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?

Australian Mathematics Competition - 2023 - Middle Primary - Grade 3 & 4 - Questions and Solutions

Problem 1:

What is the total number of petals on all 5 flowers?
(A) 10
(B) 15
(C) 20
(D) 25
(E) 50

Problem 2:

\[
2+3+7+8=
\]

(A) 10
(B) 20
(C) 30
(D) 40
(E) 50

Problem 3:

Which one of these shapes is a rectangle?

Problem 4:

Which digital clock time matches the time shown on the clock face?

Problem 5:

Emma has a bag containing 5 red, 4 yellow, 1 black and 2 blue buttons. When she chooses 1 button at random, what colour is it most likely to be?
(A) green
(B) blue
(C) black
(D) yellow
(E) red

Problem 6:

What fraction of the circle is part A ?

(A) one-half
(B) one-third
(C) two-thirds
(D) one-quarter
(E) three-quarters

Problem 7:

In a board game, Nik rolls three standard dice, one at a time. He needs his three rolls to add to 12.
His first two dice rolls are 5 and 3 . What does he need his third roll to be?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Problem 8:

Which shape has only one line of symmetry?

Problem 9:

Anastasia Ant is on a sheet of wire mesh with 10 cm squares. She can only move along the wires. She moves 10 cm east, then 30 cm north and 20 cm west. What is the least possible distance she needs to move to return to her starting point?

(A) 10 cm
(B) 20 cm
(C) 30 cm
(D) 40 cm
(E) 50 cm

Problem 10:

A piece of card is cut out and labelled as shown in the diagram. It is folded along the dotted lines to make a box without a top. Which letter is on the bottom of the box?
(A) A
(B) B
(C) C
(D) D
(E) E

Problem 11:

Jiang is threading beads onto a string to make a necklace. He decides to use a sequence, starting with a red bead, then a yellow bead and a blue bead. He repeats this pattern until he has 20 beads on the necklace.
How many red beads will he use?

(A) 5
(B) 6
(C) 7
(D) 8
(E) 9

Problem 12:

How many people played tennis on the busiest day of the week at the Paradise North Tennis Club?
(A) 45
(B) 50
(C) 55
(D) 60
(E) 65

Problem 13:

When this puzzle is complete, each of the four lines will add up to 14 . What will the four missing numbers add to?
(A) 10
(B) 14
(C) 15
(D) 16
(E) 18

Problem 14:

Which one of the following could give an answer of $25 ?$
(A) an odd number plus an odd number
(B) an even number multiplied by an even number
(C) an even number divided by an odd number
(D) an odd number minus an even number
(E) an even number multiplied by an odd number

Problem 15:

This bottle holds 4 glasses of water.

Which one of the following holds the most water?

Problem 16:

Fido needs to be weighed at the vet, but he won't sit still. So the vet weighs himself and then weighs himself holding Fido.

How much does Fido weigh?
(A) 23 kg
(B) 26 kg
(C) 28 kg
(D) 30 kg
(E) 32 kg

Problem 17:

Grandma adds 3 spoons of sugar to the juice of 10 lemons to make lemonade. How many lemons are needed if 15 spoons of sugar are used to make Grandma's recipe?
(A) 20
(B) 22
(C) 25
(D) 30
(E) 50

Problem 18:

Janus is making patterns using square tiles.
Each pattern is made by copying the previous pattern, then adding new tiles around the outside edges.
The new tiles alternate in colours, as shown.

What will be the total number of tiles in Pattern 5?
(A) 25
(B) 33
(C) 37
(D) 40
(E) 41

Problem 19:

My laptop's battery meter tells me what fraction of the battery charge is left. I used the laptop for 4 hours and 20 minutes and it went from

What would the reading be after using the laptop for another 130 minutes?

Problem 20:

Syed's mother had some money to share with her family.
She gave one-quarter of her money to Syed.
Then she gave one-third of what was left to Ahmed.
Then she gave one-half of what was left to Raiyan.
She was left with $\$ 15$, which she kept for herself.
How much money did Syed's mother have to start with?
(A) $\$ 30$
(B) $\$ 45$
(C) $\$ 60$
(D) $\$ 90$
(E) $\$ 120$

Problem 21:

In 2023, Janine's birthday is on a Thursday. Ngoc's birthday is one month later, meaning that it is on the same day-number in the next month. Ngoc's birthday is not on a Saturday or a Sunday. Which day of the week is Ngoc's birthday?
(A) Monday
(B) Tuesday
(C) Wednesday
(D) Thursday
(E) Friday

Problem 22:

Sally empties her piggy bank to see how much she has saved. She has 20 coins altogether and the total value of her coins is (\$ 3.80). She notices that she has twice as many 50 c coins as 20 c coins. The remaining coins are 10c coins. How many 10c coins does she have?
(A) 2
(B) 4
(C) 8
(D) 11
(E) 14

Problem 23:

Three different numbers from 1 to 20 are chosen. They add to 20 . Two of the numbers are 2 apart. How many possibilities are there for the largest number?
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10

Problem 24:

Estelle is making decorations shaped like the 8-pointed star shown. She folds a square of paper to make a triangle with 8 layers as shown.

How could she cut the triangle so that the unfolded shape is the star?

Problem 25:

Tom is digging holes in his garden for his new plants. As he is working he slows down. Each new hole takes a minute longer to dig than the last hole.
The first 5 holes have taken him 35 minutes in total. How long will it take him to dig the next 2 holes?
(A) 11 minutes
(B) 17 minutes
(C) 21 minutes
(D) 24 minutes
(E) 25 minutes

Problem 26:

To unlock a security door, Daniyal enters a code on the keypad shown. To make it easy to remember:

the code has 10 digits
the code uses every digit exactly once
the code steps from button to button only if the two buttons touch
the code is as large as possible.

