geometry problems from IMOC (Taiwenese IMO Camp problem colection)
with aops links in the names
2017 IMOC G1
Given $\vartriangle ABC$. Choose two points $P, Q$ on $AB, AC$ such that $BP = CQ$. Let $M, T$ be the midpoints of $BC, PQ$. Show that $MT$ is parallel to the angle bisector of $\angle BAC$
2017 IMOC G2
Given two acute triangles $\vartriangle ABC, \vartriangle DEF$. If $AB \ge DE, BC \ge EF$ and $CA \ge FD$, show that the area of $\vartriangle ABC$ is not less than the area of $\vartriangle DEF$
with aops links in the names
2017 - 2022
Given $\vartriangle ABC$. Choose two points $P, Q$ on $AB, AC$ such that $BP = CQ$. Let $M, T$ be the midpoints of $BC, PQ$. Show that $MT$ is parallel to the angle bisector of $\angle BAC$
Given two acute triangles $\vartriangle ABC, \vartriangle DEF$. If $AB \ge DE, BC \ge EF$ and $CA \ge FD$, show that the area of $\vartriangle ABC$ is not less than the area of $\vartriangle DEF$
2017 IMOC G3
Let $ABCD$ be a circumscribed quadrilateral with center $O$. Assume the incenters of $\vartriangle AOC, \vartriangle BOD$ are $I_1, I_2$, respectively. If circumcircles of $\vartriangle AI_1C$ and $\vartriangle BI_2D$ intersect at $X$, prove the following identity:
$(AB \cdot CX \cdot DX)^2 + (CD\cdot AX \cdot BX)^2 = (AD\cdot BX \cdot CX)^2 + (BC \cdot AX \cdot DX)^2$
2017 IMOC G4
Givan a acute $\vartriangle ABC$ with orthocenter $H$. Let $M_a$ be the midpoint of $BC. M_aH$ intersects the circumcircle of $\vartriangle ABC$ at $X_a$ and $AX_a$ intersects $BC$ at $Y_a$. Define $Y_b, Y_c$ in a similar way. Prove that $Y_a, Y_b,Y_c$ are collinear.
We have $\vartriangle ABC$ with $I$ as its incenter. Let $D$ be the intersection of $AI$ and $BC$ and define $E, F$ in a similar way. Furthermore, let $Y = CI \cap DE, Z = BI \cap DF$. Prove that if $\angle BAC = 120^o$, then $E, F, Y,Z$ are concyclic.
A point $P$ lies inside $\vartriangle ABC$ such that the values of areas of $\vartriangle PAB, \vartriangle PBC, \vartriangle PCA$ can form a triangle. Let $BC = a,CA = b,AB = c, PA = x,PB = y, PC = z$, prove that
$\frac{(x + y)^2 + (y + z)^2 + (z + x)^2}{x + y + z} \le a + b + c$
2017 IMOC G7
Given $\vartriangle ABC$ with circumcenter $O$. Let $D$ be a point satisfying $\angle ABD = \angle DCA$ and $M$ be the midpoint of $AD$. Suppose that $BM,CM$ intersect circle $(O)$ at another points $E, F$, respectively. Let $P$ be a point on $EF$ so that $AP$ is tangent to circle $(O)$. Prove that $A, P,M,O$ are concyclic.
Given an integer $n \ge 3$. Find the largest positive integer $k$ with the following property:
For $n$ points in general position, there exists $k$ ways to draw a non-intersecting polygon with those $n$ points as it’s vertices.
different wording
Given $n$, find the maximum $k$ so that for every general position of $n$ points , there are at least $k$ ways of connecting the points to form a polygon.
2018 IMOC G2
Given $\vartriangle ABC$ with circumcircle $\Omega$. Assume $\omega_a, \omega_b, \omega_c$ are circles which tangent internally to $\Omega$ at $T_a,T_b, T_c $ and tangent to $BC,CA,AB$ at $P_a, P_b, P_c$, respectively. If $AT_a,BT_b,CT_c$ are collinear, prove that $AP_a,BP_b,CP_c$ are collinear.
Given $\vartriangle ABC$ with circumcircle $\Omega$. Assume $\omega_a, \omega_b, \omega_c$ are circles which tangent internally to $\Omega$ at $T_a,T_b, T_c $ and tangent to $BC,CA,AB$ at $P_a, P_b, P_c$, respectively. If $AT_a,BT_b,CT_c$ are collinear, prove that $AP_a,BP_b,CP_c$ are collinear.
