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VTMO 2015 — Grade 6 — semifinal

Question 1

Evaluate $1 + 2 - 3 + 4 + 5 - 6 + 7 + 8 - 9 + 10 + 11 - 12 + 13 + 14 - 15$.
  1. $18$
  2. $30$
  3. $36$
  4. $48$
  5. $50$

Question 2

A six-digit number has digit sum $5$. Find its digit product.
  1. $0$
  2. $1$
  3. $5$
  4. $6$
  5. $720$

Question 3

Which pair has greatest common divisor $20$?
  1. $\left(30 , 40\right)$
  2. $\left(40 , 50\right)$
  3. $\left(40 , 120\right)$
  4. $\left(20 , 25\right)$
  5. $\left(20 , 80\right)$

Question 4

Find the last digit of $2015^{0} + 2015^{1} + 2015^{2} + 2015^{3} + 2015^{4}$.
  1. $1$
  2. $5$
  3. $6$
  4. $7$
  5. $9$

Question 5

Two primes sum $85$. Find their positive difference.
  1. $79$
  2. $80$
  3. $81$
  4. $82$
  5. $83$

Question 6

A perfect number equals the sum of its proper positive divisors, for example $6 = 1 + 2 + 3$. Which is perfect?
  1. $8$
  2. $18$
  3. $28$
  4. $38$
  5. $48$

Question 7

Define $a \triangle b = ab + a + b$, so $8 \triangle 5 = 53$. Given $x \triangle 3 = 39$, find $x$.
  1. $9$
  2. $10.5$
  3. $12$
  4. $13$
  5. $18$

Question 8

A bicycle’s rear-wheel diameter is $60 \,\mathrm{cm}$ and front-wheel diameter $50 \,\mathrm{cm}$. The rear wheel turns $700$ times. How many front-wheel turns occur?
  1. $500$
  2. $600$
  3. $700$
  4. $720$
  5. $840$

Question 9

Square $PQRS$ has $M , N$ the midpoints of $PS , RS$. Area $\left(SMN\right) = 18 \,\mathrm{cm}^{2}$. Find area $\left(QMN\right)$.

Question illustration
  1. $36 \,\mathrm{cm}^{2}$
  2. $48 \,\mathrm{cm}^{2}$
  3. $54 \,\mathrm{cm}^{2}$
  4. $72 \,\mathrm{cm}^{2}$
  5. $90 \,\mathrm{cm}^{2}$

Question 10

Thanh has $1024$ coins, and removes half the remaining coins each time. How many removals leave one?
  1. $10$
  2. $11$
  3. $12$
  4. $512$
  5. $1023$

Question 11

$N$ is the product of odd integers $1 \text{--} 99$ excluding those ending in $5$. Find its last digit.
  1. $1$
  2. $3$
  3. $5$
  4. $7$
  5. $9$

Question 12

Outer numbers give row or column sums in the illustrated table. Find $a + c + c$.

Question illustration
  1. $19$
  2. $21$
  3. $22$
  4. $24$
  5. Other

Question 13

Which number is neither a square nor a cube of a natural number?
  1. $2^{9}$
  2. $3^{10}$
  3. $4^{11}$
  4. $5^{12}$
  5. $6^{13}$

Question 14

A three-question test gives one point per correct answer.$5 \%$ get zero, $30 \%$ one, $45 \%$ two, $20 \%$ three correct. Find the mean.
  1. $1.8$
  2. $1.9$
  3. $2$
  4. $2.1$
  5. $2.35$

Question 15

$A = 36 \,\mathrm{cm}^{2}$ square is partitioned into four congruent rectangles and an inner square as shown. The inner square perimeter equals one rectangle perimeter. Find each rectangle area; diagram not to scale.

Question illustration
  1. $6 \,\mathrm{cm}^{2}$
  2. $6.75 \,\mathrm{cm}^{2}$
  3. $7 \,\mathrm{cm}^{2}$
  4. $7.5 \,\mathrm{cm}^{2}$
  5. $7.75 \,\mathrm{cm}^{2}$

Question 16

Sixteen circles form $a = 4 \times 4$ square with $12$ black boundary circles. If $900$ circles form a square, how many boundary circles are black?

Question illustration
  1. $104$
  2. $108$
  3. $112$
  4. $116$
  5. $120$

Question 17

Parallelogram $ABCD$ has area $40 \,\mathrm{cm}^{2}$. $E , F$ are midpoints of $AD , DC$. Find area $\left(AEFC\right)$.

Question illustration
  1. $10 \,\mathrm{cm}^{2}$
  2. $12 \,\mathrm{cm}^{2}$
  3. $15 \,\mathrm{cm}^{2}$
  4. $16 \,\mathrm{cm}^{2}$
  5. $18 \,\mathrm{cm}^{2}$

Question 18

Which number cannot be the number of black squares in a figure of the pictured sequence?

Question illustration
  1. $104$
  2. $404$
  3. $904$
  4. $1024$
  5. $1229$

Question 19

Chau lies on Friday, Saturday, Sunday and tells truth otherwise. Minh tells truth on Tuesday, Wednesday, Thursday and lies otherwise. On which day can both say“I will lie tomorrow”?
  1. Monday
  2. Tuesday
  3. Thursday
  4. Friday
  5. Sunday

Question 20

Integers $4 , 4 , x , y , 13$ satisfy $4 \le x \le y \le 13$. How many pairs make their five-number average an integer?
  1. $7$
  2. $8$
  3. $9$
  4. $10$
  5. $11$