What are the last 3 digits in Daniyal's code?

Problem 27:

In this puzzle, $\triangle$, $\square$ and represent different nonzero digits. What is the three-digit number $\triangle \square \bigcirc$ ?

Problem 28:

Ollie arranges the numbers 1 to 6 to form three 2-digit numbers.
Exactly one of his numbers divides by 3 but not divides by 4 , and exactly one other divides by 4 . None of them divide by 5 .
The three numbers add to 93 .
What is the number which does not divide by either 3 or 4 ?

Problem 29:

A table in the shape of a trapezium can seat 5 people. When two tables are put together in a row, 8 people can be seated. What is the smallest number of trapezium tables required to seat 2023 people if they are all placed in a row?

Problem 30:

Yifan has a construction set consisting of red, blue and yellow rods. All rods of the same colour are the same length, but differently coloured rods are different lengths. She wants to make quadrilaterals using these rods.

When she uses two red, one blue and one yellow rod, the perimeter of the quadrilateral is 36 cm .
When she uses two blue, one red and one yellow rod, the perimeter is 35 cm .
When she uses two yellow, one blue and one red rod, the perimeter is 33 cm .

What number do you get when you multiply the lengths of one red rod, one blue rod and one yellow rod?

Australian Mathematics Competition - 2015 - Junior years - Grade 7 & 8 - Questions and Solutions

Problem 1:

$2015+201.5$ equals
(A) 2036.5
(B) 2116.5
(C) 2225.5
(D) 2216.5
(E) 2115.5

Problem 2:

The value of $x$ in the diagram is
(A) $100^{\circ}$
(B) $130^{\circ}$
(C) $110^{\circ}$
(D) $120^{\circ}$
(E) $90^{\circ}$

Problem 3:

The trip to school takes 23 minutes. I need to be at school at 9:05 am. The latest I can leave home is
(A) 8:46 am
(B) $8: 37 \mathrm{am}$
(C) 8:52 am
(D) 8:42 am
(E) 8:48 am

Problem 4:

What is the value of 100 twenty-cent coins?
(A) $\$ 20$
(B) $\$ 10$
(C) $\$ 200$
(D) $\$ 2$
(E) $\$ 100$

Problem 5:

What is the area of this triangle in square centimetres?
(A) 10
(B) 12
(C) 14
(D) 7
(E) 6

Problem 6:

When the bell rang, there were 3 teachers and 6 students in the classroom. Several students arrived after the bell. Once everyone had arrived, there were 4 students for every teacher. How many students arrived after the bell?
(A) 18
(B) 12
(C) 6
(D) 3
(E) 9

Problem 7:

A movie lasts for $2 \frac{1}{3}$ hours. The movie is shown in two equal sessions.
For how many minutes does each session last?
(A) 85
(B) 70
(C) 80
(D) 65
(E) 75

Problem 8:

Four unit squares are laid out in five different arrangements as shown below. Which one has the largest perimeter?

Problem 9:

Ari, Bryce, Cy and Eric are members of our school's basketball team. Ari is 186 cm tall. He is 14 cm taller than Bryce who in turn is 6 cm shorter than Cy. Eric is 11 cm taller than Cy . Eric's height is
(A) 183 cm
(B) 205 cm
(C) 178 cm
(D) 189 cm
(E) 177 cm

Problem 10:

Ana, Ben, Con, Dan and Eve are sitting around a table in that order. Ana calls out the number 1, then Ben calls out the number 2, then Con calls out the number 3, and so on. After a person calls out a number, the next person around the table calls out the next number.
Anyone who calls out a multiple of 7 must immediately leave the table.
Who is the last person remaining at the table?
(A) Ana
(B) Ben
(C) Con
(D) Dan
(E) Eve

Problem 11:

$\frac{5}{19}$ of 38 is equal to
(A) 76
(B) 19
(C) $\frac{2}{5}$
(D) $2 \frac{1}{2}$
(E) 10

Problem 12:

The diagram shows a circle and a square with the same centre.
What fraction of the circle is shaded?
(A) $\frac{5}{8}$
(B) $\frac{4}{7}$
(C) $\frac{3}{5}$
(D) $\frac{6}{11}$
(E) $\frac{2}{3}$

Problem 13:

In the addition below $x, y$ and $z$ represent three different digits.

What is the value of $x+y+z$ ?
(A) 9
(B) 8
(C) 10
(D) 7
(E) 6

Problem 14:

A cube has the letters $\mathrm{A}, \mathrm{C}, \mathrm{M}, \mathrm{T}, \mathrm{H}$ and S on its six faces. Here are two views of this cube.

Which one of the following could be a third view of the same cube?

(A)

(B)

(C)

(D)

(E)

Problem 15:

Five students are to be photographed in a row with the tallest in the centre and the shortest two at the ends. If no two students are the same height, how many different arrangements are possible?
(A) 6
(B) 2
(C) 10
(D) 5
(E) 4

Problem 16:

Three boys and three girls all celebrate their birthday today, but they are each different ages. The youngest is 1 year old. The sum of the ages of the three girls is the same as the sum of the ages of the three boys. What is the smallest possible total of all six ages?
(A) 22
(B) 24
(C) 28
(D) 21
(E) 26

Problem 17:

Jenna measures three sides of a rectangle and gets a total of 80 cm . Dylan measures three sides of the same rectangle and gets a total of 88 cm . What is the perimeter of the rectangle?
(A) 112 cm
(B) 132 cm
(C) 96 cm
(D) 168 cm
(E) 156 cm

Problem 18:

