VTMO 2019 — Grade 6 — final

Place $1 \text{--} 2019$ once around the pictured circle with adjacent differences at most $2$. The positions of $2015 , x , y$ are shown. Find $x + y$.

$a$)Begin with one virus on day $1$, and multiply the previous count by $k$ on day $k$ through day $10$. Can the day $10$ count be split equally into $24$ tubes? $b$)For $13$ apples, $a$ scale measures the total of exactly two at a time. Find the total mass in eight weighings.

A six-digit integer abcdef is divisible by $7$. Prove that efabcd is divisible by $7$.

Tom trails Jerry by $100 m$ and expects to catch him in $20 s$. At $a = 15 m$ gap, Jerry triples his speed. Tom keeps his speed, and catches him two seconds late. Find Tom’s speed.

Rectangle $ABCD$ has $E$ on $AB$ and collinear $C , M , N , E$ with $CM = MN = NE$. $DN$ meets $AB$ at $F$. Shaded $ENF$ and $DNM$ have combined area $12 \,\mathrm{cm}^{2}$. Find area $\left(ABCD\right)$.

A basket has three candy types, and every $100$-candy selection contains all three. What is the maximum total?