VTMO 2026 - Grade 8 - Final - SET 1 - Self-study

In triangle $ABC$, $\angle BAC=120^\circ$. The angle bisectors are $AD$, $BE$ and $CF$. Prove that $DE\perp DF$.

a) Let $a\ne b$, $b\ne c$, $c\ne a$ and $abc\ne0$. Calculate $\frac1a+\frac1b+\frac1c$, given that $a(a-b)=b(b-c)=c(c-a)$.

b) Positive integers $a,b,c$ satisfy $a+b=b(a-c)$, where $c+1$ is the square of a prime number. Prove that at least one of $a+b$ and $ab$ is a perfect square.

Triangle $ABC$ is right-angled at $A$ and $AC=2AB$. Let $M$ be the midpoint of $AB$. The line through $A$ perpendicular to $CM$ meets $BC$ at $D$. Find $\frac{BD}{DC}$.

Four balls have masses $1$ g, $2$ g, $3$ g and $5$ g. An electronic scale can weigh any mass. Prove that exactly two weighings suffice to determine the mass of each ball.

Let $a,b,c,d$ be real numbers satisfying $(a+b)(c+d)=2$, $(a+c)(b+d)=3$ and $(a+d)(b+c)=4$. Find the minimum value of $a^2+b^2+c^2+d^2$.