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VTMO 2016 — Grade 6 — semifinal

Question 1

$A$ is the set of primes below $11$. How many subsets does A have?
  1. $32$
  2. $16$
  3. $64$
  4. $30$
  5. $6$

Question 2

Find the sum of the first $51$ positive odd integers minus the sum of the first $50$ positive even integers.
  1. $50$
  2. $51$
  3. $100$
  4. $102$
  5. $- 51$

Question 3

$A = 2^{6} \times 3^{2} \times 5^{4}$. Find its digit sum.
  1. $6$
  2. $9$
  3. $12$
  4. $18$
  5. $27$

Question 4

What is the hundreds digit of $1\times2\times3\times4\times5\times6\times7\times8\times9\times10$?
  1. $0$
  2. $2$
  3. $4$
  4. $6$
  5. $8$

Question 5

Chau has three quarters of a book’s price, Phuong one half. Reducing each book by $30000$ lets their combined money buy exactly two books. Find the original price.
  1. $40000$
  2. $160000$
  3. $120000$
  4. $100000$
  5. $80000$

Question 6

Two positive integers sum $1323$ and have $\operatorname{gcd} 147$. How many unordered pairs are possible?
  1. $3$
  2. $4$
  3. $5$
  4. $9$
  5. $10$

Question 7

The sequence is $1 , 2 , 3 , 0 , 5 , - 2 , 7 , - 4 , \ldots$, with $a_{n} = a$_$\left(n - 3\right) + a$_$\left(n - 2\right) - a$_$\left(n - 1\right)$. Find the sum of its first $2016$ terms.
  1. $3024$
  2. $2016$
  3. $1008$
  4. $3427$
  5. $504$

Question 8

Initially $70 \%$ of attendees are men.$108$ men leave, and women then number three times the remaining men. Find the initial total.
  1. $180$
  2. $126$
  3. $240$
  4. $236$
  5. $160$

Question 9

Two diagonals of $a = 3 \times 3$ grid pass through five small cells as shown. How many cells do they pass through in $a = 9 \times 9$ grid?

Question illustration
  1. $9$
  2. $11$
  3. $16$
  4. $17$
  5. $18$

Question 10

On $a = 100 m$ square, An and Binh start at opposite vertices, both counterclockwise, at $50$ and $40 m /$ min. An rests one minute at each corner, Binh two. When does An first catch Binh?
  1. $18$
  2. $17$
  3. $19$
  4. $16$
  5. $20$

Question 11

An, Binh, Chau race $100 m$ at constant speeds. When An finishes, Binh is $10 m$ short, Chau $20$. When Binh finishes, how far short is Chau?

Question 12

Twenty girls form a line, and four boys enter with $17 , 14 , 5 , 2$ girls before them. Each girl counts boys before her. Find the sum of the girls’ counts.

Question 13

Tank $A = 40 \times 30 \times 55 \,\mathrm{cm}$ is full, and tank $B = 70 \times 30 \times 45 \,\mathrm{cm}$ is empty. Transfer water until levels are equal. Find the common height.

Question illustration

Question 14

A cube net shows $1 , 5 , 6 , x , y , z$ as pictured. Both $5$ and $x$ equal the mean of their four adjacent faces. Find $x$.

Question illustration

Question 15

Among $65$ pupils wearing white or blue, there are $25$ boys in white, $23$ pupils in blue, and $28$ girls. How many girls wear blue?