VTMO 2026 - Grade 7 - Final - SET 6 - Self-study

Triangle $ABC$ is isosceles at $A$. Choose $D$ on $BC$ with $BD=BC/3$. Extend ray $AD$ beyond $D$ to $E$ so that $DE=AD$. Extend ray $BE$ beyond $E$ to $F$ so that $BE=EF$.

a) Prove that $FD\perp BC$.

b) If $ABC$ is equilateral with $AB=1$, find $FD$.

A digit sequence starts with $2,5,0,2$. From the fifth digit onward, each digit is the units digit of the sum of the preceding four digits. The sequence begins $25029674\ldots$. Can four consecutive digits ever be $2025$?

Triangle $ABC$ has $\angle ABC=40^\circ$ and $\angle ACB=20^\circ$. The segment $AD$ bisects $\angle BAC$, with $D\in BC$. Prove that $AD=BC-AC$.

a) Two soccer teams of ten players each play a match. One team consists entirely of truth-tellers, the other entirely of liars. Afterward, each player is asked how many goals he scored. Some answer “one”, Chau answers “two”, some answer “three”, and the rest answer “five”. Is Chau truthful or a liar, given that the truth-telling team wins 20–17?

b) Let $A=\overline{abcdef}$ be a six-digit number with $a\lt b\lt c\lt d\lt e\lt f$. Find the sum of the digits of $9A$.

Triangle $ABC$ has $\angle ABC=30^\circ$. Construct equilateral triangle $ACD$ externally to $ABC$. Prove that $AB^2+BC^2=BD^2$.