Taiwan APMO preliminary 2018-21 6p

  geometry problems from Taiwanese APMO preliminary with aops links in the names

2018 - 2021 

2018 Taiwan APMO Preliminary p1

Let trapezoid $ABCD$ inscribed in a circle $O$, $AB||CD$. Tangent at $D$ wrt $O$ intersects line $AC$ at $F$, $DF||BC$. If $CA=5, BC=4$, then find $AF$.

2018 Taiwan APMO Preliminary p6

Let $ABCD$ be an unit square. Let $E,F$ be the midpoints of $CD,BC$ respectively. $AE$ intersects the diagonal $BD$ at $P$. $AF$ intersects $BD,BE$ at $Q,R$ respectively. Find the area of quadrilateral $PQRE$.

2019 Taiwan APMO Preliminary p1

In $\triangle ABC$, $\angle B=90^\circ$, segment $AB>BC$. Now we have a $\triangle A_iBC(i=1,2,...,n)$ which is similiar to $\triangle ABC$ (the vertexs of them might not correspond). Find the maximum value of $n+2018$.

2019 Taiwan APMO Preliminary p3

Let $\triangle ABC$ be an acute triangle, $H$ is its orthocenter. $\overrightarrow{AH},\overrightarrow{BH},\overrightarrow{CH}$ intersect $\triangle ABC$'s circumcircle at $A',B',C'$ respectively. Find the range (minimum value and the maximum upper bound) of $\dfrac{AH}{AA'}+\dfrac{BH}{BB'}+\dfrac{CH}{CC'}$

2020 Taiwan APMO Preliminary p1

Let $\triangle ABC$ satisfies $\cos A:\cos B:\cos C=1:1:2$, then $\sin A=\sqrt[s]{t}$($s\in\mathbb{N},t\in\mathbb{Q^+}$ and $t$ is an irreducible fraction). Find $s+t$.

2020 Taiwan APMO Preliminary p7

[$XYZ$] denotes the area of $\triangle XYZ$

We have a $\triangle ABC$,$BC=6,CA=7,AB=8$

(1) If $O$ is the circumcenter of $\triangle ABC$, find [$OBC$] : [$OCA$] : [$OAB$]

(2) If $H$ is the orthocenter of $\triangle ABC$, find [$HBC$] : [$HCA$] : [$HAB$]

2021 Taiwan APMO Preliminary p2

(a) Let the incenter of $\triangle ABC$ be $I$. We connect $I$ other $3$ vertices and divide $\triangle ABC$ into $3$ small triangles which has area $2,3$ and $4$. Find the area of the inscribed circle of $\triangle ABC$.

(b) Let $ABCD$ be a parallelogram. Point $E,F$ is on $AB,BC$ respectively. If $[AED]=7,[EBF]=3,[CDF]=6$, then find $[DEF].$ (Here $[XYZ]$ denotes the area of $XYZ$)

2021 Taiwan APMO Preliminary p5

$\triangle ABC$, $\angle A=23^{\circ},\angle B=46^{\circ}$. Let $\Gamma$ be a circle with center $C$, radius $AC$. Let the external angle bisector of $\angle B$ intersects $\Gamma$ at $M,N$. Find $\angle MAN$.