Australian Mathematics Competition - 2018 - Intermediate - Grade 9 & 10 - Questions and Solutions

Problem 1:

The value of \(\frac{2018-18}{1000}\) is
(A) 0.02
(B) 0.1
(C) 1
(D) 2
(E) 2000

Problem 2:

What value is indicated on this charisma-meter?
(A) 36.65
(B) 37.65
(C) 38.65
(D) 37.15
(E) 37.3

Problem 3:

What is the difference between the sum and the product of 4 and 5 ?
(A) 1
(B) 8
(C) 9
(D) 11
(E) 20

Problem 4:

In the diagram, \(P Q R S\) is a square. What is the size of \(\angle X P Y\) ?
(A) \(25^{\circ}\)
(B) \(30^{\circ}\)
(C) \(35^{\circ}\)
(D) \(40^{\circ}\)
(E) \(45^{\circ}\)

Problem 5:

Which of the following is not a whole number?
(A) \(350 \div 2\)
(B) \(350 \div 7\)
(C) \(350 \div 5\)
(D) \(350 \div 25\)
(E) \(350 \div 20\)

Problem 6:

Nora, Anne, Warren and Andrew bought plastic capital letters to spell each of their names on their birthday cakes. Their birthdays are on different dates, so they planned to reuse letters on different cakes.
What is the smallest number of letters they needed?
(A) 8
(B) 9
(C) 10
(D) 11
(E) 12

Problem 7:

In years, 2018 days is closest to
(A) 4.5 years
(B) 5 years
(C) 5.5 years
(D) 6 years
(E) 6.5 years

Problem 8:

Two paths from \(A\) to \(C\) are pictured.
The stepped path consists of horizontal and vertical segments, whereas the dashed path is straight.
What is the difference in length between the two paths?
(A) 1 m
(B) 2 m
(C) 3 m
(D) 4 m
(E) 0 m

Problem 9:

The value of \(9 \times 1.2345-9 \times 0.1234\) is
(A) 9.9999
(B) 9
(C) 9.0909
(D) 10.909
(E) 11.1111

Problem 10:

What fraction of this regular hexagon is shaded?
(A) \(\frac{1}{2}\)
(B) \(\frac{2}{3}\)
(C) \(\frac{3}{4}\)
(D) \(\frac{3}{5}\)
(E) \(\frac{4}{5}\)

Problem 11:

The cost of feeding four dogs for three days is \(\$ 60\). Using the same food costs per dog per day, what would be the cost of feeding seven dogs for seven days?
(A) \(\$ 140\)
(B) \(\$ 200\)
(C) \(\$ 245\)
(D) \(\$ 350\)
(E) \(\$ 420\)

Problem 12:

In a certain year there were exactly four Tuesdays and exactly four Fridays in the month of December. What day of the week was 31 December?
(A) Monday
(B) Wednesday
(C) Thursday
(D) Friday
(E) Saturday

Problem 13:

Fill in this diagram so that each of the rows, columns and diagonals adds to 18 .
What is the sum of all the corner numbers?
(A) 20
(B) 22
(C) 23
(D) 24
(E) 25

Problem 14:

The sum of 4 consecutive integers is \(t\).
In terms of \(t\), the smallest of the four integers is
(A) \(\frac{t-10}{4}\)
(B) \(\frac{t-2}{4}\)
(C) \(\frac{t-3}{4}\)
(D) \(\frac{t-4}{4}\)
(E) \(\frac{t-6}{4}\)

Problem 15:

A 3-dimensional object is formed by gluing six identical cubes together. Four of the diagrams below show this object viewed from different angles, but one diagram shows a different object. Which diagram shows the different object?

