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VTMO 2014 — Grade 8 — semifinal

Question 1

Evaluate $5 \left(2 x - 1\right) - 2 \left(5 x + 3\right)$.
  1. $11$
  2. $- 11$
  3. $10$
  4. $- 10$
  5. $12$

Question 2

Simplify $\left(a - b\right) \left(a^{2} + b^{2}\right) \left(a + b\right)$.
  1. a $^{4} + b^{4}$
  2. a $^{3} + b^{3}$
  3. a $^{4} - b^{4}$
  4. a $^{3} - b^{3}$
  5. a $^{4} - b^{3}$

Question 3

Given $x - y = 5$, evaluate $x^{2} \left(x + 1\right) - y^{2} \left(y - 1\right) + xy - 3 xy \left(x - y + 1\right)$.
  1. $135$
  2. $153$
  3. $200$
  4. $150$
  5. $180$

Question 4

Simplify $\left(- a^{2} b^{4} \frac{c}{2}\right)^{3} \div \left(ab^{2} \frac{c}{4}\right)^{2}$.
  1. $2$ a $^{4} b^{2} c^{8}$
  2. $- 2$ a $^{4} b^{6} c^{2}$
  3. $- 2$ a $^{4} b^{2} c^{8}$
  4. $- 2$ a $^{4} b^{8} c$
  5. $- 2$ a $^{4} b^{8} c^{2}$

Question 5

Choose a so $4 y^{2} - 24 y - a$ is a polynomial square.
  1. $36$
  2. $- 36$
  3. $9$
  4. $16$
  5. $25$

Question 6

$x + y = 5 , x^{2} + y^{2} = 111$. Find $x^{3} + y^{3}$.
  1. $115$
  2. $227$
  3. $300$
  4. $555$
  5. $770$

Question 7

Define a＊$b = \left(a + 1\right) \left(b + 1\right)$. Natural numbers $a \gt b$ satisfy a＊$b = 13$. Find a.
  1. $0$
  2. $1$
  3. $11$
  4. $12$
  5. $13$

Question 8

Natural $a , b$ leave remainders $1 , 2$ modulo $8$. Find the remainder of a $^{2} - b^{2}$ modulo $8$.
  1. $1$
  2. $3$
  3. $5$
  4. $6$
  5. $7$

Question 9

How many occurrences of digit $5$ are in page numbers $1$ through $500$?
  1. $100$
  2. $101$
  3. $200$
  4. $201$
  5. $50$

Question 10

Square $1$ has area $64 \,\mathrm{cm}^{2}$. Each next square has diagonal equal to the preceding side, as shown. Find square $9$ area.

Question illustration
  1. $\frac{1}{8}$
  2. $\frac{1}{4}$
  3. $\frac{1}{3}$
  4. $\frac{1}{2}$
  5. $\frac{1}{9}$

Question 11

Which product is a perfect square, where $n ! = 1 \times 2 \times \ldots \times n$?
  1. $98 ! 99$!
  2. $98 ! 100$!
  3. $99 ! 100$!
  4. $99 ! 101$!
  5. $100 ! 101$!

Question 12

$ABC$ is right at $A$, with $BC = AC + \frac{AB}{2}$. Find $\frac{AB}{AC}$.
  1. $7 : 12$
  2. $12 : 5$
  3. $5 : 8$
  4. $3 : 4$
  5. $4 : 3$

Question 13

Which triangle has greatest area among sides $\left(6 , 8 , 8\right) , \left(6 , 9 , 8\right) , \left(6 , 10 , 8\right) , \left(6 , 11 , 8\right) , \left(6 , 12 , 8\right)$?
  1. $\left(6 , 8 , 8\right)$
  2. $\left(6 , 9 , 8\right)$
  3. $\left(6 , 10 , 8\right)$
  4. $\left(6 , 11 , 8\right)$
  5. $\left(6 , 12 , 8\right)$

Question 14

Which listed triple can be triangle altitudes? $A \left(1 , 2 , 3\right)$; $B \left(1 , 0.5 , 1.5\right)$; $C \left(2 , 4 , 4\right)$; $D \left(3 , 6 , 8\right)$; $E \left(3 , 6 , 9\right)$.

Question 15

Find the last digit of $1 ! + 2 ! + \ldots + 2014$!.
  1. $1$
  2. $2$
  3. $3$
  4. $4$
  5. $5$

Question 16

$f \left(x\right) = 1 + 2 x + \ldots + 2015 x^{2014}$. Find $f \left(- 1\right)$.
  1. $2014$
  2. $1009$
  3. $1008$
  4. $1007$
  5. $- 1007$

Question 17

$4^{a} = 5 , 5^{b} = 6 , 6^{c} = 7 , 7^{d} = 8$. Find $abcd$.
  1. $1$
  2. $\frac{3}{2}$
  3. $2$
  4. $\frac{5}{2}$
  5. $3$

Question 18

A team plays $90$ matches and wins $40$ of its first $50$. How many further wins give $70 \%$ overall?
  1. $20$
  2. $23$
  3. $28$
  4. $30$
  5. $35$

Question 19

Jerry runs two equal laps. The first is at $200 m /$ min. To average $240 m /$ min over both, what speed is needed for lap $2$?
  1. $200$
  2. $240$
  3. $280$
  4. $300$
  5. $320$

Question 20

$A , B , C , D$ lie in order on a line, with $\frac{AB}{AD} = \frac{3}{8}$ and $\frac{BC}{AD} = \frac{2}{5}$. Find $\frac{AB}{AC}$.

Question illustration
  1. $\frac{15}{16}$
  2. $\frac{6}{40}$
  3. $\frac{15}{31}$
  4. $\frac{3}{20}$
  5. $\frac{31}{40}$