Australian Mathematics Competition 2020, Middle Primary (Years 3-4)

Question 1

How many cubes are shown here?

(a) 6

(b) 9

(d) 12

(e) 18

Question 2

$20+20=$

(a) 40

(b) 30

(d) 220

(e) 2020

Question 3

What time is shown on this clock?

(a) 3:05

(b) $3: 50$

(d) $5: 15$

(e) 5:30

Question 4

Half of 16 is

(a) 32

(b) 4

(d) 7

(e) 8

Question 5

Today is Thursday. What is the day after tomorrow?

(a) Thursday

(b) Friday

(d) Sunday

(e) yesterday

Question 6

How many pieces have been placed in the jigsaw puzzle so far?

(a) 25

(b) 27

(d) 33

(e) 35

Question 7

What is the perimeter of this triangle?

(a) 33 m

(b) 34 m

(d) 36 m

(e) 37 m

Question 8

Tia is playing a computer game with a rabbit on a grid. Each arrow key moves the rabbit one square in the direction on the key. Starting in the centre of the grid, which sequence of moves takes Tia's rabbit back to this starting position?

(a) $\uparrow \uparrow \rightarrow \rightarrow \uparrow \uparrow \rightarrow \rightarrow$

(b) $\uparrow \uparrow \leftarrow \rightarrow \rightarrow \leftarrow \downarrow \downarrow$

(d) $\uparrow \leftarrow \downarrow \downarrow \downarrow \downarrow \rightarrow \uparrow$

(e) $\rightarrow \rightarrow \rightarrow \uparrow \leftarrow \leftarrow \leftarrow \uparrow$

Question 9

I have 10 coins in my pocket, half are 20c coins and half are 50c coins. The total value of the coins is

(a) $$ 1.50$

(b) $$ 2$

(d) $$ 3$

(e) $$ 3.50$

Question 10

The graph shows the number of eggs laid by backyard chickens Nony and Cera for the first six months of the year.

In how many months did Nony lay more eggs than Cera?

(a) 1

(b) 2

(d) 4

(e) 5

Questions 11 to 20, 4 marks each

Question 11

Micky had $$ 9.50$. He spent $$ 1.75$ on fruit for lunch and gave his two friends $$ 1.30$ each. How much money did he have left?

(a) $$ 3.35$

(b) $$ 4.35$

(d) $$ 7.75$

(e) $$ 8.20$

Question 12

At the end of a game of marbles, Lei has 15 marbles, Dora has 8 and Omar has 4. How many marbles must Lei give back to his friends if they want to start the next game with an equal number each?

(a) 5

(b) 6

(d) 8

(e) 9

Question 13

Australian $$ 1$ coins are 3 mm thick. Chris makes a stack of these coins 60 mm high. What is the stack worth?

(a) $$ 3$

(b) $$ 20$

(d) $$ 40$

(e) $$ 60$

Question 14

Ada, Billy, Con, Dee and Edie took part in a swimming race. Billy did not win or come last. Dee finished ahead of two others but did not come first. Ada finished after Dee and Con finished before Edie. Who won the race?

(a) Ada

(b) Billy

(d) Dee

(e) Edie

Question 15

At his birthday party, Ricky and his friends wear stripy paper hats in the shape of a cone, as shown on the left. After the party, Ricky makes a straight cut in one of the hats all the way up to the point at the top, as shown on the right.

Which of the following best matches what the hat will look like when Ricky flattens it out on the table?

Question 16

It is 12 km by road from Woy Woy to Gosford, as shown on this map. John lives in Tascott, 4 km north of Woy Woy. Marike lives in Wyoming, 2 km north of Gosford. How far does John have to drive to visit Marike?

(a) 10 km

(b) 18 km

(d) 6 km

(e) 20 km

Question 17

Jake is building a $3 \times 3 \times 3$ cube using small wooden cubes. The diagram shows where he is up to. How many more small cubes does he need to complete his $3 \times 3 \times 3$ cube?

(a) 5

(b) 6

(d) 8

(e) 9

Question 18

Juanita started with a square of paper, made some folds in it, then punched a single hole through all layers. The diagram shows what it looked like after she unfolded it and flattened it back out.

What was the pattern of folds she made?

Question 19

Aidan puts a range of 3D shapes on his desk at school. This is the view from his side of the desk:

Nadia is sitting on the opposite side of the desk facing Aidan. Which of the following diagrams best represents the view from Nadia's side of the desk?

Question 20

I have five 50 c coins, five $$ 1$ coins and five $$ 2$ coins. In how many different ways can I make up $$ 5$ ?

