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Australian AMC 2012 - Intermediate

Question 1

The value of $8\times3.3$ is
  1. 24.24
  2. 24.4
  3. 25.4
  4. 26.24
  5. 26.4

Question 2

Sally has \$20 of her pocket money left after 3 weeks, having spent just \$1 on a drink. How much pocket money does she get each week?
  1. \$5
  2. \$7
  3. \$9
  4. \$20
  5. \$21

Question 3

In the diagram, the size of $\angle PQR$ is

Question illustration
  1. $40^\circ$
  2. $50^\circ$
  3. $60^\circ$
  4. $70^\circ$
  5. $80^\circ$

Question 4

Three-fifths of a number is 48. What is the number?
  1. 54
  2. 60
  3. 64
  4. 80
  5. 84

Question 5

By what number must 6 be divided to obtain $\frac13$ as a result?
  1. 18
  2. $\frac12$
  3. $\frac1{18}$
  4. 2
  5. 9

Question 6

The average of the five numbers $x$, $1$, $\frac12$, $\frac13$ and $\frac14$ is $1$. The value of $x$ is
  1. $\frac15$
  2. $\frac23$
  3. $\frac{11}5$
  4. $\frac{25}{12}$
  5. $\frac{35}{12}$

Question 7

In the diagram, $PQRS$ is a square. The value of $x$ is

Question illustration
  1. 45
  2. 60
  3. 67.5
  4. 75
  5. 82.5

Question 8

Five positive integers have a mean of 10, a median of 10 and only one mode, which is 12. What is the difference between the largest and smallest of these numbers?
  1. 3
  2. 5
  3. 6
  4. 7
  5. 8

Question 9

If $750\times45=p$, then $750\times44$ equals
  1. $p-45$
  2. $p-750$
  3. $p-1$
  4. $44p$
  5. $750p$

Question 10

I can ride my bike 3 times as fast as Ted can jog. Ted starts 40 minutes before me and then I chase him. How long does it take me to catch Ted?
  1. 20 min
  2. 30 min
  3. 40 min
  4. 50 min
  5. 60 min

Question 11

If $p\%$ of $q$ is $k$, then $q\%$ of $p$ is
  1. $\frac{k}{100}$
  2. $\frac{pq}{200}$
  3. $\frac{pk}{100}$
  4. $\frac{qk}{100}$
  5. $k$

Question 12

On one side of each of the five coins below there is a number and on the other side there is a shape.

Question illustration

Peter is told that if there is a triangle on one side of a coin then there is an even number on the other. Which of the following is the fewest coins that Peter can turn over from the five to check this?
  1. S
  2. P and R
  3. Q and S
  4. P, Q, R and S
  5. all coins

Question 13

The architecture of Federation Square in Melbourne is based on frames as shown in which a large triangle is subdivided into 5 identical triangles, each similar to the large triangle.

Question illustration

If the shortest side of one of the smallest triangles is 1 m, how many metres of framing are required to construct the whole shape?
  1. 20
  2. $8+4\sqrt5$
  3. $10+4\sqrt5$
  4. $12+4\sqrt5$
  5. $15+5\sqrt5$

Question 14

If $a:b=3:2$ and $a+3b=27$, what is the value of $a+b$?
  1. 5
  2. 9
  3. 13
  4. 15
  5. 21

Question 15

This sheriff’s badge has ten equal sides, five $60^\circ$ angles and five equal reflex angles.

Question illustration

The value of $x$ is
  1. 108
  2. 132
  3. 135
  4. 138
  5. 140

Question 16

The shape shown is formed from four identical arcs, each a quarter of the circumference of a circle of radius 5 cm.

Question illustration

What is the area, in square centimetres, of the shape?
  1. 50
  2. $\frac{25\pi}{2}$
  3. $25\pi-25$
  4. $100-\frac{25\pi}{2}$
  5. $25\pi$

Question 17

The number $2012\times2013\times2014+2013$ is the cube of
  1. 2012
  2. 2013
  3. 2014
  4. 2112
  5. 2113

Question 18

A partition of a positive integer is a way of writing the integer as a sum of at least two positive integers. For example, the partitions of 4 are:

$$3+1,\quad2+2,\quad2+1+1\quad\text{and}\quad1+1+1+1.$$

How many partitions of 7 are there?
  1. 11
  2. 12
  3. 13
  4. 14
  5. 15

Question 19

These five shapes all have the same area. Which one has the largest perimeter?
  1. Question illustration
  2. Question illustration
  3. Question illustration
  4. Question illustration
  5. Question illustration

Question 20

Pippa made a litre of drink from apple juice and water in the ratio of 1 : 2. She found the taste too strong so she made a litre again in the ratio 1 : 3, but found this too weak. So she thought if she combined these two mixtures, it should be about right. What is the ratio of apple juice to water in this new mixture?
  1. 2 : 5
  2. 2 : 7
  3. 5 : 12
  4. 7 : 17
  5. 7 : 24