Jim is running five laps of the school oval. When he is $\frac{3}{4}$ of the way round his fourth lap, what fraction of his run has he completed?
(A) $\frac{2}{3}$
(B) $\frac{1}{2}$
(C) $\frac{3}{4}$
(D) $\frac{4}{5}$
(E) $\frac{5}{6}$

Problem 19:

How many two-digit numbers have the property that the sum of the digits is a perfect square?
(A) 15
(B) 18
(C) 13
(D) 19
(E) 17

Problem 20:

On this cube, opposite faces add to the same sum and all faces are prime numbers. (Note that 1 is not prime.) What is the smallest possible total of the faces which cannot be seen?
(A) 41
(B) 35
(C) 45
(D) 47
(E) 37

Problem 21:

A recipe requires 2 kg sugar, 4 kg butter, and 6 kg flour to make 8 cakes. How many cakes can you make if you have 9 kg sugar, 17 kg butter and 28 kg flour?
(A) 40
(B) 34
(C) 37
(D) 32
(E) 36

Problem 22:

Two ordinary dice are rolled. The two resulting numbers are multiplied together to create a score. The probability of rolling a score that is a multiple of six is
(A) $\frac{1}{6}$
(B) $\frac{5}{12}$
(C) $\frac{1}{4}$
(D) $\frac{1}{3}$
(E) $\frac{1}{2}$

Problem 23:

Jill and Jack are exercising at a beach. They both start from the car park at one end of the beach. Jill runs at a constant speed and Jack walks at a constant speed. When Jill turns at the end of the beach to run back, she notices that Jack is then halfway along the beach. How far along the beach will Jack be when Jill next passes him?
(A) Two-thirds of the way
(B) Five-sixths of the way
(C) Three-quarters of the way
(D) Five-eighths of the way
(E) Seven-eighths of the way

Problem 24:

The country of Numismatica has six coins of the following denominations: 1 cent, 2 cents, 4 cents, 10 cents, 20 cents and 40 cents.
Using the coins in my pocket, I can pay exactly for any amount up to and including 200 cents.
What is the smallest number of coins I could have?
(A) 12
(B) 10
(C) 11
(D) 9
(E) 8

Problem 25:

In the diagram, $P T=T S=S Q=Q R$, $\angle P Q R=90^{\circ}$ and $\angle Q P R=x^{\circ}$.
Then (x) is equal to
(A) 20
(B) 25
(C) 27.5
(D) 22.5
(E) 30

Problem 26:

I write down three different positive whole numbers that add to 96 . The sum of any two is divisible by the third.
What is the largest of these three numbers?

Problem 27:

At a football match, one-third of spectators support the Reds and the rest support the Blues. At half-time 345 Blues supporters leave because their team is losing, and the remaining Blues supporters now make up one-third of the total. How many Reds supporters are there?

Problem 28:

A $3 \times 2$ flag is divided into six squares, as shown. Each square is to be coloured green or blue, so that every square shares at least one edge with another square of the same colour.
In how many different ways can this be done?

Problem 29:

Zoltan has a list of whole numbers, all larger than 0 but smaller than 1000. He notices that every number in his list is either one-third of another number in the list or three times another number in the list. What is the largest number of different whole numbers that can be on Zoltan's list?

Problem 30:

In a stack of logs, each row has exactly one fewer log than the row below. With 9 logs, the tallest possible stack is shown.

With 2015 logs, how many rows are there in the tallest possible stack?

Australian Mathematics Competition - 2016 - Upper Primary - Grade 5 & 6 - Questions and Solutions

Problem 1:

Which of these numbers is the smallest?
(A) 655
(B) 566
(C) 565
(D) 555
(E) 556

Problem 2:

Two pizzas are sliced into quarters. How many slices will there be?
(A) 2
(B) 10
(C) 6
(D) 8
(E) 16

Problem 3:

Join the dots $P, Q, R$ to form the triangle $P Q R$.

How many dots lie inside the triangle $P Q R$ ?
(A) 13
(B) 14
(C) 15
(D) 17
(E) 18

Problem 4:

$0.3+0.4$ is
(A) 0.07
(B) 0.7
(C) 0.12
(D) 0.1
(E) 7

Problem 5:

Lee's favourite chocolates are 80 c each. He has five dollars to spend. How many of these chocolates can he buy?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

Problem 6:

Ten chairs are equally spaced around a round table. They are numbered 1 to 10 in order. Which chair is opposite chair 9 ?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Problem 7:

In a piece of music, a note like is worth one beat, is worth half a beat, is worth
2 beats and is worth 4 beats. How many beats are in the following piece of music?

(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

Problem 8:

Phoebe put her hand in her pocket and pulled out 60 cents. How many different ways could this amount be made using $10 \mathrm{c}, 20 \mathrm{c}$ and 50 c coins?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Problem 9:

Which of these containers is currently holding the most water?

Problem 10:

Which of these shapes has the most axes of symmetry (mirror lines)?

Problem 11:

A sailor coiled a rope on his ship's deck, and some paint was spilled across half of it. What did the rope look like when it was uncoiled?

Problem 12:

If the area of the tangram shown is 64 square centimetres, what is the area in square centimetres of the small square?
(A) 32
(B) 24
(C) 16
(D) 8
(E) 4

Problem 13:

For each batch of 25 biscuits, Jack uses $2 \frac{1}{2}$ packets of chocolate chips. How many packets does he need if he wants to bake 200 biscuits?
(A) 20
(B) 8
(C) 80
(D) 10
(E) 50

Problem 14:

Which one of the following is correct?
(A) Two even numbers add to an odd number.
(B) An odd number minus an odd number is always odd.
(C) Adding 2 odd numbers and an even number is always odd.
(D) Adding 3 odd numbers is always odd.
(E) An odd number multiplied by an odd number always equals an even number.

