VTMO 2026 - Grade 8 - Final - SET 2 - Self-study

Triangle $ABC$ is isosceles at $A$. The segment $BD$ bisects $\angle ABC$. The altitude $AH$ satisfies $AH=\frac12BD$. Find $\angle BAC$.

Let $ABCD$ be a square. Point $E$ lies on $BC$ with $BE=\frac14BC$. The bisector of $\angle DEB$ meets $AB$ at $F$. Prove that $DF$ bisects $\angle ADE$.

Let $ABC$ be an acute triangle with $AB\lt AC$. Its angle bisector is $AD$ and its median is $AM$, where $D,M\in BC$. From $C$, draw the perpendicular to $AD$ meeting it at $H$. The line $HM$ meets $AC$ at $N$; $CH$ meets $AM$ at $E$; and $ED$ meets $AB$ at $F$.

a) Prove that $N$ is the midpoint of $AC$.

b) Prove that triangle $DFA$ is isosceles.

Solve the system
$$\begin{cases}4-x_1x_2=0,\\4-x_2x_3=0,\\\vdots\\4-x_{2012}x_{2013}=0,\\4-x_1x_{2013}=0.\end{cases}$$

In acute triangle $ABC$, the altitudes $BD$ and $CE$ meet at $H$. Take a point $I$ inside triangle $HBC$ such that $\angle IBD=\angle ICE$. Let $M,N$ be the perpendicular projections of $I$ onto $AB,AC$ respectively. Prove that $HI$ passes through the midpoint of $MN$.