VTMO 2026 - Grade 9 - Final - SET 4 - Self-study

Let $ABCD$ be a square and consider the circle with centre $A$ and radius $AB$. Let $M$ be any point on $BC$. A tangent through $M$ to this circle meets $CD$ at $N$.

a) When $MB=MC$, find $NC:ND$.

b) Prove that as $M$ varies, the circumcentre of triangle $AMN$ lies on a fixed straight line.

Let $(I)$ be the incircle of a triangle $(S)$. Three tangents to $(I)$, parallel to the three sides of $(S)$, meet the sides of the triangle at six points $A,B,C,D,E,F$. Prove that opposite sides of hexagon $ABCDEF$ have equal lengths.

Let $a,b,c,d$ be positive integers with $a\gt b$ and $c\gt d$. Prove that if $a+b+c+d=ab-cd$, then $a+c$ is composite.

Let $x,y,z$ be real numbers satisfying $x+y+z=5$ and $x^2+y^2+z^2=9$.

a) Prove that $1\le x,y,z\le\dfrac73$.

b) Find the maximum and minimum of $P=xyz$.

Let $a,b,c$ be real numbers with $a\ge b\ge c\gt 0$. Prove that $(a-b+c)\left(\dfrac1a-\dfrac1b+\dfrac1c\right)\ge1$.