Problem 15:

The perimeter of the outer square is 36 cm , and the perimeter of the inner square is 20 cm .
If the four rectangles are all identical, what is the perimeter of the shaded rectangle in centimetres?
(A) 12
(B) 14
(C) 24
(D) 20
(E) 18

Problem 16:

George has a new lock that opens if the four numbers $1,2,3$ and 4 are pressed once each in the correct order.
If the first number must be larger than the second number, how many combinations are possible?
(A) 10
(B) 12
(C) 15
(D) 18
(E) 20

Problem 17:

A straight cut is made through the hexagon shown to create two new shapes. Which of the following could not be made?
(A) one triangle and one hexagon
(B) two pentagons
(C) two quadrilaterals
(D) one quadrilateral and one pentagon
(E) one triangle and one quadrilateral

Problem 18:

The numbers $3,9,15,18,24$ and 29 are divided into two groups of 3 numbers and each group is added. The difference between the two sums (totals) of 3 numbers is as small as possible. What is the smallest difference?
(A) 0
(B) 1
(C) 2
(D) 5
(E) 8

Problem 19:

Benny built a magic square using the numbers from 1 to 16 , where the numbers in each row, each column and each diagonal add up to the same total.
What number does he place at the X ?
(A) 16
(B) 15
(C) 17
(D) 11
(E) 14

Problem 20:

Andy has a number of red, green and blue counters.
He places eight counters equally spaced around a circle according to the following rules:

No two red counters will be next to each other.
No two green counters will be diagonally opposite each other.
As few blue counters as possible will be used.

How many blue counters will Andy need to use?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Problem 21:

I have five coloured discs in a pile as shown.
I take the top two discs and put them on the bottom (with the red dise still on top of the blue disc).
Then I again take the top two discs and put them on the bottom.
If I do this until I have made a total of 21 moves, which dise will be on the bottom?

(A) red
(B) blue
(C) green
(D) yellow
(E) orange

Problem 22:

A zoo keeper weighed some of the animals at Melbourne Zoo. He found that the lion weighs 90 kg more than the leopard, and the tiger weighs 50 kg less than the lion. Altogether the three animals weigh 310 kg . How much does the lion weigh?
(A) 180 kg
(B) 150 kg
(C) 140 kg
(D) 130 kg
(E) 100 kg

Problem 23:

Adrienne, Betty and Cathy were the only three competitors participating in a series of athletic events. In each event, the winner gets 3 points, second gets 2 points and third gets 1 point. After the events, Adrienne has 8 points, Betty has 11 points and Cathy has 5 points. In how many events did Adrienne come second?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Problem 24:

Jane and Tom are comparing their pocket money. Jane has as many 5c coins as Tom has 10 c coins and as many 10 c coins as Tom has 20 c coins. However, Jane has as many 50c coins as Tom has 5 c coins.
They have no other coins and they find that they each have the same amount of money.
What is the smallest number of coins they each can have?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

Problem 25:

A tuckshop has two jars of cordial mixture.
Jar A is $30 \%$ cordial, while Jar B is $60 \%$ cordial.
Some of Jar A is mixed with some of Jar B to make 18 litres of $50 \%$ cordial.
How many litres from Jar A are used?
(A) 9
(B) 12
(C) 4
(D) 3
(E) 6

Problem 26:

Qiang, Rory and Sophia are each wearing a hat with a number on it. Each adds the two numbers on the other two hats, giving totals of 11,17 and 22 . What is the largest number on a hat?

Problem 27:

The number 840 is the 3 -digit number with the most factors. How many factors does it have?

Problem 28:

A class has 2016 matchsticks. Using blobs of modelling clay to join the matches together, they make a long row of cubes. This is how their row starts.

They keep adding cubes to the end of the row until they don't have enough matches left for another cube. How many cubes will they make?

Problem 29:

You have an unlimited supply of five different coloured pop-sticks, and want to make as many different coloured equilateral triangles as possible, using three sticks.
One example is shown here.
Two triangles are not considered different if they are rotations or reflections of each other.
How many different triangles are possible?

Problem 30:

Today my three cousins multiplied their ages together and it came to 2016. This day last year their ages multiplied to 1377 .
When they multiplied their ages together 2 years ago today, what was their answer?

Australian Mathematics Competition - 2015 - Upper Primary - Grade 5 & 6 - Questions and Solutions

Problem 1:

What does the digit 1 in 2015 represent?

(A) One
(B) Ten
(C) One hundred
(D) One thousand
(E) Ten thousand

Problem 2:

What is the value of 10 twenty-cent coins?
(A) $\$ 1$
(B) $\$ 2$
(C) $\$ 5$
(D) $\$ 20$
(E) $\$ 50$

Problem 3:

What temperature does this thermometer show?
(A) $25^{\circ}$
(B) $38^{\circ}$
(C) $27^{\circ}$
(D) $32^{\circ}$
(E) $28^{\circ}$

Problem 4:

Which number do you need in the box to make this number sentence true?

(A) 34
(B) 44
(C) 46
(D) 64
(E) 84

Problem 5:

Which number has the greatest value?
(A) 1.3
(B) 1.303
(C) 1.31
(D) 1.301
(E) 1.131

Problem 6:

The perimeter of a shape is the distance around the outside. Which of these shapes has the smallest perimeter?

Problem 7:

The class were shown this picture of many dinosaurs. They were asked to work out how many there were in half of the picture.

