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VTMO 2016 — Grade 8 — semifinal

Question 1

For positive integer $n$, which expression is always divisible by $3$?
  1. $\left(n + 1\right) \left(n + 4\right)$
  2. $n \left(n + 2\right) \left(n + 6\right)$
  3. $n \left(n + 2\right) \left(n + 4\right)$
  4. $\left(n + 3\right) \left(n - 3\right)$
  5. $\left(n + 2\right) \left(n + 3\right) \left(n + 5\right)$

Question 2

Given a $^{2} - b^{2} = 6$ and $\left(a - 2\right)^{2} - \left(b - 2\right)^{2} = 18$, find $a + b$.
  1. $- 2$
  2. $\frac{2}{3}$
  3. $\frac{3}{2}$
  4. $2$
  5. $\frac{1}{3}$

Question 3

A bus has single and double seats. With $13$ passengers, $9$ seats are empty; with $10$ passengers, $6$ seats are empty. How many seats are there, excluding the driver?
  1. $15$
  2. $16$
  3. $14$
  4. $18$
  5. $22$

Question 4

Given $x \ne 2 t , y \ne 2 t$ and $x + y - 4 t = 0$, evaluate $Q = \frac{x}{x - 2 t} + 2 \frac{t}{y - 2 t}$.
  1. $1$
  2. $- 1$
  3. $0$
  4. $2$
  5. $- 2$

Question 5

How many odd numbers must be selected from $123 , 124 , \ldots , 2016$ to guarantee two summing to $2016$?
  1. $503$
  2. $504$
  3. $505$
  4. $506$
  5. $507$

Question 6

Find the units digit of $1 \cdot 3 \cdot 5 \cdot 7 + 3 \cdot 5 \cdot 7 \cdot 9 + \ldots + 2011 \cdot 2013 \cdot 2015 \cdot 2017$.
  1. $0$
  2. $2$
  3. $4$
  4. $6$
  5. $8$

Question 7

The four sides and one diagonal of a quadrilateral have lengths $1 , 2 , 2.8 , 5 , 7.5$. Which length is the diagonal?
  1. $1$
  2. $2$
  3. $2.8$
  4. $5$
  5. $7.5$

Question 8

Dividing $A \left(x\right) = x^{10} - x^{2} - 1$ by $B \left(x\right) = x^{3} - 2$ gives remainder $ax^{2} + bx + c$. Find $a + b + c$.
  1. $3$
  2. $4$
  3. $5$
  4. $6$
  5. $7$

Question 9

Hai scores $50 \% : 13$ of the first $20$ questions and $25 \%$ of the rest are correct. How many questions are there?
  1. $36$
  2. $38$
  3. $24$
  4. $32$
  5. $40$

Question 10

How many triangles have perimeter $27 \,\mathrm{cm}$ and distinct integer sides $x \lt y \lt z$?
  1. $9$
  2. $10$
  3. $11$
  4. $12$
  5. $15$

Question 11

Twenty girls and four boys line up. The boys count $17 , 14 , 5 , 2$ girls before them. Each girl counts the boys before her. What is the girls’ total?

Question 12

Let $A = \frac{1}{\frac{1}{2011} + \frac{1}{2012} + \ldots + \frac{1}{2016}}$. Find the greatest integer $B \le A$.

Question 13

In the figure $ABC$ is right at $A , ADEF$ is a rectangle with $E$ on $BC , D$ on $AB , F$ on $AC$. $BD = 10 \,\mathrm{cm} , FC = 15 \,\mathrm{cm}$. Find the rectangle’s area.

Question illustration

Question 14

Notebooks cost $10000 VND$. A offers one free per five purchased (nine purchased earns one). $B$ discounts $15 \%$ on purchases of at least four. What minimum cost buys $14$ notebooks, allowing both stores?

Question 15

$A = 15 \times 15$ table has entry $ij$ in row $i$, column $j$. Find the absolute difference of its diagonal sums.

Question illustration