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VTMO 2014 — Grade 9 — semifinal

Question 1

Given $f \left(4 x + 3\right) = 2 x + 1$, find $f \left(- 9\right)$.
  1. $- 9$
  2. $5$
  3. $- 5$
  4. $9$
  5. $3$

Question 2

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 3

Let $x _{1} , x _{2}$ be the roots of $\left(x + 1\right) \left(2 x - 3\right) + \left(x - 4\right) \left(x + 11\right) + 38 = 0$. Find $x _{1}^{2} + x _{2}^{2}$.
  1. $- 2$
  2. $10$
  3. $4$
  4. $12$
  5. $2$

Question 4

$f \left(x + y\right) = x + f \left(y\right)$ for all real $x , y$, and $f \left(0\right) = 2$. Find $f \left(2014\right)$.
  1. $0$
  2. $2$
  3. $2012$
  4. $2014$
  5. $2016$

Question 5

Find the area of the triangle formed by $y = 3 , y = 2 x + 1 , y = - 2 x + 1$.
  1. $2$
  2. $2.5$
  3. $3$
  4. $3.5$
  5. $4$

Question 6

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

Question 7

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 8

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

Question 9

Traveling at $40 \frac{\,\mathrm{km}}{h}$ arrives $1$ minute late; $45 \frac{\,\mathrm{km}}{h}$ arrives $1$ minute early. Find the distance.
  1. $72 \,\mathrm{km}$
  2. $48 \,\mathrm{km}$
  3. $24 \,\mathrm{km}$
  4. $18 \,\mathrm{km}$
  5. $12 \,\mathrm{km}$

Question 10

$A , B , C , D$ 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. $\frac{31}{40}$

Question 11

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 12

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

Question 13

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

Question 14

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 15

Among $2014$ natural numbers, every product of $15$ selected numbers is even. What is the minimum number of even numbers?
  1. $2000$
  2. $15$
  3. $1$
  4. $1999$
  5. $16$

Question 16

A square of area $10 \,\mathrm{cm}^{2}$ is inscribed in a circle, which is inscribed in a larger square. Find the larger area.

Question illustration
  1. $20 \,\mathrm{cm}^{2}$
  2. $50 \,\mathrm{cm}^{2}$
  3. $15 \,\mathrm{cm}^{2}$
  4. $\sqrt{150} \,\mathrm{cm}^{2}$
  5. $\sqrt{120} \,\mathrm{cm}^{2}$

Question 17

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 18

At $3 : 15$, what is the smaller angle between the clock hands?
  1. $0 ^{\circ}$
  2. $7.5 ^{\circ}$
  3. $14 ^{\circ}$
  4. $12 ^{\circ}$
  5. $10 ^{\circ}$

Question 19

Three circles of radii $2 , 3 , 3 \,\mathrm{cm}$ are pairwise externally tangent. Find the inradius of their center triangle.

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

Question 20

Circles $\left(A\right) , \left(B\right)$ touch both sides of the right angle xOy. Center $A$ lies on circle $\left(B\right)$, whose circle is larger as pictured. Find rB/rA.

Question illustration
  1. $2 - \sqrt{2}$
  2. $\sqrt{2} - 1$
  3. $\frac{1}{\sqrt{2} - 1}$
  4. $\frac{1}{\sqrt{2} + 1}$
  5. $2 + \sqrt{2}$