geometry problems from (Brazilian) Olympic Revenge
with aops links in the names
\(ABCD\) is a inscribed quadrilateral. \(P\) is the intersection point of its diagonals. \(O\) is its circumcenter. \(\Gamma\) is the circumcircle of \(ABO\). \(\Delta\) is the circumcircle of \(CDO\).
\(M\) is the midpoint of arc \(AB\) on \(\Gamma\) who doesn't contain \(O\). \(N\) is the midpoint of arc \(CD\) on \(\Delta\) who doesn't contain \(O\). Show that \(M,N,P\) are collinear.
2003 Olympic Revenge P1
Let $ABC$ be a triangle with circumcircle $\Gamma$. $D$ is the midpoint of arc $BC$ (this arc does not contain $A$). $E$ is the common point of $BC$ and the perpendicular bisector of $BD$. $F$ is the common point of $AC$ and the parallel to $AB$ containing $D$. $G$ is the common point of $EF$ and $AB$. $H$ is the common point of $GD$ and $AC$. Show that $GAH$ is isosceles.
2003 Olympic Revenge P3
Let $\Gamma$ be a circumference, and $A,B,C,D$ points of $\Gamma$ (in this order).$r$ is the tangent to $\Gamma$ at point A.$s$ is the tangent to $\Gamma$ at point D. Let $E=r \cap BC,F=s \cap BC$. Let $X=r \cap s,Y=AF \cap DE,Z=AB \cap CD$. Show that the points $X,Y,Z$ are collinear.
Note: assume the existence of all above point
2006 missing
2007 Olympic Revenge P3
The triangles $BCD$ and $ACE$ are externally constructed to sides $BC$ and $CA$ of a triangle $ABC$ such that $AE = BD$ and $\angle BDC+\angle AEC = 180^\circ$. Let $F$ be a point on segment $AB$ such that ${AF\over FB}={CD\over CE}$. Prove that ${DE\over CD+CE}={EF\over BC}={FD\over AC}$.
2008 might be missing
2008 Olympic Revenge P
Let $ABC$ be a triangle and $M$ the midpoint of $BC$. Let $D$ and $E$ be points such that $ABC$, $DBA$ and $EAC$ are similar (in this order). The symmedians of $D$ with respect to $DBA$ and $E$ with respect to $EAC$ meet at $K$. Suppose that AM/BC=√3/2 Prove that $KAB$ and $KCA$ are similar (in this order).
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.
2020 Olympic Revenge P3
Let $ABC$ be a triangle and $\omega$ its circumcircle. Let $D$ and $E$ be the feet of the angle bisectors relative to $B$ and $C$, respectively. The line $DE$ meets $\omega$ at $F$ and $G$. Prove that the tangents to $\omega$ through $F$ and $G$ are tangents to the excircle of $\triangle ABC$ opposite to $A$.
with aops links in the names
It is a competition for teachers, written by students.
inside aops here
2002 - 2021
missing 2006 , 2008
2002 Olympic Revenge P2missing 2006 , 2008
\(ABCD\) is a inscribed quadrilateral. \(P\) is the intersection point of its diagonals. \(O\) is its circumcenter. \(\Gamma\) is the circumcircle of \(ABO\). \(\Delta\) is the circumcircle of \(CDO\).
\(M\) is the midpoint of arc \(AB\) on \(\Gamma\) who doesn't contain \(O\). \(N\) is the midpoint of arc \(CD\) on \(\Delta\) who doesn't contain \(O\). Show that \(M,N,P\) are collinear.
2003 Olympic Revenge P1
Let $ABC$ be a triangle with circumcircle $\Gamma$. $D$ is the midpoint of arc $BC$ (this arc does not contain $A$). $E$ is the common point of $BC$ and the perpendicular bisector of $BD$. $F$ is the common point of $AC$ and the parallel to $AB$ containing $D$. $G$ is the common point of $EF$ and $AB$. $H$ is the common point of $GD$ and $AC$. Show that $GAH$ is isosceles.
