VTMO 2026 - Grade 7 - Final - SET 9 - Self-study

Triangle $ABC$ has $\angle BAC=120^\circ$. Choose $D$ on the angle bisector of $\angle BAC$ such that $AD=AB+AC$. Prove that $BCD$ is equilateral.

Arrange the integers from 1 to 2015 in any order. Subtract 1 from the first, 2 from the second, and so on, subtracting 2015 from the last. Let $P$ be the product of the resulting 2015 numbers. Prove that $P^2+2P$ is divisible by 8.

Suppose
$$\frac{a_1}{a_2^2}=\frac{a_2}{a_3^2}=\frac{a_3}{a_4^2}=\cdots=\frac{a_{2023}}{a_{2024}^2}=\frac{a_{2024}}{a_1^2}.$$
Evaluate
$$V=\frac{(a_1+a_2+\cdots+a_{2024})^2}{a_1^2+a_2^2+\cdots+a_{2024}^2}.$$
Assume all the displayed ratios are defined.

Numbers $a,b,c$ satisfy $a+b+c=3$, $ab+bc+ca=1$, and $abc=-1$. Prove that exactly one of them equals 1.

Replace the letters in MATEMATIKA with digits and the signs $+,-$ to obtain an expression equal to 2014. Different letters must receive different digits or signs, and repeated letters must receive the same symbol. Give a valid assignment.