VTMO 2013 — Grade 6 — semifinal

Question 1

How many integer divisors does $20$ have?

$4$
$5$
$6$
$10$
$12$

Question 2

Hoang multiplies an integer by $8$. Which cannot be the result?

$- 24$
$- 48$
$84$
$48$
$32$

Question 3

Distinct shapes represent distinct digits. In the illustrated addition, square+square+circle-circle=triangle-triangle-triangle. Find square.

Question illustration

Question 4

Reduce every side of a square garden by $10 \%$. What percentage decrease in area results?

$20 \%$
$9 \%$
$81 \%$
$19 \%$
$10 \%$

Question 5

A father is $36$ and child $6$. After how many years will the father be three times the child’s age?

$30$
$15$
$6$
$9$
$12$

Question 6

How many three-digit numbers are divisible by $8$?

$26$
$111$
$112$
$124$
$125$

Question 7

Nam writes consecutive integers $1$ through $120$. How many digits are written?

$242$
$247$
$249$
$252$
$360$

Question 8

Find the shaded-to-square area ratio in the $6 \,\mathrm{cm}$ square, with adjacent sides split into $2 \,\mathrm{cm}$ and $4 \,\mathrm{cm}$ as shown.

Question illustration

$\frac{1}{2}$
$\frac{1}{3}$
$\frac{1}{4}$
$\frac{1}{6}$
$\frac{2}{9}$

Question 9

Women average $18$ years, men $24$, and men are twice as numerous. Find the group average.

$20$
$18$
$19$
$22$
$21$

Question 10

For each integer $30$ through $50$, subtract its units digit from its tens digit. Find the total.

$15$
$- 15$
$5$
$- 5$
$0$

Question 11

A boat takes $2$ hours downstream $A$→$B$ and $4$ upstream. How long does a drifting raft take $A$→$B$?

Question 12

Use each digit $1 \text{--} 8$ once to form two four-digit numbers. Find their minimum sum.

$2468$
$3333$
$3825$
$6912$
$5590$

Question 13

Ten distinct positive integers sum $100$. What maximum largest term is possible?

$45$
$55$
$65$
$75$
$91$

Question 14

Pi cuts one sheet into $10$ pieces, then cuts one piece into $10$ smaller pieces, and repeats the latter operation twice more. How many pieces result?

$20$
$27$
$30$
$37$
$40$

Question 15

In which expression can every $6$ be replaced by another integer while preserving the result?

$\left(6 + 6\right) \div 6 + 6$
$6 \times \left(6 + 6\right) \div 6$
$6 + 6 - 6 + 6$
$\left(6 + 6 - 6\right) \times 6$
$\left(6 + 6 - 6\right) \div 6$

Question 16

Five boxes contain $A , B , O , R , V$ as shown. Remove letters so each retains one, and all five retained letters differ. Which letter remains in box $5$?

Question illustration

Question 17

Rectangles $8 \times 10 \,\mathrm{cm}$ and $9 \times 12 \,\mathrm{cm}$ overlap. The solid shaded region is $37 \,\mathrm{cm}^{2}$. Find the striped area.

Question illustration

$60$
$62$
$64$
$65$
$71$

Question 18

A box has $10$ blue, $13$ red, $12$ yellow balls. What minimum blind draw guarantees $10$ of one color?

$28$
$27$
$13$
$12$
$10$

Question 19

There are $60$ seats around a circular table. Seat students so each has exactly one occupied neighboring seat. What is the maximum number?

$10$
$20$
$30$
$40$
$60$

Question 20

Join every pair of illustrated points. How many resulting segments cross line $d$?

Question illustration

$6$
$12$
$3$
$7$
$1$