2018 IMOC G3
Given an acute $\vartriangle ABC$ whose orthocenter is denoted by $H$. A line $\ell$ passes $H$ and intersects $AB,AC$ at $P ,Q$ such that $H$ is the mid-point of $P,Q$. Assume the other intersection of the circumcircle of $\vartriangle ABC$ with the circumcircle of $\vartriangle APQ$ is $X$. Let $C'$ is the symmetric point of $C$ with respect to $X$ and $Y$ is the another intersection of the circumcircle of $\vartriangle ABC$ and $AO$, where O is the circumcenter of $\vartriangle APQ$. Show that $CY$ is tangent to circumcircle of $\vartriangle BCC'$.
Given an acute $\vartriangle ABC$ whose orthocenter is denoted by $H$. A line $\ell$ passes $H$ and intersects $AB,AC$ at $P ,Q$ such that $H$ is the mid-point of $P,Q$. Assume the other intersection of the circumcircle of $\vartriangle ABC$ with the circumcircle of $\vartriangle APQ$ is $X$. Let $C'$ is the symmetric point of $C$ with respect to $X$ and $Y$ is the another intersection of the circumcircle of $\vartriangle ABC$ and $AO$, where O is the circumcenter of $\vartriangle APQ$. Show that $CY$ is tangent to circumcircle of $\vartriangle BCC'$.
2018 IMOC G4
Given an acute $\vartriangle ABC$ with incenter $I$. Let $I'$ be the symmetric point $I$ with respect to the midpoint of $B,C$ and $D$ is the foot of $A$. If $DI$ and the circumcircle of vartriangle $BI'C$ intersect at $T$ and $TI' $ intersects the circumcircle of $\vartriangle ATI$ at $X$. Furthermore, $E,F$ are tangent points of the incircle and $AB,AC, P$ is the another intersection of the circumcircles of $\vartriangle ABC, \vartriangle AEF$. Show that $AX \parallel PI$.
Given an acute $\vartriangle ABC$ with incenter $I$. Let $I'$ be the symmetric point $I$ with respect to the midpoint of $B,C$ and $D$ is the foot of $A$. If $DI$ and the circumcircle of vartriangle $BI'C$ intersect at $T$ and $TI' $ intersects the circumcircle of $\vartriangle ATI$ at $X$. Furthermore, $E,F$ are tangent points of the incircle and $AB,AC, P$ is the another intersection of the circumcircles of $\vartriangle ABC, \vartriangle AEF$. Show that $AX \parallel PI$.
2018 IMOC G5
Suppose $I,O,H$ are incenter, circumcenter, orthocenter of $\vartriangle ABC$ respectively. Let $D = AI \cap BC$,$E = BI \cap CA$, $F = CI \cap AB$ and $X$ be the orthocenter of $\vartriangle DEF$. Prove that $IX \parallel OH$.
Suppose $I,O,H$ are incenter, circumcenter, orthocenter of $\vartriangle ABC$ respectively. Let $D = AI \cap BC$,$E = BI \cap CA$, $F = CI \cap AB$ and $X$ be the orthocenter of $\vartriangle DEF$. Prove that $IX \parallel OH$.
2019 IMOC G1
Let $I$ be the incenter of a scalene triangle $\vartriangle ABC$. In other words, $\overline{AB},\overline{BC},\overline{CA}$ are distinct. Prove that if $D,E$ are two points on rays $\overrightarrow{BA},\overrightarrow{CA}$, satisfying $\overline{BD}=\overline{CA},\overline{CE}=\overline{BA}$ then line $DE$ pass through the orthocenter of $\vartriangle BIC$.
Let $I$ be the incenter of a scalene triangle $\vartriangle ABC$. In other words, $\overline{AB},\overline{BC},\overline{CA}$ are distinct. Prove that if $D,E$ are two points on rays $\overrightarrow{BA},\overrightarrow{CA}$, satisfying $\overline{BD}=\overline{CA},\overline{CE}=\overline{BA}$ then line $DE$ pass through the orthocenter of $\vartriangle BIC$.
2019 IMOC G2
Given a scalene triangle $\vartriangle ABC$ with orthocenter $H$. The midpoint of $BC$ is denoted by $M$. $AH$ intersects the circumcircle at $D \ne A$ and $DM$ intersects circumcircle of $\vartriangle ABC$ at $T\ne D$. Now, assume the reflection points of $M$ with respect to $AB,AC,AH$ are $F,E,S$. Show that the midpoints of $BE,CF,AM,TS$ are concyclic.
