Australian AMC 2019 - Intermediate

Question 1

The value of $20.19-19$ is

39.19
20.38
20
1.19
1

Question 2

Sharyn’s piano lesson was 40 minutes long, and finished at 4.10 pm. When did it start?

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3.30 pm
3.40 pm
3.50 pm
4.40 pm
4.50 pm

Question 3

Which of the following is closest to $7\times1.8$?

Question 4

What is the area of the shaded triangle?

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$8\text{ m}^2$
$12\text{ m}^2$
$14\text{ m}^2$
$20\text{ m}^2$
$24\text{ m}^2$

Question 5

Five-eighths of a number is 200. What is the number?

Question 6

Every row and every column of this $3\times3$ square must contain each of the numbers 1, 2 and 3.
What is the value of $N+M$?

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Question 7

A piece of paper is folded in three, then a semi-circular cut and a straight cut are made, as shown in the diagram.
When the paper is unfolded, what does it look like?

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Question 8

When a rectangle is cut in half, two squares are formed. If each square has a perimeter of 48, what is the perimeter of the original rectangle?

Consider the undulating number sequence

$1,4,7,4,1,4,7,4,1,4,\ldots,$

which repeats every four terms. The running total of the first 3 terms is 12. The running total of the first 7 terms is 28.
Which one of the following is also a running total of this sequence?

Question 10

Sebastien has a Personal Identification Number (PIN) consisting of four digits. The first three digits in order are 591. If Sebastien’s PIN is divisible by 3, then how many possibilities are there for the final digit?

Question 11

A quadrilateral $ABCD$ has $AD\parallel BC$, $AB=BC$ and $AC=CD$. The external $\angle CDE=140^\circ$. What is the value, in degrees, of $\angle ABC$?

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Question 12

In my dance class, 14 students are taller than Bob, and 12 are shorter than Alice. Four students are both shorter than Alice and taller than Bob. How many students are in my dance class?

Question 13

Three equilateral triangles are joined to form the quadrilateral $ABCD$ shown.
The point $X$ is halfway along $CD$.
What fraction of the area of $ABCD$ is shaded?

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$\frac12$
$\frac34$
$\frac23$
$\frac56$
$\frac35$

Question 14

In a year 10 Maths class, there are 30 students. Each student is either 15 or 16 years old, and either left- or right-handed.
The ratio of right-handed students to left-handed students is 4 : 1, the ratio of 15 year olds to 16 year olds is 1 : 2 and the ratio of left-handed 15 year olds to left-handed 16 year olds is 1 : 5.
If the names of the students in this class are placed in a hat and one is selected at random, what is the probability that the student selected is 15 years old and right-handed?

$\frac1{30}$
$\frac16$
$\frac3{10}$
$\frac12$
$\frac45$

Question 15

The set of four rollers shown has fixed axles and transfers rotation from each roller to the next without slipping. Their diameters are 21 cm, 32 cm, 50 cm and 14 cm respectively.
While the 21 cm roller makes a full rotation $(360^\circ)$, through which angle does the 14 cm roller rotate?

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$180^\circ$
$310^\circ$
$360^\circ$
$540^\circ$
$620^\circ$

Question 16

In a box of apples, $\frac37$ of the apples are red and the rest are green. Five more green apples are added to the box. Now $\frac58$ of the apples are green.
How many apples are there now in the box?

Question 17

Asha chooses a whole number from 1 to 5 and announces it. Then Richy chooses a whole number from 1 to 5 and announces it. Finally, Asha chooses a whole number from 1 to 5 and announces it.
If the sum of the three numbers announced is a multiple of 7, then Asha wins; otherwise, Richy wins.
What number should Asha choose on her first turn to guarantee that she can win?

Question 18

The area of the shaded triangle inside this rectangle is

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$\frac12(x+y)^2$
$x(x+y)$
$y(x+y)$
$\frac12(y^2-x^2)$
$\frac12(x^2+y^2)$

Question 19

These three cubes are labelled in exactly the same way, with the 6 letters A, M, C, D, E and F on their 6 faces:

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The cubes are now placed in a row so that the front looks like this:

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When we look at the cubes from the opposite side, we will see

Question 20

Five numbers are placed in a row. From the third number on, each number is the average of the previous two numbers. The first number is 12 and the last number is 7. What is the third number?