Let $ABC$ be a triangle with $\angle BAC =60^\circ$. $A'$ is the symmetric point of $A$ wrt $\overline{BC}$. $D$ is the point in $\overline{AC}$ such that $\overline{AB}=\overline{AD}$. $H$ is the orthocenter of triangle $ABC$. $l$ is the external angle bisector of $\angle BAC$. $\{M\}=\overline{A'D}\cap l$,$\{N\}=\overline{CH} \cap l$. Show that $\overline{AM}=\overline{AN}$
2004 Olympic Revenge P1
$ABC$ is a triangle and $D$ is an internal point such that $\angle DAB=\angle DBC =\angle DCA$. $O_a$ is the circumcenter of $DBC$. $O_b$ is the circumcenter of $DAC$. $O_c$ is the circumcenter of $DAB$. Show that if the area of $ABC$ and $O_aO_bO_c$ are equal then $ABC$ is equilateral.
2004 Olympic Revenge P3$ABC$ is a triangle and $D$ is an internal point such that $\angle DAB=\angle DBC =\angle DCA$. $O_a$ is the circumcenter of $DBC$. $O_b$ is the circumcenter of $DAC$. $O_c$ is the circumcenter of $DAB$. Show that if the area of $ABC$ and $O_aO_bO_c$ are equal then $ABC$ is equilateral.
$ABC$ is a triangle and $\omega$ its incircle. Let $P,Q,R$ be the intersections with $\omega$ and the sides $BC,CA,AB$ respectively. $AP$ cuts $\omega$ in $P$ and $X$. $BX,CX$ cut $\omega$ in $M,N$ respectively. Show that $MR,NQ,AP$ are parallel or concurrent.
Note: assume the existence of all above point
2006 missing
2007 Olympic Revenge P3
The triangles $BCD$ and $ACE$ are externally constructed to sides $BC$ and $CA$ of a triangle $ABC$ such that $AE = BD$ and $\angle BDC+\angle AEC = 180^\circ$. Let $F$ be a point on segment $AB$ such that ${AF\over FB}={CD\over CE}$. Prove that ${DE\over CD+CE}={EF\over BC}={FD\over AC}$.
2008 might be missing
2008 Olympic Revenge P
Let $ABC$ be a triangle and $M$ the midpoint of $BC$. Let $D$ and $E$ be points such that $ABC$, $DBA$ and $EAC$ are similar (in this order). The symmedians of $D$ with respect to $DBA$ and $E$ with respect to $EAC$ meet at $K$. Suppose that AM/BC=√3/2 Prove that $KAB$ and $KCA$ are similar (in this order).
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 $D`F$ with $E`G$, $Z$ is the intersection of $D`G$ with $E`F$ 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 $\angle BAX = \angle CAZ$.
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 $T$.
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 $D`F$ with $E`G$, $Z$ is the intersection of $D`G$ with $E`F$ 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 $\Gamma$.
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 $IH$.
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 $BC$.
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 $\Gamma$.
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 $\Gamma$.
Given a triangle $A_1 A_2 A_3$, let $a_i$ denote the side opposite to $A_i$, where indices are taken modulo 3. Let $D_1 \in a_1$. For $D_i \in A_i$, let $\omega_i$ be the incircle of the triangle formed by lines $a_i, a_{i+1}, A_iD_i$, and $D_{i+1} \in a_{i+1}$ with $A_{i+1} D_{i+1}$ tangent to $\omega_i$. Show that the set $\{D_i: i \in \mathbb{N}\}$ is finite.
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 $\Gamma$ 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 $PA=PI$
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 $\angle BAC = 60^{\circ}.$
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.
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.
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.
2020 Olympic Revenge P3
Let $ABC$ be a triangle and $\omega$ its circumcircle. Let $D$ and $E$ be the feet of the angle bisectors relative to $B$ and $C$, respectively. The line $DE$ meets $\omega$ at $F$ and $G$. Prove that the tangents to $\omega$ through $F$ and $G$ are tangents to the excircle of $\triangle ABC$ opposite to $A$.
Let $I, C, \omega$ and $\Omega$ be the incenter, circumcenter, incircle and circumcircle, respectively,
of the scalene triangle $XYZ$ with $XZ > YZ > XY$. The incircle $\omega$ is tangent to the sides
$YZ, XZ$ and $XY$ at the points $D, E$ and $F$. Let $S$ be the point on $\Omega$ such that $XS,
CI$ and $YZ$ are concurrent. Let $(XEF) \cap \Omega = R$, $(RSD) \cap (XEF) = U$, $SU \cap CI = N$,
$EF \cap YZ = A$, $EF \cap CI = T$ and $XU \cap YZ = O$.
Prove that $NARUTO$ is cyclic.
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