VTMO 2015 — Grade 6 — final

Balls $0 \text{--} 9$ are split into An’s three, Binh’s three, and Chau’s four. Their digit products are $0 , 90 , 72$ respectively. Find An’s sum.

$S \left[A\right]$ is the sum of all elements over all subsets of $A$, including the empty set. For $A = \{ 1 , 2 \} , S \left[A\right] = 6$. $a$)Find $S \left[\{ 1 , 2 , 3 \}\right]$. $b$)For four distinct natural numbers $C = \{ a , b , c , d \}$, prove $8$ divides $S \left[C\right]$.

Cars $1 , 2$ leave A toward $B$ at $75$ and $50 \frac{\,\mathrm{km}}{h}$. After $6$ hours, car $1$ meets an opposite-bound car $3$, which meets car $2$ after another $1.5$ hours. Find car $3$’s speed.

A tank fills with A for $7$ hours then $B$ for $14$, or A for $10$ then $B$ for $2$. If A runs $5$ hours, how long must $B$ run?

A palindrome reads identically in both directions, for example $7 , 11 , 121 , 1221$. For a six-digit palindrome abccba: $a$)prove divisibility by $11$; $b$)count those divisible by $91$.