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

Given $x=\dfrac{b^2+c^2-a^2}{2bc}$ and $y=\dfrac{a^2-(b-c)^2}{(b+c)^2-a^2}$, calculate $P=x+y+xy$.

Numbers $a,b,c,x,y,z$ satisfy $a+b+c=0$, $x+y+z=0$ and $\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=0$. Calculate $A=a^2x+b^2y+c^2z$.

In rectangle $ABCD$, $AB=5$ and $BC=2$. Points $M$ on $AB$ and $N$ on $DC$ satisfy $AM=1$ and $DN=2$. Let $E$ be the reflection of $D$ in $M$, and let $F=EN\cap CM$. (a) Prove that $CM\perp DM$. (b) Let $K=DF\cap CE$. Find $DK$.

A string consists of letters $A$ and $B$. Any of these replacements is allowed: $BBA\to A$, $ABA\to B$, $A\to BBA$, $B\to ABA$. Starting with $BA\cdots A$ (one $B$ followed by $2019$ letters $A$), can one obtain $A\cdots AB$ ($2019$ letters $A$ followed by one $B$) after finitely many replacements? Explain.

Find $(x,y)$ satisfying $(4x^2+6x+4)(4y^2-12y+25)=28$.