Simon wrote $6 \times 10$.
Carrie wrote $5 \times 12$.
Brian wrote $10 \times 12 \div 2$.
Rémy wrote $10 \div 2 \times 12$. Who was correct?
(A) All four were correct
(B) Only Simon
(C) Only Carrie
(D) Only Brian
(E) Only Rémy

Problem 8:

In the diagram, the numbers $1,3,5,7$ and 9 are placed in the squares so that the sum of the numbers in the row is the same as the sum of the numbers in the column.
The numbers 3 and 7 are placed as shown. What could be the sum of the row?
(A) 14
(B) 15
(C) 12
(D) 16
(E) 13

Problem 9:

To which square should I add a counter so that no two rows have the same number of counters, and no two columns have the same number of counters?
(A) A
(B) B
(C) C
(D) D
(E) E

Problem 10:

A half is one-third of a number. What is the number?
(A) three-quarters
(B) one-sixth
(C) one and a third
(D) five-sixths
(E) one and a half

Problem 11:

The triangle shown is folded in half three times without unfolding, making another triangle each time.

Which figure shows what the triangle looks like when unfolded?

Problem 12:

If $L=100$ and $M=0.1$, which of these is largest?
(A) $L+M$
(B) $L \times M$
(C) $L \div M$
(D) $M \div L$
(E) $L-M$

Problem 13:

You want to combine each of the shapes $A$ to $E$ shown below separately with the shaded shape on the right to make a rectangle.
You are only allowed to turn and slide the shapes, not flip them over. The finished pieces will not overlap and will form a rectangle with no holes.
For which of the shapes is this not possible?

Problem 14:

A plumber has 12 lengths of drain pipe to load on his ute. He knows that the pipes won't come loose if he bundles them so that the rope around them is as short as possible. How does he bundle them?

Problem 15:

The numbers 1 to 6 are placed in the circles so that each side of the triangle has a sum of 10 . If 1 is placed in the circle shown, which number is in the shaded circle?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Problem 16:

Follow the instructions in this flow chart.

(A) 57
(B) 63
(C) 75
(D) 81
(E) 84

Problem 17:

A square piece of paper is folded along the dashed lines shown and then the top is cut off.

The paper is then unfolded. Which shape shows the unfolded piece?

(A)

(B)

(C)

(D)

(E)

Problem 18:

Sally, Li and Raheelah have birthdays on different days in the week beginning Sunday 2 August. No two birthdays are on following days and the gap between the first and second birthday is less than the gap between the second and third. Which day is definitely not one of their birthdays?
(A) Monday
(B) Tuesday
(C) Wednesday
(D) Thursday
(E) Friday

Problem 19:

A square of side length 3 cm is placed alongside a square of side 5 cm .

What is the area, in square centimetres, of the shaded part?
(A) 22.5
(B) 23
(C) 23.5
(D) 24
(E) 24.5

Problem 20:

A cube has the letters $A, C, M, T, H$ and $S$ on its six faces. Here are two views of this cube.

Which one of the following could be a third view of the same cube?

(A)

(B)

(C)

(D)

(E)

Problem 21:

A teacher gives each of three students Asha, Betty and Cheng a card with a 'secret' number on it. Each looks at her own number but does not know the other two numbers. Then the teacher gives them this information.
All three numbers are different whole numbers and their sum is 13 . The product of the numbers is odd. Betty and Cheng now know what the numbers are on the other two cards, but Asha does not have enough information. What number is on Asha's card?
(A) 9
(B) 7
(C) 5
(D) 3
(E) 1

Problem 22:

In this multiplication, $L, M$ and $N$ are different digits. What is the value of $L+M+N$ ?
(A) 13
(B) 15
(C) 16
(D) 17
(E) 20

Problem 23:

A scientist was testing a piece of metal which contains copper and zinc. He found the ratio of metals was 2 parts copper to 3 parts zinc. Then he melted this metal and added 120 g of copper and 40 g of zinc into it, forming a new piece of metal which weighs 660 g .
What is the ratio of copper and zinc in the new metal?
(A) 1 part copper to 3 parts zinc
(B) 2 parts copper to 3 parts zinc
(C) 16 parts copper to 17 parts zinc
(D) 8 parts copper to 17 parts zinc
(E) 8 parts copper to 33 parts zinc

Problem 24:

Jason had between 50 and 200 identical square cards. He tried to arrange them in rows of 4 but had one left over. He tried rows of 5 and then rows of 6 , but each time he had one card left over. Finally, he discovered that he could arrange them to form one large solid square. How many cards were on each side of this square?
(A) 8
(B) 9
(C) 10
(D) 11
(E) 12

Problem 25:

Eve has $\$ 400$ in Australian notes in her wallet, in a mixture of 5,10 , 20 and 50 dollar notes.
As a surprise, Viv opens Eve's wallet and replaces every note with the next larger note. So, each $\$ 5$ note is replaced by a $\$ 10$ note, each $\$ 10$ note is replaced by a $\$ 20$ note, each $\$ 20$ note is replaced by a $\$ 50$ note and each $\$ 50$ note is replaced by a $\$ 100$ note.
Eve discovers that she now has $\$ 900$. How much of this new total is in $\$ 50$ notes?
(A) $\$ 50$
(B) $\$ 100$
(C) $\$ 200$
(D) $\$ 300$
(E) $\$ 500$

Problem 26:

Alex is designing a square patio, paved by putting bricks on edge using the basketweave pattern shown.
She has 999 bricks she can use, and designs her patio to be as large a square as possible. How many bricks does she use?

Problem 27:

There are many ways that you can add three different positive whole numbers to get a total of 12 . For instance, $1+5+6=12$ is one way but $2+2+8=12$ is not, since 2,2 and 8 are not all different.
If you multiply these three numbers, you get a number called the product.
Of all the ways to do this, what is the largest possible product?

Problem 28:

I have 2 watches with a 12 hour cycle. One gains 2 minutes a day and the other loses 3 minutes a day. If I set them at the correct time, how many days will it be before they next together tell the correct time?

