VTMO 2019 — Grade 6 — semifinal

Question 1

Evaluate $\left(1 + 1\right) \left(1 + \frac{1}{2}\right) \left(1 + \frac{1}{3}\right) \ldots \left(1 + \frac{1}{100}\right)$.

$\frac{99}{100}$
$\frac{101}{100}$
$\frac{1}{100}$
$100$
$101$

Question 2

How many positive divisors does $2019$ have?

Question 3

Minh’s first two math tests average $88$, and last three average $93$. Find the five-test average.

$88$
$90$
$93$
$91$
$92$

Question 4

A map scale gives $1 \,\mathrm{cm}$ for $8 \,\mathrm{km}$. What real area corresponds to $15 \,\mathrm{cm}^{2}$?

$120$
$225$
$180$
$960$
Other

Question 5

Three circles with radii $7 , 8 , 9 \,\mathrm{cm}$ are externally tangent as pictured. Find the perimeter of the center triangle.

Question illustration

$30$
$24$
$32$
$36$
$48$

Question 6

$97$ and $79$ are prime. How many two-digit primes remain prime when their digits are reversed?

Question 7

$A = 3.75 \times 2.5 m$ room is tiled with congruent squares without cuts. Find the minimum tile count.

$4$
$6$
$8$
$10$
$12$

Question 8

Start with $6$; an even term $t$ is followed by $\frac{t}{2}$, an odd $s$ by $3 s + 1$. The first terms are $6 , 3 , 10 , 5$. Find term $100$.

Question 9

Cars simultaneously leave $HCMC$ and Vung Tau toward each other, and meet one third of the route from Vung Tau. Find the speed ratio $HCMC$-car: Vung-Tau-car.

$1 : 1$
$3 : 2$
$2 : 1$
$5 : 2$
$3 : 1$

Question 10

Twenty students have distinct correct-answer counts, and each question is answered correctly by at most three students. What minimum question count is possible?

$63$
$64$
$66$
$67$
$70$

Question 11

In the illustrated sequence, which figure contains $66$ triangles?

Question illustration

$15$
$16$
$17$
$18$
$19$

Question 12

The pictured shape is formed from four semicircles of radius $3 \,\mathrm{cm}$. Find its area, taking $\pi = \frac{22}{7}$.

Question illustration

$9 \,\mathrm{cm}^{2}$
$24 \,\mathrm{cm}^{2}$
$36 \,\mathrm{cm}^{2}$
$27 \,\mathrm{cm}^{2}$
$18 \,\mathrm{cm}^{2}$

Question 13

$1 ABCDE \times 3 = ABCDE 1$, where letters are digits. Find $A + B + C + D + E$.

$29$
$26$
$22$
$28$
$30$

Question 14

$A = 2 \,\mathrm{cm}$ wooden cube weighs $100 g$. Find the mass of $a = 6 \,\mathrm{cm}$ cube of the same material.

$800 g$
$1000 g$
$2700 g$
$300 g$
$900 g$

Question 15

Repeatedly subdivide the upper-left subsquare into four congruent squares as illustrated. Find the shaded fraction of the original.

Question illustration

$1 : 4$
$17 : 64$
$85 : 256$
$7 : 16$
$65 : 128$

Question 16

Three dice each have faces $1 \text{--} 6$ as shown. Find the total dots on unseen faces.

Question illustration

Question 17

Use digits $0 , 1 , 2 , 3 , 4 , 5$ once each in six-digit numbers divisible by $1 , 2 , 3 , 4 , 5 , 6$. How many are there?

A worker paints four walls and the ceiling of a room in $20$ hours. Doubling its length and width while keeping height unchanged requires $50$ hours at the same rate. How long does the original ceiling take?

Question 19

Extend each square side $2 \,\mathrm{cm}$ in the pictured direction, and connect the new endpoints into a larger square. Its area exceeds the original by $36 \,\mathrm{cm}^{2}$. Find the original area.

Question illustration

Question 20

Two brothers’ schools lie on opposite sides of home. The cyclist reaches his school $2.2$ km away, then returns after taking the wrong bag. The younger has covered two fifths of his walk by then. The cyclist catches him at his school. Both speeds are constant. Find the younger’s school distance in meters.