(a) 4

(b) 6

(d) 10

(e) 12

Questions 21 to 25, 5 marks each

Question 21

After the first kilometre of the school cross-country run, Petra was second last. In the next kilometre she managed to overtake seven runners. In the third kilometre, two runners overtook her. In the final kilometre, she passed eight runners, but four other runners overtook her. She finished ninth. How many were in the race?

(a) 15

(b) 18

(d) 20

(e) 21

Question 22

I fold up this net to make a cube. I then multiply the numbers on opposite faces to get three numbers. The largest of these is

(a) 12

(b) 15

(d) 24

(e) 30

Question 23

Emanuel works in a busy restaurant washing dishes. Each dirty plate from the stack on the left takes 1 minute to wash and dry, before being placed on top of the clean stack on the right. After 7 minutes, and every 7 minutes from then on, a waiter brings 4 more dirty plates and adds them to the top of the dirty stack.

How high is the stack of clean plates when the coloured plate is being washed?

(a) 14

(b) 16

(d) 20

(e) 22

Question 24

A primary school has 400 students and they each have one vote for a school captain. They voted for Jordan, Evie and Emily. Jordan got 3 times as many votes as Emily. Evie got 20 fewer votes than Jordan. How many votes did Evie get?

(a) 20

(b) 60

(d) 140

(e) 160

Question 25

Karl likes to avoid walking on the cracks in the footpath by taking three equally spaced steps for every two blocks. Every third block of the footpath is darker than the others, as shown.

In his first 100 steps, how many times does Karl's left foot step on a darker block?

(a) 11

(b) 16

(d) 25

(e) 33 For questions 26 to 30 , shade the answer as a whole number 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.

Question 26

Janine thinks of three numbers. Between them, they use the digits $1,3,5,6,7,8$ and 9 , with each digit being used exactly once. The second number is 2 times the first number. The third number is 4 times the first number. What is the third number?

Question 27

In the following diagram, you enter at the square labelled entry and exit at the square labelled exit. You can move horizontally and vertically along the white squares, but must stay off the coloured squares. Each square can only be visited once. By moving this way and adding the numbers in the squares you pass through, what is the highest sum you can get?

Question 28

A bale of hay can be eaten by a horse in 2 days, by a cow in 3 days and by a sheep in 12 days. A farmer has 22 bales of hay and one horse, one cow and one sheep to feed. How many days will his bales last?

Question 29

A number is oddtastic if all of its digits are odd. For example, 9, 57 and 313 are oddtastic. However, 50 and 787 are not oddtastic, since 0 and 8 are even digits. How many of the numbers from 1 to 999 are oddtastic?

Question 30

Oliver used small cubes to build a set of solid shapes as shown. In the first shape, he used 1 cube; in the second shape, he used 6 cubes; in the third shape, he used 19 cubes. How many cubes did Oliver use to build his fifth shape?

Australian Mathematics Competition 2021, Middle Primary (Years 3-4)

Question 1

How many dots are on this domino?

(a) 5

(b) 7

(d) 10

(e) 11

Question 2

What is the difference between 14 and 2 ?

(a) 28

(b) 16

(d) 10

(e) 7

Question 3

This Nigerian flag is white and green. What fraction of it is green?

(a) one-third

(b) one-quarter

(d) two-fifths

(e) two-thirds

Question 4

$234+100=$

(a) 23400

(b) 1234

(d) 334

(e) 244

Question 5

How many minutes are in a quarter of an hour?

(a) 4

(b) 10

(d) 20

(e) 40

Question 6

My tank can hold 80 kL of water. The indicator on the tank shows the water level inside the tank. Which of the following is closest to the amount of water in the tank?

(a) 35 kL

(b) 45 kL

(d) 65 kL

(e) 75 kL

Question 7

Which number makes this number sentence true? $$ \square-5=9 $$

(a) 0

(b) 4

(d) 9

(e) 14

Question 8

Each face of this cube is divided into 4 small squares. How many small squares are there on the outside of the cube altogether?

(a) 16

(b) 18

(d) 24

(e) 30

Question 9

A cross country track is marked out with a number of flags as shown. How many of the flags will be on the left of the runners when they pass them?

(a) 7

(b) 8

(d) 10

(e) 11

Question 10

Which one of these shaded areas is the largest?

Questions 11 to 20, 4 marks each

Question 11

Leo is waiting in line at school. There are four students ahead of him and twice as many behind him. How many students are in this line?

(a) 4

(b) 8

(d) 12

(e) 13

Question 12

I am shuffling a deck of cards but I accidentally drop a card on the ground every now and then. After a while, I notice that I have dropped five cards. From above, the five cards look like one of the following pictures. Which picture could it be?