Given a scalene triangle $\vartriangle ABC$ with orthocenter $H$. The midpoint of $BC$ is denoted by $M$. $AH$ intersects the circumcircle at $D \ne A$ and $DM$ intersects circumcircle of $\vartriangle ABC$ at $T\ne D$. Now, assume the reflection points of $M$ with respect to $AB,AC,AH$ are $F,E,S$. Show that the midpoints of $BE,CF,AM,TS$ are concyclic.
2019 IMOC G3
Given a scalene triangle $\vartriangle ABC$ has orthocenter $H$ and circumcircle $\Omega$. The tangent lines passing through $A,B,C$ are $\ell_a,\ell_b,\ell_c$. Suppose that the intersection of $\ell_b$ and $\ell_c$ is $D$. The foots of $H$ on $\ell_a,AD$ are $P,Q$ respectively. Prove that $PQ$ bisects segment $BC$
2019 IMOC G4
$\vartriangle ABC$ is a scalene triangle with circumcircle $\Omega$. For a arbitrary $X$ in the plane, define $D_x,E_x, F_x$ to be the intersection of tangent line of $X$ (with respect to $BXC$) and $BC,CA,AB$, respectively. Let the intersection of $AX$ with $\Omega$ be $S_x$ and $T_x = D_xS_x \cap \Omega$. Show that $\Omega$ and circumcircle of $\vartriangle T_xE_xF_x$ are tangent to each other.
Given a scalene triangle $\vartriangle ABC$ has orthocenter $H$ and circumcircle $\Omega$. The tangent lines passing through $A,B,C$ are $\ell_a,\ell_b,\ell_c$. Suppose that the intersection of $\ell_b$ and $\ell_c$ is $D$. The foots of $H$ on $\ell_a,AD$ are $P,Q$ respectively. Prove that $PQ$ bisects segment $BC$
$\vartriangle ABC$ is a scalene triangle with circumcircle $\Omega$. For a arbitrary $X$ in the plane, define $D_x,E_x, F_x$ to be the intersection of tangent line of $X$ (with respect to $BXC$) and $BC,CA,AB$, respectively. Let the intersection of $AX$ with $\Omega$ be $S_x$ and $T_x = D_xS_x \cap \Omega$. Show that $\Omega$ and circumcircle of $\vartriangle T_xE_xF_x$ are tangent to each other.
2019 IMOC G5
Given a scalene triangle $\vartriangle ABC$ with orthocenter $H$ and circumcenter $O$. The exterior angle bisector of $\angle BAC$ intersects circumcircle of $\vartriangle ABC$ at $N \ne A$. Let $D$ be another intersection of $HN$ and the circumcircle of $\vartriangle ABC$. The line passing through $O$, which is parallel to $AN$, intersects $AB,AC$ at $E, F$, respectively. Prove that $DH$ bisects the angle $\angle EDF$.
Given a scalene triangle $\vartriangle ABC$ with orthocenter $H$ and circumcenter $O$. The exterior angle bisector of $\angle BAC$ intersects circumcircle of $\vartriangle ABC$ at $N \ne A$. Let $D$ be another intersection of $HN$ and the circumcircle of $\vartriangle ABC$. The line passing through $O$, which is parallel to $AN$, intersects $AB,AC$ at $E, F$, respectively. Prove that $DH$ bisects the angle $\angle EDF$.
IMOC 2020 G1 (ltf0501)
Let $O$ be the circumcenter of triangle $ABC$. Choose a point $X$ on the circumcircle ⊙ $(ABC)$ such that $OX\parallel BC$. Assume that ⊙ $(AXO)$ intersects $AB, AC$ at $E, F$, respectively, and $OE, OF$ intersect $BC$ at $P, Q$, respectively. Furthermore, assume that ⊙ $(XP Q)$ and ⊙$ (ABC)$ intersect at $R$. Prove that $OR$ and ⊙$(XP Q)$ are tangent to each other.
IMOC 2020 G2 (Li4)
Let $O$ be the circumcenter of triangle $ABC$. Define $O_{A0} = O_{B0} = O_{C0} = O$. Recursively, define $O_{An}$ to be the circumcenter of $\vartriangle BO_{A(n-1)}C$. Similarly define $O_{Bn}, O_{Cn}$. Find all $n \ge 1$ so that for any triangle $ABC$ such that $O_{An}, O_{Bn}, O_{Cn}$ all exist, it is true that $AO_{An}, BO_{Bn}, CO_{Cn}$ are concurrent.