Problem 29:

Problem 30:

The squares in a $25 \times 25$ grid are painted black or white in a spiral pattern, starting with black at the centre $\boldsymbol{*}$ and spiralling out.
The diagram shows how this starts. How many squares are painted black?

Australian Mathematics Competition - 2014 - Upper Primary - Grade 5 & 6 - Questions and Solutions

Problem 1:

What is the value of $7+6+4+3$ ?
(A) 20
(B) 19
(C) 18
(D) 17
(E) 16

Problem 2:

Which shape can make a pyramid if you fold along the dotted lines?

Problem 3:

A square of paper is folded in half to make a triangle, then in half to make a smaller triangle, then in half again to make an even smaller triangle.

How many layers of paper are in the final triangle?
(A) 3
(B) 4
(C) 6
(D) 8
(E) 12

Problem 4:

The year 5 students at my local school were surveyed to find which one of the four teams in the local football competition they followed.

How many more students followed the most popular team than followed the least popular team?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

Problem 5:

This week at my lemonade stand I sold $\$ 29$ worth of lemonade, but I had spent $\$ 34$ on lemons and $\$ 14$ on sugar. My total loss for the week was
(A) $\$ 1$
(B) $\$ 9$
(C) $\$ 19$
(D) $\$ 21$
(E) $\$ 29$

Problem 6:

A piece of paper is cut out and labelled as shown. It is folded along the dashed lines to form an open box and placed so that the top is open. Which letter is on the bottom of the box?
(A) A
(B) B
(C) C
(D) D
(E) E

Problem 7:

Which of the following gives an odd number?
(A) $12^2$
(B) $141-57$
(C) $36 \times 9$
(D) $308 \div 7$
(E) $\frac{1}{3}$ of 123

Problem 8:

A new block of flats is being built and Trudy is buying the letterbox numbers. The letterboxes are to be labelled from 190 to 212 . How many zeros will she need to buy?
(A) 22
(B) 13
(C) 12
(D) 10
(E) 4

Problem 9:

This $4 \times 4$ square grid can be covered by three shapes made from $1 \times 1$ squares. None of the shapes overlap.

If two of the shapes are

then the third shape is

Problem 10:

Given that $\frac{1001}{77}=13$, what is the value of $\frac{100.1}{770} ?$
(A) 0.13
(B) 1.3
(C) 13
(D) 130
(E) 1300

Problem 11:

These two squares, each with a side length of 10 cm , overlap as shown in the diagram. The shape of the overlap is also a square which has an area of 16 square centimetres. In centimetres, what is the perimeter of the combined shape?
(A) 40
(B) 56
(C) 64
(D) 80
(E) 92

Problem 12:

Six boys looked in the fridge and found this much leftover pizza. They shared the leftover pizza equally. How much of the whole pizza did each boy get?
(A) $\frac{1}{12}$
(B) $\frac{1}{8}$
(C) $\frac{1}{6}$
(D) $\frac{1}{4}$
(E) $\frac{1}{3}$

Problem 13:

Which of the shaded areas below is the largest?

(A) A
(B) B
(C) C
(D) D
(E) E

Problem 14:

A map, 40 cm wide and 20 cm high, is folded along the dashed lines indicated to form a $10 \mathrm{~cm} \times 10 \mathrm{~cm}$ square so that it just fits in its envelope. It is then pinned to a notice board.

Which one of the following could be the pattern of pinholes on the map?

Problem 15:

Sally thinks of a number, multiplies it by 2 , adds 2 , divides by 2 and then subtracts 2 . Her answer is 2 . What was her original number?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Problem 16:

The paint colour 'Roebourne Red' is made by mixing 2 parts yellow, 5 parts red and 1 part black.
If a batch of paint was made using 3 litres of yellow, how many litres of paint would be in the whole batch?
(A) 6
(B) 8
(C) 9
(D) 10 ,
(E) 12

Problem 17:

A square of paper has its corners folded in as shown to make a smaller square with an internal square, as shown on the right. What is the area of this internal square?

(A) $4 \mathrm{~cm}^2$
(B) $9 \mathrm{~cm}^2$
(C) $16 \mathrm{~cm}^2$
(D) $49 \mathrm{~cm}^2$
(E) $58 \mathrm{~cm}^2$

Problem 18:

Ancient Egyptian mathematicians used fractions but only with a numerator of 1 , such as $\frac{1}{2}$ or $\frac{1}{7}$ or $\frac{1}{14}$. They made other fractions by adding these fractions together. For example, $\frac{3}{4}$ was written as $\frac{1}{2}+\frac{1}{4}$. Which of the following is an Ancient Egyptian way of writing $\frac{11}{16}$ ?
(A) $\frac{1}{2}+\frac{1}{4}+\frac{1}{10}$
(B) $\frac{1}{2}+\frac{1}{8}+\frac{1}{16}$
(C) $\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$
(D) $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$
(E) $\frac{1}{3}+\frac{1}{6}+\frac{1}{2}$

Problem 19:

This symmetrical star is made up of two overlapping equilateral triangles of the same size. The area of the star is 60 square centimetres. What is the area of the shaded region in square centimetres?
(A) 30
(B) 36
(C) 42
(D) 45
(E) 48

Problem 20:

In this correctly worked addition, $P, Q, R$ and $S$ represent different digits, and all seven digits are different. What is the sum of $P, Q, R$ and $S$ ?
(A) 23
(B) 22
(C) 18
(D) 16
(E) 14

Problem 21:

In a competition between four people, Sally scored twice as many points as Brian and 30 points more than Corrie. Donna scored 50 points more than Brian. Which of the following statements is definitely true?
(A) Sally won the competition.
(B) Brian came last in the competition.
(C) Donna won the competition.
(D) Corrie beat Brian.
(E) Sally and Donna together scored more than Brian and Corrie.

