Australian Mathematics Competition - 2018 - Senior years 11 -12 Questions and Solutions

Problem 1:

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 2:

The Great North Walk is a 250 km long trail from Sydney to Newcastle. If you want to complete it in 8 days, approximately how far do you need to walk each day?
(A) 15 km
(B) 20 km
(C) 30 km
(D) 40 km
(E) 80 km

Problem 3:

Half of a number is 32 . What is twice the number?
(A) 16
(B) 32
(C) 64
(D) 128
(E) 256

Problem 4:

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 5:

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 6:

What is \(2^0-1^8\) ?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 10

Problem 7:

\(1000 \%\) of a number is 100 . What is the number?
(A) 0.1
(B) 1
(C) 10
(D) 100
(E) 1000

Problem 8:

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 9:

In the triangle \(A B C, M\) is the midpoint of \(A B\).
Which one of the following statements must be true?
(A) \(\angle C A M=\angle A C M\)
(B) \(\angle C M B=2 \angle C A M\)
(C) \(A C=2 B C\)
(D) \(C M=B C\)
(E) Area \(\triangle A M C=\) Area \(\triangle M B C\)

Problem 10:

The sum of the numbers from 1 to 100 is 5050 . What is the sum of the numbers from 101 to 200 ?
(A) 15050
(B) 50500
(C) 51500
(D) 150500
(E) 505000

Problem 11:

Leila has a number of identical equilateral triangle shaped tiles. How many of these must she put together in a row (edge to edge) to create a shape which has a perimeter ten times that of a single tile?
(A) 14
(B) 20
(C) 25
(D) 28
(E) 30

Problem 12:

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 13:

Instead of multiplying a number by 4 and then subtracting 330 , I accidentally divided that number by 4 and then added 330 . Luckily, my final answer was correct. What was the original number?
(A) 220
(B) 990
(C) 144
(D) 374
(E) 176

Problem 14:

The diagram shows a regular octagon of side length 1 metre. In square metres, what is the area of the shaded region?
(A) 1
(B) \(\sqrt{2}\)
(C) 2
(D) \(3-\sqrt{2}\)
(E) \(\frac{1+\sqrt{2}}{2}\)

Problem 15:

A netball coach is planning a train trip for players from her two netball clubs, Panthers and Warriors.
The two clubs are in different towns, so the train fares per player are different. For the same cost she can either take 6 Panthers and 7 Warriors or she can take 8 Panthers and 4 Warriors.
If she takes only members of the Warriors on the train journey, the number she could take for the same cost is
(A) 11
(B) 13
(C) 16
(D) 20
(E) 25

Problem 16:

The triangle \(P Q R\) shown has a right angle at \(P\). Points \(T\) and \(S\) are the midpoints of the sides \(P R\) and \(P Q\), respectively. Also \(\angle Q T P=\alpha\) and \(\angle S R P=\beta\).
The ratio \(\tan \alpha: \tan \beta\) equals
(A) \(3: 1\)
(B) \(4: 1\)
(C) \(5: 1\)
(D) \(7: 2\)
(E) \(9: 2\)

Problem 17:

Three fair 6 -sided dice are thrown. What is the probability that the three numbers rolled are three consecutive numbers, in some order?
(A) \(\frac{1}{6}\)
(B) \(\frac{1}{9}\)
(C) \(\frac{1}{27}\)
(D) \(\frac{7}{36}\)
(E) \(\frac{1}{54}\)

Problem 18:

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

Problem 19:

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

\(\underbrace{111 \ldots .111}{100 \text { digits }}-\underbrace{222 \ldots 222}{50 \text { digits }}\)
What is the sum of the digits in the result?
(A) 375
(B) 420
(C) 429
(D) 450
(E) 475

Problem 20:

I have two regular polygons where the larger polygon has 5 sides more than the smaller polygon. The interior angles of the two polygons differ by \(1^{\circ}\). How many sides does the larger polygon have?
(A) 30
(B) 40
(C) 45
(D) 50
(E) 60

Problem 21:

How many solutions \((m, n)\) exist for the equation \(n=\sqrt{100-m^2}\) where both \(m\) and \(n\) are integers?
(A) 4
(B) 6
(C) 7
(D) 8
(E) 10

Problem 22:

A tetrahedron is inscribed in a cube of side length 2 as shown. What is the volume of the tetrahedron?
(A) \(\frac{8}{3}\)
(B) 4
(C) \(\frac{16}{3}\)
(D) \(\sqrt{6}\)
(E) \(8-2 \sqrt{2}\)

Problem 23:

A rectangle has sides of length 5 and 12 units.
A diagonal is drawn and then the largest possible circle is drawn in each of the two triangles.
What is the distance between the centres of these two circles?

(A) \(\sqrt{60}\)
(B) 8
(C) \(\sqrt{65}\)
(D) \(\sqrt{68}\)
(E) 9

Problem 24:

In the equation \(\underbrace{\sqrt{\sqrt{\ldots \sqrt{256}}}}_{60}=2^{\left(8^x\right)}\) the value of \(x\) is
(A) -17
(B) -19
(C) -21
(D) -23
(E) 16

Problem 25:

A right-angled triangle with sides of length 3,4 and 5 is tiled by infinitely many rightangled triangles, as shown.
What is the shaded area?
(A) \(\frac{18}{7}\)
(B) \(\frac{54}{25}\)
(C) \(\frac{8}{3}\)
(D) \(\frac{27}{17}\)
(E) \(\frac{96}{41}\)

Problem 26:

Let \(A\) be a 2018-digit number which is divisible by 9 . Let \(B\) be the sum of all digits of \(A\) and \(C\) be the sum of all digits of \(B\). Find the sum of all possible values of \(C\).

Problem 27:

The trapezium \(A B C D\) has \(A B=100, B C=130\), \(C D=150\) and

\(D A=120\), with right angles at \(A\) and D.

An interior point \(Q\) is joined to the midpoints of all 4 sides. The four

quadrilaterals formed have equal areas. What is the length \(A Q\) ?

Problem 28:

Donald has a pair of blue shoes, a pair of red shoes, and a pair of white shoes. He wants to put these six shoes side by side in a row. However, Donald wants the left shoe of each pair to be somewhere to the left of the corresponding right shoe. How many ways are there to do this?

Problem 29:

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 \(n\) ?

Problem 30:

Consider an \(n \times n\) grid filled with the numbers \(1, \ldots, n^2\) in ascending order from left to right, top to bottom. A shuffle consists of the following two steps:

• Shift every entry one position to the right. An entry at the end of a row moves to the beginning of the next row and the bottom-right entry moves to the top-left position.
• Then shift every entry down one position. An entry at the bottom of a column moves to the top of the next column and again the bottom-right entry moves to the top-left position.

An example for the \(3 \times 3\) grid is shown. Note that the two steps shown constitute one shuffle.

What is the smallest value of \(n\) for which the \(n \times n\) grid requires more than 20000 shuffles for the numbers to be returned to their original order?