IMOC 2020 G3 (houkai)
Triangle $ABC$ has incenter $I$ and circumcenter $O$. $AI, BI, CI$ intersect the circumcircle of $ABC$ again at $M_A, M_B, M_C$, respectively. Show that the Euler line of $BIC$ passes through the circumcenter of $OM_BM_C$.
IMOC 2020 G4 (ltf0501)
Let $I$ be the incenter of triangle $ABC$. Let $BI$ and $AC$ intersect at $E$, and $CI$ and $AB$ intersect at $F$. Suppose that $R$ is another intersection of ⊙ $(ABC)$ and ⊙ $(AEF)$. Let $M$ be the midpoint of $BC$, and $P, Q$ are the intersections of $AI, MI$ and $EF$, respectively. Show that $A, P, Q, R$ are concyclic.
IMOC 2020 G5 (ltf0501)
Let $O, H$ be the circumcentor and the orthocenter of a scalene triangle $ABC$. Let $P$ be the reflection of $A$ w.r.t. $OH$, and $Q$ is a point on ⊙$(ABC)$ such that $AQ, OH, BC$ are concurrent. Let $A'$ be a points such that $ABA'C$ is a parallelogram. Show that $A', H, P, Q$ are concylic.
IMOC 2020 G6 (Li4)
Let $ABC$ be a triangle, and $M_a, M_b, M_c$ be the midpoints of $BC, CA, AB$, respectively. Extend $M_bM_c$ so that it intersects ⊙ $(ABC)$ at $P$. Let $AP$ and $BC$ intersect at $Q$. Prove that the tangent at $A$ to ⊙ $(ABC)$ and the tangent at $P$ to ⊙ $(P QM_a)$ intersect on line $BC$.
Let $\overline{BE}$ and $\overline{CF}$ be altitudes of triangle $ABC$, and let $D$ be the antipodal point of $A$ on the circumcircle of $ABC$. The lines $\overleftrightarrow{DE}$ and $\overleftrightarrow{DF}$ intersect $\odot(ABC)$ again at $Y$ and $Z$, respectively. Show that $\overleftrightarrow{YZ}$, $\overleftrightarrow{EF}$ and $\overleftrightarrow{BC}$ intersect at a point.
Let the midline of $\triangle ABC$ parallel to $BC$ intersect the circumcircle $\Gamma$ of $\triangle ABC$ at $P$, $Q$, and the tangent of $\Gamma$ at $A$ intersects $BC$ at $T$. Show that $\measuredangle BTQ = \measuredangle PTA$.
Let $I$ be the incenter of the acute triangle $\triangle ABC$, and $BI$, $CI$ intersect the altitude of $\triangle ABC$ through $A$ at $U$, $V$, respectively. The circle with $AI$ as a diameter intersects $\odot(ABC)$ again at $T$, and $\odot(TUV)$ intersects the segment $BC$ and $\odot(ABC)$ at $P$, $Q$, respectively. Let $R$ be another intersection of $PQ$ and $\odot(ABC)$. Show that $AR\parallel BC$.
Let $D$ be a point on the side $AC$ of a triangle $ABC$. Suppose that the incircle of triangle $BCD$ intersects $BD$ and $CD$ at $X$, $Y$, respectively. Show that $XY$ passes through a fixed point when $D$ is moving on the side $AC$.
The incircle of a cyclic quadrilateral $ABCD$ tangents the four sides at $E$, $F$, $G$, $H$ in counterclockwise order. Let $I$ be the incenter and $O$ be the circumcenter of $ABCD$. Show that the line connecting the centers of $\odot(OEG)$ and $\odot(OFH)$ is perpendicular to $OI$.
Let $\Omega$ be the circumcircle of triangle $ABC$. Suppose that $X$ is a point on the segment $AB$ with $XB=XC$, and the angle bisector of $\angle BAC$ intersects $BC$ and $\Omega$ at $D$, $M$, respectively. If $P$ is a point on $BC$ such that $AP$ is tangent to $\Omega$ and $Q$ is a point on $DX$ such that $CQ$ is tangent to $\Omega$, show that $AB$, $CM$, $PQ$ are concurrent.
The incircle of triangle $ABC$ tangents $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Let the tangents of $E$, $F$ with respect to $\odot(AEF)$ intersect at $P$, and $X$ be a point on $BC$ such that $EF$, $DP$, $AX$ are concurrent. Define $Q$, $Y$ and $R$, $Z$ similarly. Show that $X$, $Y$, $Z$ are collinear.