Problem 22:

A plane was travelling at an altitude of 4500 metres for 20 minutes. It then climbed at a rate of 500 metres per minute for 5 minutes before descending at 350 metres per minute for 8 minutes. Which of the following graphs best describes the altitude of the plane over this time?

Problem 23:

Mrs Graham wants to fill her swimming pool for the summer. She knows that if she uses the backyard hose it will take 12 hours, or if she uses the frontyard hose it will take 4 hours. How long will it take if she uses both hoses together?
(A) 3 hours
(B) 2.5 hours
(C) 2 hours
(D) 1.5 hours
(E) 1 hour

Problem 24:

Four soccer teams played each other four times in a season of competition. Each winning team was awarded 3 points. Teams that had a draw were awarded 1 point each, and losing teams were awarded no points. The season ended with Kangaroos on 21 points, Kookaburras on 17 points, Koalas on 16 points and Crocodiles on 12 points. How many games ended in a draw?
(A) 7
(B) 6
(C) 5
(D) 4
(E) 3

Problem 25:

Five different whole numbers, chosen from the numbers from 1 to 30 , add up to 30 . What is the greatest possible value of the largest of these numbers?
(A) 6
(B) 10
(C) 15
(D) 20
(E) 26

Problem 26:

A 3 by 5 grid of dots is set out as shown. How many straight line segments can be drawn that join two of these dots and pass through exactly one other dot?

Problem 27:

A cube is made up of $1 \mathrm{~cm} \times 1 \mathrm{~cm} \times 1 \mathrm{~cm}$ blocks and measures $12 \mathrm{~cm} \times 12 \mathrm{~cm} \times 12 \mathrm{~cm}$. Sharyn is using the same set of blocks to make a set of stairs.
The picture shows how she started, making a set of stairs 4 blocks high, 4 blocks from front to back and 5 blocks wide.
Her finished set of stairs will use all the blocks and be 8 blocks high and 8 blocks from front to back. How many blocks wide will they be?

Problem 28:

Three whole numbers add up to 149 and multiply to give 987 . What is the largest of the three numbers?

Problem 29:

Which three-digit number is 11 times the sum of its digits?

Problem 30:

How many three-digit numbers are there in which one of the digits is the sum of the other two?

Australian Mathematics Competition - 2013 - Upper Primary - Grade 5 & 6 - Questions and Solutions

Problem 1:

What fraction of this rectangle is shaded?

(A) one-fifth
(B) two-fifths
(C) two-thirds
(D) one-third
(E) three-fifths

Problem 2:

Which of the following numbers is closest to zero?
(A) 0.03
(B) 0.048
(C) 0.009
(D) 0.005
(E) 0.02

Problem 3:

A 737 passenger aircraft has 3 seats on each side of a centre aisle in each row. It is designed to carry 150 passengers. How many rows of seats does it have?
(A) 50
(B) 37
(C) 33
(D) 32
(E) 25

Problem 4:

Alice has two 50c coins, three 20c coins and eight 5c coins. David has four 20c coins and six 10c coins. How much more money does Alice have than David?
(A) 40 c
(B) 60 c
(C) 80 c
(D) $\$ 1.40$
(E) $\$ 2.00$

Problem 5:

What is the difference between the largest and smallest 5-digit numbers which can be made from rearranging the 5 digit cards below?

(A) 41967
(B) 41976
(C) 44444
(D) 42024
(E) 41076

Problem 6:

At the supermarket, a regular packet of chips is 75 g . A packet of chips on special is advertised at one-third extra. How many grams does the special packet have?
(A) 50
(B) 78
(C) 100
(D) 125
(E) 150

Problem 7:

How many triangles are in the following picture?

(A) 9
(B) 10
(C) 13
(D) 14
(E) 17

Problem 8:

Jan doubles a number then adds two. Then she halves that number and subtracts two. Her final answer is six. What was her original number?
(A) 1
(B) 6
(C) 7
(D) 14
(E) 16

Problem 9:

On the number line, where should the fraction $\frac{1}{3}$ be placed?

(A) between 0 and 0.3
(B) between 0.3 and 0.4
(C) between 0.4 and 0.7
(D) between 0.7 and 0.8
(E) between 0.8 and 1

Problem 10:

Each triangle in the diagram is equilateral. What fraction of the largest triangle is shaded?

(A) $\frac{1}{4}$
(B) $\frac{15}{64}$
(C) $\frac{1}{3}$
(D) $\frac{3}{16}$
(E) $\frac{7}{32}$

Problem 11:

In the number sentence below, three of the digits are missing, as shown by the boxes. If the number sentence is correct, what is the sum of the three missing digits?

(A) 7
(B) 8
(C) 10
(D) 11
(E) 17

Problem 12:

In the square shown, the length of the diagonal is 8 cm . What is the area of the square?

(A) $28 \mathrm{~cm}^2$
(B) $32 \mathrm{~cm}^2$
(C) $49 \mathrm{~cm}^2$
(D) $64 \mathrm{~cm}^2$
(E) $81 \mathrm{~cm}^2$

Problem 13:

A bus leaves a bus stop at $10: 35 \mathrm{am}$ and then stops at 4 other bus stops. If neighbouring stops are 10 minutes apart and the bus remains at each stop for 3 minutes, at what time does the bus arrive at the last stop?