Question 13

Kayla had six apples. She cut them all into quarters and shared them equally between her three brothers and herself. How many apples do they each receive?

(a) 1

(b) 3

(d) $1 \frac{1}{3}$

(e) $1 \frac{1}{2}$

Question 14

Five boxes are compared on a balance.

Which of the five boxes is lightest?

(c)

(d) D

(e) E

Question 15

Lydia is saving for a cricket bat. The sports shop has the bat she wants for $$ 56$ and her grandfather has promised to pay half the price. She has saved $$ 16$. How much more does she need to save before she can buy the bat?

(a) $$ 4$

(b) $$ 12$

(d) $$ 28$

(e) $$ 36$

Question 16

Five cards with digits $1,2,3,4$ and 9 are arranged to form the largest possible 5 -digit even number. Which digit is in the tens place?

(a) 1

(b) 2

(d) 4

(e) 9

Question 17

Each letter in this grid stands for a number from 1 to 6 . The numbers outside the grid are the sums of the values of all the letters in each row or column. For example, in the first column, the values of $M, L, L$ and $R$ add to 16 .

$M$	$M$	$F$	$F$	\multirow{2}{*}{\begin{tabular}{c} 16
10

$L$ & $H$ & $U$ & $H$ & \cline { 1 - 4 } $L$ & $F$ & $U$ & $H$ & $R$ & $R$ & $R$ & $H$ & \cline { 1 - 4 } 11 & & & & \cline { 1 - 3 } 16 & 13 & 16 & 5 & \cline { 1 - 3 } What is the value of the letter $L$ ?

(a) 1

(b) 2

(d) 5

(e) 6

Question 18

Greg is 19 years old, Karin is 26 and Anthony is 31 . In how many years from now will their ages add to 100 ?

(a) 6

(b) 8

(d) 24

(e) 26

Question 19

Mr Northrop's class has students from Ainslie, Turner, Downer, Watson and Dickson. He made a chart showing how many live in each suburb. Unfortunately his dog tore the bottom of the chart, leaving only the last few letters of each suburb. He forgot the order of the suburbs on the chart, but he remembered that more students live in Downer than Watson. How many students live in Turner?

(a) 3

(b) 5

(d) 7

(e) 9

Question 20

Alexander's pen leaked on his addition homework, covering up three of the digits in the calculation shown. How many different possibilities are there for the correct working?

(a) 2

(b) 3

(d) 5

(e) 6

Questions 21 to 25, 5 marks each

Question 21

Here are four sentences and their translations into Windarian, an invented language. The two lists are not in the same order.

English

Mum likes apples.

Dad likes oranges.

Brother loves apples.

Sister loves apples.

Windarian

Ato bem kito.

Awe tum kete.

Eke bem kete.

Alo tum kete.

How should we translate the sentence 'Mum loves oranges'?

(a) Awe tum kete

(b) Ato bem kito

(d) Awe bem kete

(e) Eke bem kito

Question 22

The biscuit section in a cookbook has 6 pages. The sum of all the page numbers in this section is 147 . What is the number of the last page in this section of the book?

(a) 26

(b) 27

(d) 29

(e) 30

Question 23

Six white cubes are joined together as shown. The model is then painted blue all over. When the model is pulled apart, how many faces of these cubes are still white?

(a) 4

(b) 5

(d) 10

(e) 13

Question 24

Three gears are connected as shown. The two larger gears have 20 teeth each and the smaller gear has 10 teeth. The middle gear is rotated half a turn in the

direction of the arrows, turning the M upside down. What do the three gears look like after this rotation?

Question 25

In a dice game, Yasmin rolls 5 standard dice, all at once. She needs to roll a full house, which has a triple of one number and a pair of a different number. How many different full house rolls are possible?

(a) 2

(b) 5

(d) 25

(e) 30 For questions 26 to 30 , shade the answer as a whole number 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.

Question 26

This is a magic square, so that all rows, columns and diagonals add up to the same sum. Some numbers are already filled in. When we complete it and multiply the numbers in the three shaded squares, what do we get?

16	a	2
10	c	8
b	7	12
4	15	1

Question 27

Hayden saved $$ 1420$ and Mitchell saved $$ 505$. After they each spent an equal amount of money, Hayden had 4 times as much money as Mitchell. In dollars, how much did each of them spend?

Question 28

The block pattern below has 1 block in the first tower, 4 blocks in the second tower, 9 blocks in the third tower and so on. How many blocks are needed to make all of the first ten towers in this pattern?

