VTMO 2017 — Grade 8 — final

$a$)Find the smallest positive $n$ such that $\left(n + 1\right) \left(n + 2\right) \left(n + 3\right) \left(n + 4\right)$ is divisible by $1000$. $b$)Prove a $^{4} + 324 b^{4}$ is not prime for positive integers $a , b$.

Real $a , b , c , d$ satisfy $\left(a + b\right) \left(c + d\right) = 2 , \left(a + c\right) \left(b + d\right) = 3 , \left(a + d\right) \left(b + c\right) = 4$. Find the minimum sum of their squares.

A train travels $A$→$B$. Reducing speed $25 \%$ when a fault is found would delay arrival $1 h = 20$ min. Keeping the old speed for another $40 \,\mathrm{km}$ before reducing $25 \%$ delays it $1 h$. Find the original speed.

In parallelogram $ABCD , E , F$ lie on $AB$ with $E$ between $A , F$, and $M = DE \cap CF$. Prove $a ) S_{ADM} + S_{BCM} = S_{DFC}$; $b ) S_{MAB} = S_{MDF}$.

In parallelogram $ABCD$ with $AB \gt AD$, take $E$ on $AB$ with $AE = AD , F = DE \cap AC , K$ on $EF , M = AK \cap BC , N = CK \cap AB$. Prove $AN = CM$.

Fill $a = 10 \times 10$ grid with integers $1 , \ldots , 10$ so that any two cells sharing an edge or vertex have coprime entries. Prove some number appears at least $17$ times.