Problem 16:

In the circle shown, \(C\) is the centre and \(A, B, D\) and \(E\) all lie on the circumference.
Reflex \(\angle B C D=200^{\circ}, \angle D C A=x^{\circ}\) and \(\angle B C A=3 x^{\circ}\) as shown.
The ratio of \(\angle D A C: \angle B A C\) is
(A) \(3: 1\)
(B) \(5: 2\)
(C) \(8: 3\)
(D) \(7: 4\)
(E) \(7: 3\)

Problem 17:

Allan and Zarwa are playing a game tossing a coin. Allan wins as soon as a head is tossed and Zarwa wins if two tails are tossed. The probability that Allan wins is
(A) \(\frac{1}{2}\)
(B) \(\frac{3}{5}\)
(C) \(\frac{5}{8}\)
(D) \(\frac{2}{3}\)
(E) \(\frac{3}{4}\)

Problem 18:

In this expression

we place either a plus sign or a minus sign in each box so that the result is the smallest positive number possible. The result is
(A) between 0 and \(\frac{1}{100}\)
(B) between \(\frac{1}{100}\) and \(\frac{1}{50}\)
(C) between \(\frac{1}{50}\) and \(\frac{1}{20}\)
(D) between \(\frac{1}{20}\) and \(\frac{1}{10}\)
(E) between \(\frac{1}{10}\) and 1

Problem 19:

A town is laid out in a square of side 1 kilometre, with six straight roads as shown.
Each day the postman must walk the full length of every road at least once, starting wherever he likes and ending wherever he likes. How long is the shortest route he can take, in kilometres?

(A) \(4+\frac{\sqrt{2}}{2}\)
(B) \(4+\sqrt{2}\)
(C) \(4+2 \sqrt{2}\)
(D) \(4+3 \sqrt{2}\)
(E) \(5+2 \sqrt{2}\)

Problem 20:

A rectangle with integer sides has a diagonal stripe which starts 1 unit from the diagonal corners, as in the diagram. The area of the stripe is exactly half of the area of the rectangle. What is the perimeter of this rectangle?
(A) 14
(B) 16
(C) 18
(D) 20
(E) 22

Problem 21:

How many digits does the number \(20^{18}\) have?
(A) 24
(B) 38
(C) 18
(D) 36
(E) 25

Problem 22:

In this subtraction, the first number has 100 digits and the second number has 50 digits.

\(\underbrace{111 \ldots \ldots 111}{100 \text { digits }}-\underbrace{222 \ldots 222}{50 \text { digits }}\)

What is the sum of the digits in the result?
(A) 375
(B) 420
(C) 429
(D) 450
(E) 475

Problem 23:

Suppose \(p\) is a two-digit number and \(q\) has the same digits, but in reverse order. The number \(p^2-q^2\) is a non-zero perfect square. The sum of the digits of \(p\) is
(A) 7
(B) 9
(C) 11
(D) 12
(E) 13

Problem 24:

In triangle \(\triangle P Q R, U\) is a point on \(P R, S\) is a point on \(P Q, T\) is a point on \(Q R\) with \(U S | R Q\), and \(U T | P Q\).
The area of \(\triangle P S U\) is \(120 \mathrm{~cm}^2\) and the area of \(\triangle T U R\) is \(270 \mathrm{~cm}^2\).
The area of \(\triangle Q S T\), in square centimetres, is
(A) 150
(B) 160
(C) 170
(D) 180
(E) 200

Problem 25:

This year Ann's age is the sum of the digits of her maths teacher's age. In five years Ann's age will be the product of the digits of her maths teacher's age at that time.
How old is Ann now?
(A) 11
(B) 13
(C) 15
(D) 14
(E) 16

Problem 26:

I have a three-digit number, and I add its digits to create its digit sum. When the digit sum of my number is subtracted from my number, the result is the square of the digit sum. What is my three-digit number?

Problem 27:

A road from Tamworth to Broken Hill is 999 km long. There are road signs each kilometre along the road that show the distances (in kilometres) to both towns as shown in the diagram.

How many road signs are there that use exactly two different digits?

Problem 28:

In the division shown, \(X, Y\) and \(Z\) are different non-zero digits.\(\begin{array}{r}ZX\\ 8\enclose{longdiv}{XYZ}\end{array}\) rem Y
What is the three-digit number \(X Y Z\) ?

Problem 29:

An infinite increasing list of numbers has the property that the median of the first \(n\) terms equals the \(n^{\text {th }}\) odd positive integer. How many numbers in the list are less than 2018 ?

Problem 30

For \(n \geq 3\), a pattern can be made by overlapping \(n\) circles, each of circumference 1 unit, so that each circle passes through a central point and the resulting pattern has order- \(n\) rotational symmetry.
For instance, the diagram shows the pattern where \(n=7\). If the total length of visible arcs is 60 units, what is \(n\) ?