Question 29

Verity has 6 cards with digits $1,2,3,4,5$ and 6 . She arranges them to form three 2-digit numbers. Only her first number is a multiple of 4 . Only her second number is a multiple of 5 . Only her third number is a multiple of 6 . What is the answer when she multiplies her first two numbers and then adds her third number?

Question 30

I want to place the numbers 1 to 10 in this diagram, with one number in each circle. On each of the three sides, the four numbers add to a side total, and the three side totals are all the same. What is the smallest number that this side total could be?

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Australian Mathematics Competition - 2013 - Junior years - Grade 7 & 8 - Questions and Solutions

Problem 1:

$1999+24$ is equal to
(A) 1923
(B) 2003
(C) 2013
(D) 2023
(E) 2113

Problem 2:

$P Q R$ is a straight line. Find the value of $x$.

(A) 40
(B) 90
(C) 100
(D) 110
(E) 120

Problem 3:

The value of the fraction $\frac{1}{2}$ is closest to
(A) 0.45
(B) 0.6
(C) $\frac{1}{3}$
(D) $\frac{5}{8}$
(E) $\frac{2}{5}$

Problem 4:

Which of the following is equal to 20 ?
(A) $3+2 \times 4$
(B) $(9+5) \times 2-4 \times 2$
(C) $10^2$
(D) $20+20 \div 2$
(E) $10 \div 2$

Problem 5:

How many minutes are there between $8: 37 \mathrm{am}$ and $10: 16 \mathrm{am}$ ?
(A) 39
(B) 79
(C) 99
(D) 141
(E) 179

Problem 6:

Three squares each with an area of $25 \mathrm{~cm}^2$ are placed side by side to form a rectangle. The perimeter, in centimetres, of the rectangle is
(A) 20
(B) 36
(C) 40
(D) 75
(E) 100

Problem 7:

If every digit of a whole number is either a 3 or a 5 , the number will always be
(A) divisible by 3
(B) divisible by 5
(C) prime
(D) even
(E) odd

Problem 8:

(P) is the point at 0.56 and $Q$ is the point at 1.2 on a number line. The point which is halfway between $P$ and $Q$ is at
(A) 0.34
(B) 0.64
(C) 0.83
(D) 0.88
(E) 0.93

Problem 9:

If triangle $A B C$ is isosceles with $\angle A=40^{\circ}$, what are all of the possible values for $\angle B$ ?
(A) $40^{\circ}$
(B) $40^{\circ}$ and $70^{\circ}$
(C) $40^{\circ}$ and $100^{\circ}$
(D) $70^{\circ}$ and $100^{\circ}$
(E) $40^{\circ}, 70^{\circ}$ and $100^{\circ}$

Problem 10:

In Gwen's classroom, the desks are arranged in a grid. Each row has the same number of desks. Gwen's desk is third from the front, second from the back and has one desk to the left and four to the right. How many desks are there?
(A) 20
(B) 24
(C) 25
(D) 28
(E) 30

Problem 11:

William travels to school in two different ways. Either he walks to school and takes the bus home, or he takes the bus to school and walks home. In each case his total travelling time is 40 minutes. If he were to take the bus both ways, his total travelling time would be 20 minutes. How many minutes would it take if he walked both ways?
(A) 30
(B) 40
(C) 50
(D) 60
(E) 80

Problem 12:

The opposite faces on a standard dice add to give a total of 7. The game of Corners is played by rolling a dice and then choosing a vertex of the dice with your eyes closed. For example, the score for the vertex chosen below would be $4+5+6=15$.

Which of the following scores is NOT possible when playing Corners?
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10

Problem 13:

A piece of paper in the shape of an equilateral triangle has one corner folded over, as shown.

What is the value of $x$ ?
(A) 60
(B) 70
(C) 80
(D) 90
(E) 100

Problem 14:

Beginning at the point $A$, Joel draws the spiral pattern of line segments below on a 1 cm grid. If he continues this pattern, how long, in centimetres, is the 97 th segment?

(A) 46
(B) 47
(C) 48
(D) 49
(E) 50

Problem 15:

Sixteen discs are arranged in four rows of four. The discs have a number on one side and are either red or green on the other. The number shows how many discs touching that disc have green on the other side.

Which of the following statements is true?
(A) All of the rows have the same number of green discs.
(B) Row one has more green discs than any other row.
(C) Row two has more green discs than any other row.
(D) Row three has fewer green discs than any other row.
(E) Row four has fewer green discs than any other row.

