VTMO 2026 - Grade 8 - Final - SET 6 - Self-study

A $10\times10$ board consists of $100$ unit squares. Each square contains a positive integer not exceeding $10$, such that the numbers in any two squares sharing an edge or a vertex are coprime. Prove that some number occurs at least $17$ times.

a) Find the smallest positive integer $n$ such that $(n+1)(n+2)(n+3)(n+4)$ is divisible by $1000$.

b) Prove that there are no positive integers $a,b$ for which $a^4+324b^4$ is prime.

How many natural numbers $n\lt 2024$ have the property that $\frac1n+\frac15$, written in lowest terms, has denominator smaller than $n$?

A rectangular floor of dimensions $15\times7$ is tiled with the two types shown: L-shaped tiles made of three unit squares and $2\times2$ square tiles. Prove that at most $9$ square tiles are used.

In parallelogram $ABCD$, take points $E,F$ on side $AB$, with $E$ between $A$ and $F$. Let $M=DE\cap CF$, and use $S$ to denote area.

a) Prove $S_{ADM}+S_{BCM}=S_{DFC}$.

b) Prove $S_{MAB}=S_{MDF}$.