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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 $PC = QC$.

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$.

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 $ABCD$.

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 $a^2 =\frac{a^3 + b^3 + c^3}{a + b + c}$

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

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.
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 $\angle BCM$

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  $\vartriangle ABC$

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.

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.

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 $XY$

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