VTMO 2013 — Grade 8 — semifinal

Question 1

Find the perimeter of the shown staircase, with three horizontal steps a and vertical steps $b , 2 b , b$.

Question illustration

$3 a + 4 b$
$3 a + 8 b$
$6 a + 4 b$
$6 a + 6 b$
$6 a + 8 b$

Question 2

Expand $\left(3 x - \frac{1}{2}\right)^{3}$.

$27 x^{3} - \frac{1}{8}$
$27 x^{3} - \left(\frac{27}{2}\right) x^{2} - \left(\frac{9}{4}\right) x + \frac{1}{8}$
$27 x^{3} - \left(\frac{27}{2}\right) x^{2} + \left(\frac{9}{4}\right) x - \frac{1}{8}$
$27 x^{3} - \left(\frac{9}{2}\right) x^{2} + \left(\frac{9}{2}\right) x - \frac{1}{8}$
$9 x^{3} - \left(\frac{9}{x}\right) x^{2} + \left(\frac{9}{4}\right) x - \frac{1}{8}$

Question 3

Solve $\frac{x + 1}{x - 2} + \frac{3 - 2 x}{2 - x} = 0$.

$\frac{2}{3}$
$1$
$4$
$2$
$\frac{2}{3}$ and $2$

Question 4

Find the remainder when $P \left(x\right) = x^{3} + x^{2} + x + 1$ is divided by $x - 3$.

$- 17$
$- 27$
$40$
$17$
$27$

Question 5

How many three-digit numbers leave remainder $1$ upon division by $9$?

$102$
$98$
$101$
$99$
$100$

Question 6

For $P \left(x\right) = \left(x - 2\right) \left(x - 1\right) \left(x + 1\right) \left(x + 2\right)$, which listed $x$ gives $P \left(x\right) \gt 0$?

$x = 1.5$
$x = - 1$
$x = - 0.5$
$x = - 1.5$
$x = 2$

Question 7

$ABCD$ has $AB = 11 \,\mathrm{cm} , BC = 7 \,\mathrm{cm} , CD = 9 \,\mathrm{cm}$, and right angles at $A , C$. Find its area.

Question illustration

$30$
$44$
$48$
$60$
$96$

Question 8

A grandmother may share a cake among $3 , 5$, or $6$ grandchildren, with equal shares in each case. What is the least number of equal pieces needed?

$12$
$15$
$18$
$30$
$45$

Question 9

The three parallel cross-sections have lengths $8 , x , 16$, and the corresponding consecutive side lengths are $y , 4 , 4$. Find $x , y$.

Question illustration

$x = 12 , y = 8$
$x = 10 , y = 6$
$x = 11 , y = 8$
$x = 12 , y = 4$
$x = 14 , y = 16$

Question 10

A boat takes $3 h$ downstream from A to $B$ and $4 h$ upstream. How many hours does drifting vegetation take from A to $B$?

$3$
$4$
$7$
$12$
$24$

Question 11

Find the area of the shown room, whose adjacent walls are perpendicular and dimensions are marked $a , b$.

Question illustration

$2 ab + a \left(b - a\right)$
$3 a \left(a + b\right) - a^{2}$
$3 a \left(b - a\right) + a^{2}$
$3 ab$
$2 \left(a^{2} + ab\right)$

Question 12

$AM$ is a median of $ABC , I$ is midpoint $AM$, and extended $BI$ meets $AC$ at $N$. Find $AN : AC$.

$1 : 4$
$1 : 3$
$1 : 2$
$1$
$2 : 3$

Question 13

The $1500$-digit number $20142014 \ldots 2014$ is shortened by deleting digits. What maximum deletions leave digit sum $2014$?

$860$
$868$
$996$
$611$
$375$

Question 14

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 15

All balances shown are in equilibrium; ignore the bars and strings. Hanging objects total $112 g$. What does one star weigh?

Question illustration

$7$
$12$
$15$
$16$
$14$

Question 16

$ABCD$ is a square, $M$ midpoint $AD$, and $MN \perp AC$ at $N$. Find area $MNC$ divided by the square area.

Question illustration

$1 : 5$
$1 : 6$
$7 : 36$
$3 : 16$
$7 : 40$

Question 17

Given $9$ divides $2 a + 7 b$, which $n$ ensures $9$ divides $5 a$+nb?

Question 18

Write $1 , 2 , 3 , \ldots , 2014$ consecutively. How many digits are written?

$6760$
$6234$
$6949$
$6849$
$6939$

Question 19

For a three-digit number, deleting either the first or last digit leaves a perfect square. Find the sum of all such numbers.

$1013$
$1177$
$1465$
$1993$
$2016$

Question 20

Two parallel lines contain six and four marked points. How many triangles have vertices among them?

Question illustration

$6$
$15$
$36$
$60$
$96$