VTMO 2026 - Grade 9 - Final - SET 12 - Self-study

Let $ABC$ be a non-isosceles triangle whose shortest side is $BC$. Choose $D$ on $AB$ and $E$ on $AC$ with $BD=CE=BC$. Let $M=BE\cap CD$, let $MH\perp BC$, and set $K=AM\cap BC$.

a) Prove $BH=\dfrac{BC+BA-AC}{2}$.

b) If $AB+AC=3BC$, prove $CK=BH$.

Let $ABCD$ be a quadrilateral. Choose $M$ on $AB$ and $N$ on $CD$ with $\dfrac{BM}{BA}=\dfrac{DN}{DC}$. Let $O$ be the midpoint of $MN$. Prove that $O$ moves on a fixed line.

Let $AD,BE$ be angle bisectors of triangle $ABC$. The circumcircles of triangles $ADC,BEC$ meet again at $F$ on $AB$. Find $\angle ACB$.

Let $G$ be an interior point of triangle $ABC$, and let $M,N,P$ be the intersections of $AG,BG,CG$ with $BC,CA,AB$, respectively. Suppose
$$\frac{AG}{GM}+\frac{BG}{GN}+\frac{CG}{GP}=6.$$
Prove that $G$ is the centroid of $ABC$.

Let $ABCD$ be a parallelogram with $AB=2BC$ and $\angle DAB\gt 90^\circ$. Let $M$ be the midpoint of $DC$. Choose $K$ on line $AC$ such that $\angle MKC=\angle CKB$. The circumcircle of $DKM$ meets $AC$ again at $N$. Find $NA/NC$.