Problem 16:

While shopping this week I misread my shopping list and bought 5 loaves of bread and 2 bottles of milk. So I went back to the supermarket, got a full refund, and bought 2 loaves of bread and 5 bottles of milk. This cost me $\$ 4.20$ less than my first purchase. How do the prices of bread and milk compare?
(A) A loaf of bread costs $\$ 1.40$ more than a bottle of milk.
(B) A loaf of bread costs $\$ 0.60$ more than a bottle of milk.
(C) A loaf of bread costs $\$ 0.42$ more than a bottle of milk.
(D) A loaf of bread costs $\$ 0.60$ less than a bottle of milk.
(E) A loaf of bread costs $\$ 1.40$ less than a bottle of milk.

Problem 17:

Starting with the number 0 on my calculator, I do a calculation in five steps. At each step, I either add 1 or multiply by 2 . What is the smallest number that cannot be the final result?
(A) 11
(B) 10
(C) 9
(D) 8
(E) 7

Problem 18:

The three squares in the figure below are the same size. Find the value, in degrees, of $\angle A M T$.

(A) $45^{\circ}$
(B) $50^{\circ}$
(C) $55^{\circ}$
(D) $60^{\circ}$
(E) $75^{\circ}$

Problem 19:

Eight $1 \times 1$ square tiles are laid as shown.

Two more $1 \times 1$ tiles are added, so that at least one side of each new tile is shared with a side of the original shape. Several different perimeter lengths are now possible. What is the sum of the shortest and longest possible perimeter of the modified shape?
(A) 28
(B) 30
(C) 32
(D) 34
(E) 36

Problem 20:

In the triangle $P Q R, S$ is a point on $P R$ such that $P Q S$ and $S Q R$ are both isosceles triangles (as shown). Angle $Q P S$ is equal to angle $S Q R$.

What is the value of $x$ ?
(A) 30
(B) 36
(C) 40
(D) 45
(E) 48

Problem 21:

A biologist has a set of cages in a $4 \times 4$ array. He wants to put one mouse (black or white) into each cage in such a way that each mouse has at least one neighbour of each colour (neighbouring cages share a common wall).

The black mice are more expensive, so he wants to use as few of them as possible. What is the smallest number of black mice that he needs?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

Problem 22:

Two discs have different numbers on each side as shown.

The discs are flipped and they land on a table. The two numbers on the sides that are showing are added. If the possible sums that can be obtained in this way are $8,9,10$ and 11 , the sum $b+c+d$ is
(A) 8
(B) 18
(C) 20
(D) 27
(E) 30

Problem 23:

An oddie number is a 3 -digit number with all three digits odd. The number of oddie numbers divisible by 3 is
(A) 20
(B) 26
(C) 29
(D) 41
(E) 42

Problem 24:

Consider the following $4 \times 4$ squares with a $1 \times 1$ square deleted $shown in black$.

Consider tiling the squares $\mathrm{P}, \mathrm{Q}$ and R using tiles like the one below.

Which of the following statements is true?
(A) Only P can be tiled this way.
(B) Only Q can be tiled this way.
(C) Only R can be tiled this way.
(D) Only P and Q can be tiled this way.
(E) All the shapes can be tiled this way.

Problem 25:

A number is formed by writing the numbers 1 to 30 in order as shown.

\[
\text { 12345678910111213……… } 2930
\]

Simeon removed 45 of these 51 digits leaving 6 in their original order to make the largest 6-digit number possible. What is the sum of the digits of this number?
(A) 33
(B) 38
(C) 41
(D) 43
(E) 51

Problem 26:

Consider a sequence of letters where each letter is $A$ or $B$. We call the sequence stable if, when we tally the number of $A$ s and the number of $B \mathrm{~s}$ in the sequence, working from left to right, the difference is never greater than one. For example, the sequence $A B B A B A$ is stable but the sequence $A A B B A B$ is not, because after counting the first two letters, the difference is two. How many stable sequences with eighteen letters are there?

Problem 27:

Whenever Callum reads a date like $1 / 8 / 2013$, he incorrectly interprets it as two divisions, with the second one evaluated before the first one:

\[
1 \div(8 \div 2013)=251 \frac{5}{8}
\]

For some dates, like this one, he does not get an integer, while for others, like $28 / 7 / 2013$, he gets $28 \div(7 \div 2013)=8052$, an integer. How many dates this year $day/month/year$ give him an integer?

Problem 28:

What is the smallest positive integer that can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers and the sum of eleven consecutive integers?

Problem 29:

Each of the four circles below has a whole number value. $X$ is the value of the top-left circle. A number written on the figure indicates the product of the values of the circles it lies within. What is the value of $X+k$ ?

Problem 30:

Three different non-zero digits are used to form six different 3-digit numbers. The sum of five of them is 3231 . What is the sixth number?