2020 AMC

AUSTRALIAN MATHEMATICS COMPETITION

Intermediate Years 9–10

(Australian school years)

THURSDAY 30 JULY 2020

NAME

TIME ALLOWED: 75 MINUTES

INSTRUCTIONS AND INFORMATION

General
1. Do not open the booklet until told to do so by your teacher.
2. NO calculators, maths stencils, mobile phones or other calculating aids are permitted.
Scribbling paper, graph paper, ruler and compasses are permitted, but are not essential.
3. Diagrams are NOT drawn to scale. They are intended only as aids.
4. There are 25 multiple-choice questions, each requiring a single answer, and 5 questions that
require a whole number answer between 0 and 999. The questions generally get harder as
you work through the paper. There is no penalty for an incorrect response.
5. This is a competition not a test; do not expect to answer all questions. You are only
competing against your own year in your own country/Australian state so different years
doing the same paper are not compared.
6. Read the instructions on the answer sheet carefully. Ensure your name, school name and
school year are entered. It is your responsibility to correctly code your answer sheet.
7. When your teacher gives the signal, begin working on the problems.

The answer sheet

1. Use only lead pencil.
2. Record your answers on the reverse of the answer sheet (not on the question paper) by
FULLY colouring the circle matching your answer.
3. Your answer sheet will be scanned. The optical scanner will attempt to read all markings
even if they are in the wrong places, so please be careful not to doodle or write anything
extra on the answer sheet. If you want to change an answer or remove any marks, use a
plastic eraser and be sure to remove all marks and smudges.

Integrity of the competition

The AMT reserves the right to re-examine students before deciding whether to grant official
status to their score.
Reminder: You may sit this competition once, in one division only, or risk no score.

Copyright © 2020 Australian Mathematics Trust

ACN 083 950 341
 Intermediate Division
Questions 1 to 10, 3 marks each

1. 2 − (0 − (2 − 0)) =
(A) −4 (B) −2 (C) 0 (D) 2 (E) 4

2. 2 1000%
1000% of 2 is equal to
(A) 0.002 (B) 20 (C) 200 (D) 1002 (E) 2000

3. x y
In the diagram provided, find the sum of x and y.
(A) 30 (B) 75 (C) 95 y◦
(D) 105 (E) 180

|
|

x◦
105◦

1+2+3+4+5 1+2
4. − =
1+2+3+4 1+2+3
5 7
(A) 3 (B) (C) 1 (D) (E) 2
6 6

5. 26 14

Sebastien is thinking of two numbers whose sum is 26 and whose difference is 14.
The product of Sebastien’s two numbers is
(A) 80 (B) 96 (C) 105 (D) 120 (E) 132
6.

(A)
All of the shapes have equal area.
(B) Q S
Only Q and S have equal area.
(C) R T
Only R and T have equal area.

(D) P R T
Only P, R and T have equal area.
(E) P R T Q S
P, R and T have equal area, and Q and S have equal area.

7. 123456 − 12345 + 1234 − 123 + 12 − 1 =

(A) 33333 (B) 101010 (C) 111111 (D) 122223 (E) 112233

8.

(A) 6 (B) 8 (C) 10 (D) 12 (E) 14

9.

A piece of paper is folded twice as shown and cut along the dotted lines.


 I 22020 Australian Mathematics Competition — Intermediate

Once unfolded, which letter does the piece of paper most resemble?
(A) M (B) O (C) N (D) B (E) V

10.

(A) 24 (B) 27 (C) 30 (D) 33 (E) 36

Questions 11 to 20, 4 marks each

11.

(A) 82 (B) 88 (C) 94 (D) 112 (E) 130

12

2 1 1 1 2
(A) (B) (C) (D) (E)
99 11 10 2 9
2020 Australian Mathematics Competition — IntermediateI 3

13. 25 20 25 = 5+7 +13

10 20

The number 25 can be written as the sum of three different primes less than 20. For
instance, 25 = 5 + 7 + 13.
How many multiples of 10 can be written as the sum of three different primes less t
han 20?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

14.

(A) 10 (B) 15 (C) 18 (D) 24 (E) 36

15. 10

There are 10 children in a classroom. The ratio of boys to girls increases when another
girl and another boy enter the room. What is the greatest number of boys that could
have been in the room at the beginning?
(A) 1 (B) 4 (C) 5 (D) 6 (E) 9
16. A B A B
A B
Two triangles, A and B, have the same area. Triangle A is isosceles and triangle B
is right-angled.

5 cm 5 cm
A B

6 cm 6 cm

The difference between the perimeters of triangle A and triangle B is

(A) nothing (B) between 0 cm and 1 cm (C) between 1 cm and 2 cm
(D) between 2 cm and 3 cm (E) more than 3 cm

17. 2 5
2, 5, 10, 50, 500, . . .

A list of numbers has first term 2 and second term 5. The third term, and each term
after this, is found by multiplying the two preceding terms together:

2, 5, 10, 50, 500, . . .

The value of the eighth term is

(A) 25 × 58 (B) 28 × 59 (C) 28 × 513 (D) 29 × 515 (E) 213 × 521

18.

(A) 4 (B) 6 (C) 8 (D) 9 (E) 12

 I4

19.

(A) 7 (B) 14 (C) 21 (D) 35 (E) 70

20.

(A) 30 (B) 35 (C) 40 (D) 45 (E) 50

21.

A scientist measured the amount of bacteria in a Petri dish over several weeks and
also recorded the temperature and humidity for the same time period. The results
are summarised in the following graphs.

A B C D E
Bacteria

Humidity
Temperature

Week Humidity Temperature

During which week was the bacteria population highest?

(A) week A (B) week B (C) week C (D) week D (E) week E
 I5
22. A B C D E 40
A E D B
C D E
Five friends read a total of 40 books between them over the holidays. Everyone read
at least one book but no-one read the same book as anyone else.
Asilata read twice as many books as Eammon. Dane read twice as many as Bettina.
Collette read as many as Dane and Eammon put together.
Who read exactly eight books?
(A) Asilata (B) Bettina (C) Colette (D) Dane (E) Eammon

23. 5 2 cm 3 cm 4 cm 5 cm 8 cm

There are 5 sticks of length 2 cm, 3 cm, 4 cm, 5 cm and 8 cm. Three sticks are chosen
randomly. What is the probability that a triangle can be formed with the chosen
sticks?
(A) 0.25 (B) 0.3 (C) 0.4 (D) 0.5 (E) 0.6

24.
Five squares of unit area are circumscribed by a circle
as shown.What is the radius of the circle?
√ √
3 2 5 10
(A) (B) (C)
2 3 2
√ √
13 185
(D) (E)
2 8

25.

(A) 0 (B) 2016 (C) 2018 (D) 2020 (E) 2021

For questions 26 to 30, shade the answer as an integer from 0 to 999

in the space provided on the answer sheet.
Questions 26–30 are worth 6, 7, 8, 9 and 10 marks, respectively.
 I 62020 Australian Mathematics Competition — Intermediate

26. n n! 1 n 4! = 4×3×2×1 = 24
1! + 2! + 3! + ... + 2020!
If n is a positive integer, n! is found by multiplying the integers from 1 to n. For
example, 4! = 4 × 3 × 2 × 1 = 24.
What are the three rightmost digits of the sum 1! + 2! + 3! + · · · + 2020! ?

27.

28.
29.

30.