VTMO 2026 - Grade 8 - Final - SET 10 - Self-study

Find all integer solutions of $2^a+8b^2-3^c=283$.

In quadrilateral $ABCD$, points $M$ on $AB$ and $N$ on $CD$ satisfy $BM/BA=DN/DC=1/3$. Construct parallelogram $ABCE$. Let $I,J,O$ be the midpoints of $AC,BD,MN$, respectively. (a) Prove $DE=2IJ$. (b) Prove that $O,I,J$ are collinear.

A ship travels from $A$ to $B$ at a planned constant speed. After part of the journey, an engine problem is discovered and the driver proposes reducing the speed by $25\%$ of its original value. The captain calculates that doing so immediately would make the ship arrive $1$ hour $20$ minutes late, beyond the allowed delay. Instead, he orders it to travel another $40$ km at its original speed before making the proposed reduction; it then arrives $1$ hour late. Find the ship\'s planned speed.

Let $ABCD$ be a square. Equilateral triangles $AMD$ and $DCN$ are drawn with $M$ outside the square and $N$ inside it. Prove that $B,M,N$ are collinear.

Find all integers $a,b$ such that $a^4+4b^4$ is prime.