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

Let $(O)$ be a circle with diameter $AB$ and let $C$ lie on the circle. The line through $C$ parallel to $AB$ meets the circle at $D$. Choose $E$ on $AB$ with $AE=BD$. Extend $CE$ to meet the circle at $M$, and extend $DO$ to meet it at $N$. Let $F=AN\cap BM$. Prove that $EF\perp AB$.

Let square $ABCD$ be inscribed in circle $(O)$. Let $M$ be the midpoint of the minor arc $CD$, and choose $N$ on $AB$ with $NA=2NB$. Line $MN$ meets the circle again at $E$. Reflect $E$ across $AB$ to obtain $F$.

a) Prove $FA=2FB$.

b) Prove $DF=3FB$.

Find all real $a$ such that $x^2-ax+a+2=0$ has an integer root.

Triangle $ABC$ is isosceles at $A$, with altitude $AH$. Choose $M$ on segment $HC$, distinct from $H,C$. Let $E,F$ be the perpendicular projections of $M$ onto $AC,AB$, respectively, and let $O$ be the midpoint of $AM$. Prove $HO\perp EF$.

Let $a,b$ be positive odd integers satisfying $b\mid a^2+2$ and $a\mid b^2+2$.

a) Prove $ab\mid a^2+b^2+2$.

b) Prove that $\dfrac{a^2+b^2+2}{ab}$ is a perfect square.