VTMO 2016 — Grade 6 — final

$A = 3 \times 4$ grid has $- 8 , 1 , - 1 , - 2$ prefilled as shown. Remove one of $0 , 2 , 3 , 4 , 5 , 7 , 8 , 10 , 11$, and fill the eight remaining cells so every row and column has the same average. Explain.

$a$)Find a six-digit number $N$ using only $8 , 9$ divisible by $2^{6}$. $b$)Use each digit $1 \text{--} 9$ once in three three-digit integers $a : b : c = 1 : 3 : 5$. Find $a , b , c$.

$25 \%$ of contestants receive awards. Two sevenths of boys receive awards, and $420$ boys do not. There are $648$ girls. How many girls receive awards?

Mother walks a lake lap in $12$ minutes. Binh skating in the same direction overtakes her every $18$ minutes. If he reverses direction, how often do they meet?

Find unordered positive-integer pairs with $\operatorname{lcm} 108$ and $\operatorname{gcd} 6$.

$A , B , C , D$ wear $6 , 7$, or $8$. $A = 7$ always lies, $a = 6$ or $8$ tells truth. Asked their total, A says $28 , B = 27 , C = 26 , D = 25$. Find each shirt number.