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Australian AMC 2013 - Senior

Question 1

The expression $\frac{0.6}{12}$ is equal to
  1. 5
  2. 0.5
  3. 0.05
  4. 0.005
  5. 0.0005

Question 2

Which of the following is a right-angled triangle?
  1. Question illustration
  2. Question illustration
  3. Question illustration
  4. Question illustration
  5. Question illustration

Question 3

The yield from a farm increased by 20% this season to 114 tonnes. Last season the number of tonnes yielded was
  1. 90
  2. 91
  3. 93
  4. 94
  5. 95

Question 4

The cube root of 201 320 132 013 is closest to
  1. 600
  2. 5000
  3. 6000
  4. 50 000
  5. 60 000

Question 5

If $p=4b+26$ and b is a positive integer, then p could not be divisible by
  1. 2
  2. 4
  3. 5
  4. 6
  5. 7

Question 6

The first five terms of a sequence are shown.

$$1,2,1,-1,-2,\ldots$$

After the second term, each term can be obtained by subtracting from the previous term the term before that. What is the sum of the first 42 terms of this sequence?
  1. 0
  2. 4
  3. 12
  4. 24
  5. 30

Question 7

A line on a number plane passes through the vertical axis at 1 and the horizontal axis between 2 and 3, as shown. Which of the following equations could be the equation of this line?

Question illustration
  1. $5x+12y-12=0$
  2. $4x+y+1=0$
  3. $3x-7y+7=0$
  4. $3x-2y+2=0$
  5. $2x+3y-3=0$

Question 8

When Cody races Chris over 100 metres, Cody wins by 5 metres (that is, when Cody finishes, Chris still has 5 metres to run). When Chris races Joseph over 100 metres, Chris wins by 10 metres. If Cody races Joseph over 100 metres, what will Cody’s winning margin be, in metres?
  1. 15.5
  2. 15
  3. 14.5
  4. 14
  5. 13.5

Question 9

On a ranch in Peru, there are two types of animals, llamas and alpacas. The ranch also has a number of managers. The ratio of llamas to alpacas is 2 : 3 and the ratio of alpacas to managers is 8 : 1. The ratio of animals to managers is
  1. 16 : 3
  2. 13 : 1
  3. 12 : 1
  4. 40 : 3
  5. 20 : 3

Question 10

Starting with the number 0 on my calculator, I do a calculation in five steps. At each step, I either add 1 or multiply by 2. What is the smallest number that cannot be the final result?
  1. 11
  2. 10
  3. 9
  4. 8
  5. 7

Question 11

A box contains three bags. One bag contains one white pebble and three black pebbles. Another bag also contains one white pebble and three black pebbles. The remaining bag contains one white pebble and four black pebbles. You reach into the box and randomly pull out a bag, then reach into the bag and randomly pull out a pebble. What is the probability that this pebble is white?
  1. $\frac7{30}$
  2. $\frac3{13}$
  3. $\frac19$
  4. $\frac14$
  5. $\frac{11}{36}$

Question 12

A region on a number plane is enclosed by the x-axis, the y-axis, the line $x=2$ and the graph of the function $y=f(x)$ where

$$f(x)=\begin{cases}1&\text{for }0\le x\le1,\\1+\sqrt{1-(x-2)^2}&\text{for }1\lt x\le2.\end{cases}$$

The area of this region, in square units, is
  1. $\frac52$
  2. $1+\frac\pi4$
  3. $1+\frac\pi2$
  4. $2+\pi$
  5. $2+\frac\pi4$

Question 13

Three squares, with side lengths 3 cm, 5 cm and 8 cm, are arranged in a row as shown.

Question illustration

The area, in square centimetres, of the shaded trapezium is
  1. 12
  2. $\frac{73}{6}$
  3. $\frac{55}{4}$
  4. 14
  5. $\frac{25}{2}$

Question 14

A square of perimeter 20 cm is inscribed in a square of perimeter 28 cm. What is the greatest distance between a vertex of the inner square and a vertex of the outer square, in centimetres?

Question illustration
  1. $\sqrt{58}$
  2. $\frac{7\sqrt5}{2}$
  3. 8
  4. $\sqrt{65}$
  5. $5\sqrt3$

Question 15

Six different two-digit numbers are formed using three different non-zero digits. The sum of five of those six numbers equals 100. What is the sixth number?
  1. 23
  2. 32
  3. 45
  4. 54
  5. 67

Question 16

Three darts are thrown at the $3\times3$ grid, each landing in a different small square. After each throw, each of the remaining small squares is equally likely to be hit.

Question illustration

What is the probability that the squares in which they land form a horizontal, vertical or diagonal row?
  1. $\frac1{63}$
  2. $\frac2{21}$
  3. $\frac19$
  4. $\frac1{42}$
  5. $\frac8{81}$

Question 17

What is the largest possible volume of a box with rectangular faces of integer dimensions where the long diagonal $XY=9$?

Question illustration
  1. 32
  2. 81
  3. 90
  4. 108
  5. 112

Question 18

The number of different ordered pairs $(x,y)$ which satisfy the equation

$$2x^2-2xy+y^2=169$$

where x and y are integers and $x\ge0$ is
  1. 2
  2. 4
  3. 5
  4. 7
  5. 8

Question 19

A running track encloses an area formed by a rectangle joined to two semicircles, as shown in the diagram below. If the length of the track must be 400 metres, then what is the maximum possible area of the rectangle, in square metres?

Question illustration
  1. $\frac{160\,000}{(\pi+2)^2}$
  2. $\frac{20\,000}{\pi}$
  3. $\frac{30\,000}{\pi}$
  4. 10 000
  5. $\frac{40\,000}{\pi}$

Question 20

In how many ways can 2013 be expressed as the sum of two or more consecutive positive integers?
  1. 5
  2. 6
  3. 7
  4. 8
  5. 9