Let $P$ be an arbitrary interior point of $\triangle ABC$, and $AP$, $BP$, $CP$ intersect $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Suppose that $M$ be the midpoint of $BC$, $\odot(AEF)$ and $\odot(ABC)$ intersect at $S$, $SD$ intersects $\odot(ABC)$ at $X$, and $XM$ intersects $\odot(ABC)$ at $Y$. Show that $AY$ is tangent to $\odot(AEF)$.
Let the incenter and the $A$-excenter of $\triangle ABC$ be $I$ and $I_A$, respectively. Let $BI$ intersect $AC$ at $E$ and $CI$ intersect $AB$ at $F$. Suppose that the reflections of $I$ with respect to $EF$, $FI_A$, $EI_A$ are $X$, $Y$, $Z$, respectively. Show that $\odot(XYZ)$ and $\odot(ABC)$ are tangent to each other.
Let $O$, $I$ be the circumcenter and the incenter of triangle $ABC$, respectively, and let the incircle tangents $BC$ at $D$. Furthermore, suppose that $H$ is the orthocenter of triangle $BIC$, $N$ is the midpoint of the arc $BAC$, and $X$ is the intersection of $OI$ and $NH$. If $P$ is the reflection of $A$ with respect to $OI$, show that $\odot(IDP)$ and $\odot(IHX)$ are tangent to each other.
The incircle of $\triangle ABC$ tangents $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. The projections of $B$, $C$ to $AD$ are $U$, $V$, respectively; the projections of $C$, $A$ to $BE$ are $W$, $X$, respectively; and the projections of $A$, $B$ to $CF$ are $Y$, $Z$, respectively. Show that the circumcircle of the triangle formed by $UX$, $VY$, $WZ$ is tangent to the incircle of $\triangle ABC$.
The circumcenter and orthocenter of $ABC$ are $O$ and $H$, respectively. Let $XACH$ be a
parallelogram. Show that if $OH$ is parallel to $BC$, then $OX$ and $AB$ intersect at some point
on the perpendicular bisector of $AH$.
2022 IMOC G2
The incenter of triangle $ABC$ is $ I$. the circumcircle of $ABC$ is tangent to $BC$, $CA$, $AB$
at $T, E, F$. $R$ is a point on $BC$ . Let the $C$-excenter of $\vartriangle CER$ be $L$. Prove that
points $L,T,F$ are collinear if and only if $B,E,A,R$ are concyclic.
2022 IMOC G3
Let $\vartriangle ABC$ be an acute triangle. $R$ is a point on arc $BC$. Choose two points $P, Q$ on
$AR$ such that $B,P,C,Q$ are concyclic. Let the second intersection of $BP$, $CP$, $BQ$, $CQ$ and
the circumcircle of $\vartriangle ABC$ is $P_B$, $P_C$, $Q_B$, $Q_C$, respectively. Let the
circumcenter of $\vartriangle P P_BP_C$ and $\vartriangle QQ_BQ_C$ are $O_P$ and $O_Q$,
respectively. Prove that $A,O_P,O_Q,R$ are concylic.
2022 IMOC G4
Let $\vartriangle ABC$ be an acute triangle with circumcircle $\Omega$. A line passing through $A$
perpendicular to $BC$ meets $\Omega$ again at $D$. Draw two circles $\omega_b$, $\omega_c$ with
$B, C$ as centers and $BD$, $CD$ as radii, respectively, and they intersect $AB$, $AC$ at $E, F,$
respectively. Let $K\ne A$ be the second intersection of $(AEF)$ and $\Omega$, and let $\omega_b$,
$\omega_c$ intersect $KB$, $KC$ at $P, Q$, respectively. The circumcenter of triangle $DP Q$ is $O$,
prove that $K, O, D$ are collinear.
2022 IMOC G5
$P$ is a point inside $ABC$. $BP$, $CP$ intersect $AC, AB$ at $E, F$, respectively. $AP$ intersect
$\odot (ABC)$ again at X. $\odot (ABC)$ and $\odot (AEF)$ intersect again at $S$. $T$ is a point on
$BC$ such that $P T \parallel EF$. Prove that $\odot (ST X)$ passes through the midpoint of $BC$.
2022 IMOC G6Let $D$ be a point on the circumcircle of some triangle $ABC$. Let $E, F$ be points on $AC$, $AB$,
respectively, such that $A,D,E,F$ are concyclic. Let $M$ be the midpoint of $BC$. Show that if $DM$,
$BE$, $CF$ are concurrent, then either $BE \cap CF$ is on the circle $ADEF$, or $EF$ is parallel to
$BC$.
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