Philippines Area II 2008-20 (PMO) 12p

 geometry problems from Area (part 2)  Round of Philippine Mathematical Olympiads (PMO)

with aops links in the names

                                          2008 - 2020

[year x stands for school year x-1 to x]

2008 Philippine MO Areas P2

Let $ABC$ be an acute-angled triangle. Let $D$ and $E$ be points on $\overline{BC}$ and  $\overline{AC}$, respectively, such that  $\overline{AD}  \perp \overline{BC}$ and $\overline{BE} \perp  \overline{AC}$. Let $P$ be the point where $\overrightarrow{AD}$ meets the semicircle constructed outwardly on  $\overline{BC}$, and $Q$ the point where $\overrightarrow{BE}$ meets the semicircle constructed outwardly on  $\overline{AC}$. Prove that $PC = QC$.

2009 Philippine MO Areas P2  (corrected) 

In triangle $\triangle ABC,$ the bisector of angle $\angle B$ intersect the circumcircle at $D$ and $AC$ at $E$ so that $2BD^2=AB^2+ BC^2.$ The midpoint of $AC$ is $F.$ Prove that $DF=EF$.

2010 Philippine MO Areas P2

Let $E$ and $F$ be points on the sides $AB$ and $AD$ of a convex quadrilateral $ABCD$ such that $EF$ is parallel to the diagonal $BD$. Let the segments $CE$ and $CF$ intersect $BD$ at points $G$ and $H$, respectively. Prove that if the quadrilateral $AGCH$ is a parallelogram, then so is $ABCD$.

2011 Philippine MO Areas P2

Denote by $a, b$ and $c$ the sides of a triangle, opposite the angles $\alpha, \beta$ and $\gamma$, respectively. If $\alpha$ is sixty degrees, show that $a^2 =\frac{a^3 + b^3 + c^3}{a + b + c}$

2012 Philippine MO Areas P2

In rectangle $ABCD$, $E$ and $F$ are chosen on $\overline{AB}$ and $\overline{CD}$, respectively, so that $AEFD$ is a square.  If $\frac{AB}{BE} = \frac{BE}{BC}$ , determine the value of $\frac{AB}{BC}$

2013 didn't have geometry

2014 Philippine MO Areas P2

Two circle of radius 12 have their centers on each other as shown in the figure. A smaller circle is constructed tangent to $AB$ and the two given circles, internally to the circle below and externally to the circle above, as shown. Find the radius of the smaller circle.

2015 Philippine MO Areas P2

Points $A, M, N$ and $B$ are collinear, in that order, and $AM = 4, MN = 2, NB = 3$. 

If point $C$ is not collinear with these four points, and $AC = 6$, prove that CN bisects $\angle BCM$

2016 Philippine MO Areas P2

Point $P$ on side $BC$ of triangle $ABC$ satisfies $|BP|: |PC| = 2:1$.

Prove that the line $AP$ bisects the median of triangle $ABC$ drawn from vertex $C$.

2017 Philippine MO Areas P2 (IGO 2015 Intermediate 2)

Let $BH$ be the altitude from the vertex $B$ to the side $AC$ of an acute-angled triangle $ABC$. Let $D$ and $E$ be the midpoints of $AB$ and $AC$, respectively, and $F$ the reflection of $H$ across the line segment $ED$. Prove that the line BF passes through the circumcenter of  $\vartriangle ABC$

2018 Philippine MO Areas P2

A point $P$ is chosen randomly inside the triangle with sides $13, 20$, and $21$. Find the probability that the circle centered at $P$ with radius $1$ will intersect at least one of the sides of the triangle.

2019 Philippine MO Areas P2

In $\Delta ABC$, $AB>AC$ and the incenter is $I$. The incircle of the triangle is tangent to sides $BC$ and $AC$ at points $D$ and $E$, respectively. Let $P$ be the intersection of the lines $AI$ and $DE$, and let $M$ and $N$ be the midpoints of sides $BC$ and $AB$, respectively. Prove that $M$, $N$, and $P$ are collinear.

2020 Philippine MO Areas P2

In $\vartriangle ABC$, $AB = AC$. A line parallel to $BC$ meets sides $AB$ and $AC$ at $D$ and $E$, respectively. The angle bisector of $\angle BAC$ meets the circumcircles of $\vartriangle ABC$ and $\vartriangle ADE$ at points $X$ and $Y$ , respectively. Let $F$ and $G$ be the midpoints of$BY$ and $XY$ , respectively. Let $T$ be the intersection of lines $CY$ and $DF$. Prove that the circumcenter of $\vartriangle FGT$ lies on line $XY$

sources:

cjquines.com
pmo.ph