Philippines 2008-22 (PMO) 15p (-14,-15)

geometry problems from Final Round of Philippine Mathematical Olympiads (PMO)
with aops links in the names

2008 - 2022
missing 2014, 2015

2008 Philippine MO P3 
Let $P$ be a point outside a circle $\Gamma$, and let the two tangent lines through $P$ touch $\Gamma$ at $A$ and $B$. Let $C$ be on the minor arc $AB$, and let ray $PC$ intersect $\Gamma$ again at $D$. Let $\ell$ be the line through $B$ and parallel to $PA$. $\ell$ intersects $AC$ and $AD$ at $E$ and $F$, respectively. Prove that $B$ is the midpoint of $EF$.

2009 Philippine MO P5
Segments $AC$ and $BD$ intersect at point $P$ such that $PA = PD$ and $PB = PC$. Let $E$ be the foot of the perpendicular from $P$ to the line $CD$. Prove that the line $PE$ and the perpendicular bisectors of the segments $PA$ and $PB$ are concurrent.

2010 Philippine MO P2
On a cyclic quadrilateral $ABCD$, there is a point $P$ on side $AD$ such that the triangle $CDP$ and the quadrilateral $ABCP$ have equal perimeters and equal areas. Prove that two sides of $ABCD$ have equal lengths.

2011 Philippine MO P2
In triangle $ABC$, let $X$ and $Y$ be the midpoints of $AB$ and $AC$, respectively. On segment $BC$, there is a point $D$, different from its midpoint, such that $\angle{XDY}=\angle{BAC}$. Prove that $AD\perp BC$.

2013 Philippine MO P2
Let $P$ be a point in the interior of triangle $ABC$ . Extend $AP, BP,$ and $CP$ to meet $BC, AC$, and $AB$ at $D, E$, and $F$, respectively. If triangle $APF$, triangle $BPD$ and triangle $CPE$ have equal areas, prove that $P$ is the centroid of triangle $ABC$ .

(16th) 2014 missing
(17th) 2015 missing

2016 Philippine MO P5
Pentagon $ABCDE$ is inscribed in a circle. Its diagonals $AC$ and $BD$ intersect at $F$. The bisectors of $\angle BAC$ and $\angle CDB$ intersect at $G$. Let $AG$ intersect $BD$ at $H$, let $DG$ intersect $AC$ at $I$, and let $EG$ intersect $AD$ at $J$. If $FHGI$ is cyclic and $$JA \cdot FC \cdot GH = JD \cdot FB \cdot GI,$$ prove that $G$, $F$ and $E$ are collinear.

2017 Philippine MO P4
Circles $\mathcal{C}_1$ and $\mathcal{C}_2$ with centers at $C_1$ and $C_2$ respectively, intersect at two points $A$ and $B$. Points $P$ and $Q$ are varying points on $\mathcal{C}_1$ and $\mathcal{C}_2$, respectively, such that $P$, $Q$ and $B$ are collinear and $B$ is always between $P$ and $Q$. Let lines $PC_1$ and $QC_2$ intersect at $R$, let $I$ be the incenter of $\Delta PQR$, and let $S$ be the circumcenter of $\Delta PIQ$. Show that as $P$ and $Q$ vary, $S$ traces the arc of a circle whose center is concyclic with $A$, $C_1$ and $C_2$.

2018 Philippine MO P1
In triangle $ABC$ with $\angle ABC = 60^{\circ}$ and $5AB = 4BC$, points $D$ and $E$ are the feet of the altitudes from $B$ and $C$, respectively. $M$ is the midpoint of $BD$ and the circumcircle of triangle $BMC$ meets line $AC$ again at $N$. Lines $BN$ and $CM$ meet at $P$. Prove that $\angle EDP = 90^{\circ}$.

2019 Philippine MO P4
In acute triangle $ABC$with $\angle BAC > \angle BCA$, let $P$ be the point on side $BC$ such that $\angle PAB = \angle BCA$. The circumcircle of triangle $AP B$ meets side $AC$ again at $Q$. Point $D$ lies on segment $AP$ such that $\angle QDC = \angle CAP$.
Point $E$ lies on line $BD$ such that $CE = CD$. The circumcircle of triangle $CQE$ meets segment $CD$ again at $F$, and line $QF$ meets side $BC$ at $G$. Show that $B, D, F$, and $G$ are concyclic.

2020 Philippine MO P4
Let $\triangle ABC$ be an acute triangle with circumcircle $\Gamma$ and $D$ the foot of the altitude from $A$. Suppose that $AD=BC$. Point $M$ is the midpoint of $DC$, and the bisector of $\angle ADC$ meets $AC$ at $N$. Point $P$ lies on $\Gamma$ such that lines $BP$ and $AC$ are parallel. Lines $DN$ and $AM$ meet at $F$, and line $PF$ meets $\Gamma$ again at $Q$. Line $AC$ meets the circumcircle of $\triangle PNQ$ again at $E$. Prove that $\angle DQE = 90^{\circ}$.

2021 Philippine MO P1

In convex quadrilateral $ABCD$, $\angle CAB = \angle BCD$. $P$ lies on line $BC$ such that

$AP = PC$, $Q$ lies on line $AP$ such that $AC$ and $DQ$ are parallel, $R$ is the point of

intersection of lines $AB$ and $CD$, and $S$ is the point of intersection of lines $AC$ and $QR$.

Line $AD$ meets the circumcircle of $AQS$ again at $T$. Prove that $AB$ and $QT$ are parallel. 2021 Philippine MO P8

In right triangle $ABC$, $\angle ACB = 90^{\circ}$ and $\tan A > \sqrt{2}$. $M$ is the midpoint of

$AB$, $P$ is the foot of the altitude from $C$, and $N$ is the midpoint of $CP$. Line $AB$ meets the

circumcircle of $CNB$ again at $Q$. $R$ lies on line $BC$ such that $QR$ and $CP$ are parallel, $S$

lies on ray $CA$ past $A$ such that $BR = RS$, and $V$ lies on segment $SP$ such that $AV = VP$.

Line $SP$ meets the circumcircle of $CPB$ again at $T$. $W$ lies on ray $VA$ past $A$ such that

$2AW = ST$, and $O$ is the circumcenter of $SPM$. Prove that lines $OM$ and $BW$ are

perpendicular.

2022 Philippine MO P4

Let $\triangle ABC$ have incenter $I$ and centroid $G$. Suppose that $P_A$ is the foot of the perpendicular from $C$ to the exterior angle bisector of $B$, and $Q_A$ is the foot of the perpendicular from $B$ to the exterior angle bisector of $C$. Define $P_B$, $P_C$, $Q_B$, and $Q_C$ similarly. Show that $P_A, P_B, P_C, Q_A, Q_B,$ and $Q_C$ lie on a circle whose center is on line $IG$.

2022 Philippine MO P6

In $\triangle ABC$, let $D$ be the point on side $BC$ such that $AB+BD=DC+CA.$ The line $AD$ intersects the circumcircle of $\triangle ABC$ again at point $X \neq A$. Prove that one of the common tangents of the circumcircles of $\triangle BDX$ and $\triangle CDX$ is parallel to $BC$.

sources: cjquines.com ,pmo.ph