let $ABC$ an acute triangle and $\Gamma$ its circumcircle. The bisector of $BAC$ intersects $\Gamma$ at $M\neq A$. A line $r$ parallel to $BC$ intersects $AC$ at $X$ and $AB$ at $Y$. Also, $MX$ and $MY$ intersect $\Gamma$ again at $S$ and $T$, respectively. If $XY$ and $ST$ intersect at $P$, prove that $PA$ is tangent to $\Gamma$.

2016 Olympic Revenge P3
Let $\Gamma$ be a fixed circle. Find all finite sets $S$ of points on $\Gamma$ with the following property: For any point $P$ on $\Gamma$, one can partition $S$ into two sets $A$ and $B$ such that the sum of distances from $P$ to the points in $A$ equals the sum of distances from $P$ to the points in $B$.

2016 Olympic Revenge P4
Let $\Omega$ and $\Gamma$ be two circles such that $\Omega$ is inside $\Gamma$. Let $P$ be a point in $\Gamma$. Let the the two lines tangent to $\Omega$ passing through $P$ meet again $\Gamma$ at $A$ and $B$ ($A,B \neq P$). Prove that as we vary the point $P$ in $\Gamma$, the line $AB$ remains tangent to a fixed circle.

2017 Olympic Revenge P2
Let $\triangle$$ABC a triangle with circumcircle \Gamma. Suppose there exist points R and S on sides AB and AC, respectively, such that BR=RS=SC. A tangent line through A to \Gamma meet the line RS at P. Let I the incenter of triangle \triangle$$ARS$. Prove that $PA=PI$

2018 Olympic Revenge P2
Let $ABC$ be a scalene triangle with incenter $I$ and circumcircle $\Gamma$, of center $O$. Let the inscribed circle to $ABC$ touch the sides at $D$, $E$ and $F$. Let $AI$ intersect $EF$ at $N$, and $\Gamma$ at $M \neq A$. Let $MD$ intersect $\Gamma$ at $L\neq M$. Let $IL$ intersect $EF$ at $K$. And let the circle with diameter $MN$ intersect $\Gamma$ at $P \neq M$. Prove that $AK, PN$ and $OI$ meet at a point.

2018 Olympic Revenge P4
Let $ABC$ be a triangle with acute angles with incircle $\omega$ and incenter $I$. Let $T_A,T_B$ and $T_C$ be the points where $\omega$ touches the sides of $ABC$. Let $l_A$ be a line through $A$ parallel to $BC$, and let $l_B$ and $l_C$ be analogously defined. Let $L_A$ be the second intersection of $AI$ to the circumcircle of $ABC$, and define $L_B$ and $L_C$ analogously. Let $P_A = T_BT_C \cap l_A$ and $P_B, P_C$ be defined analogously. Let $S_A = P_BT_B \cap P_CT_C$ and define $S_B, S_C$ analogously. Prove that $S_AL_A, S_BL_B$ and $S_CL_C$ meet at a point.

2019 Olympic Revenge P1
Let $ABC$ be a scalene acute-angled triangle and $D$ be the point on its circumcircle such that $AD$ is a symmedian of triangle $ABC$. Let $E$ be the reflection of  $D$ about $BC$, $C_0$ the reflection of $E$ about $AB$ and $B_0$ the reflection of $E$ about $AC$. Prove that the lines $AD$, $BB_0$ and $CC_0$ are concurrent if and only if $\angle BAC = 60^{\circ}.$

2019 Olympic Revenge P3
Let $\Gamma$ be a circle centered at $O$ with radius $R$. Let $X$ and $Y$ be points on $\Gamma$ such that $XY<R$. Let $I$ be a point such that $IX = IY$ and $XY = OI$. Describe how to construct with ruler and compass a triangle which has circumcircle $\Gamma$, incenter $I$ and Euler line $OX$. Prove that this triangle is unique.