VTMO 2026 - Grade 9 - Final - SET 1 - Self-study

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

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

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

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

Find all integers $k$ for which the following equation in $x$ has integer roots: $$x^2-2kx+3k+2=0.$$