(A) 11:11 am
(B) 11:14 am
(C) 11:21 am
(D)11:24 am
(E) 11:27 am

Problem 14:

A muffin recipe which makes 12 muffins requires $\frac{2}{3}$ of a cup of milk. How many muffins can be made using 18 cups of milk?
(A) 90
(B) 108
(C) 144
(D) 216
(E) 324

Problem 15:

Two identical equilateral triangles, each with an area equal to $36 \mathrm{~cm}^2$, are placed one on top of the other so that the overlap forms a regular hexagon.

What is the area of the hexagon?
(A) $18 \mathrm{~cm}^2$
(B) $20 \mathrm{~cm}^2$
(C) $24 \mathrm{~cm}^2$
(D) $30 \mathrm{~cm}^2$
(E) $36 \mathrm{~cm}^2$

Problem 16:

A ten-sided dice (numbers $0-9) $ and a six-sided dice (numbers $1-6) $ are thrown at the same time. What is the chance of displaying a total of 6 ?
(A) one in ten
(B) one in eight
(C) one in six
(D) one in five
(E) one in two

Problem 17:

Each of the faces of 2 discs has a different whole number on it. The numbers on two of the faces are shown.

If the discs are tossed, the possible sums of the numbers showing are $10,11,12$ and 13 . What is the product of the two numbers that are on the other side of these two discs?
(A) 24
(B) 25
(C) 30
(D) 32
(E) 35

Problem 18:

A school builds a raised vegetable patch in its garden by joining four panels to form an open-ended prism and filling it with soil. Each of the thin plastic panels has sides of length 40 cm and 80 cm . Jessie wants to stand the panels on their long side and Tom wants to stand them on their short side as shown.

Which statement is true?
(A) It is impossible to calculate the volume of soil needed to fill the vegetable patch.
(B) It is impossible to compare the volumes of soil needed to fill the vegetable patch.
(C) Both vegetable patches need the same volume of soil.
(D) Jessie's vegetable patch needs more soil than Tom's.
(E) Tom's vegetable patch needs more soil than Jessie's.

Problem 19:

Aditya's dad is one year older than his mum and next year the product of his parents' ages will be over 1000 for the first time. What is the product of their ages now?
(A) 930
(B) 961
(C) 992
(D) 995
(E) 999

Problem 20:

Jake and Joe wanted to buy the same magazine. Jake needed $\$ 2.80$ more to buy it, while Joe needed $\$ 2.60$ more. So they put their money together and bought the magazine. They had $\$ 2.60$ left. How much was the magazine?
(A) $\$ 10$
(B) $\$ 9$
(C) $\$ 8$
(D) $\$ 7$
(E) $\$ 6$

Problem 21:

There is a shaded square inside a rectangle as shown. From $A$ to $B$ is 6 cm and from $C$ to $D$ is 8 cm . What is the perimeter of the large rectangle?

(A) 28 cm
(B) 27 cm
(C) 26 cm
(D) 25 cm
(E) 24 cm

Problem 22:

Karen's class and Jacqui's class are sharing some apples donated by a local farmer and everyone in each classroom will have 6 apples. If Karen's class shared all the apples, each student would have 10 apples. If Jacqui's class shared all the apples, how many apples would each student in the class have?
(A) 5
(B) 8
(C) 10
(D) 12
(E) 15

Problem 23:

Eight $1 \times 1$ square tiles are laid to form a shape 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 perimeters of the modified shape?
(A) 28
(B) 30
(C) 32
(D) 34
(E) 36

Problem 24:

On a special cubic dice the numbers $1,2,3,4,5$ and 6 are on the faces: 1 is on the face opposite 2,3 is on the face opposite 4 , and 5 is on the face opposite 6 .

Each vertex is given a vertex number, which is the sum of the numbers on the three faces that form the vertex. If I subtract the smallest vertex number from the largest vertex number, what number will I get?
(A) 1
(B) 3
(C) 5
(D) 6
(E) 7

Problem 25:

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

\[
12345678910111213 \ldots \ldots . . . .2930
\]

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:

Dried dog food is available in three sizes: 1 kg bags, which cost $\$ 6$; 3 kg bags, which cost $\$ 15$; and 8 kg bags, which cost $\$ 25$. What is the smallest number of bags you can buy such that the average price per kilogram is exactly $\$ 4$ ?

Problem 27:

In how many ways can three different numbers be selected from the numbers 1 to 12 , so that their sum can be exactly divided by 3 ?

Problem 28:

Adam, Barney and Joe carry 999 books out of the library. Adam works for 3 hours, Barney works for 4 hours and Joe works for 5 hours. They work at different speeds, with Adam carrying 5 books for every 3 books Barney carries and every 2 books Joe carries. How many books did Adam carry?

Problem 29:

Find the largest 3-digit number, with no two digits the same and with its digits in ascending order, which when multiplied by 5 has its digits in descending order?

Problem 30:

A hockey game between two teams is 'relatively close' if the number of goals scored by the two teams never differ by more than two. In how many ways can the first 12 goals of a game be scored if the game is 'relatively close'?

Australian Mathematics Competition - 2012 - Upper Primary - Grade 5 & 6 - Questions and Solutions

Problem 1:

What is the value of $101-2+1+102$ ?
(A) 0
(B) 100
(C) 198
(D) 200
(E) 202

Problem 2:

Mary's soccer team wins a game by two goals. Between them the two teams scored 8 goals. How many goals did Mary's team score?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

Problem 3:

Which of these spinners would be more likely to spin a rabbit?

Problem 4:

A small can of lemonade holds 250 mL . How many of these cans would fill a 1.5 L jug?
(A) 3
(B) 4
(C) 6
(D) 8
(E) 10

Problem 5:

Which of the following numbers has a value between $\frac{1}{5}$ and $\frac{1}{4}$ ?
(A) 0.26
(B) 0.15
(C) 0.21
(D) 0.19
(E) 0.3

Problem 6:

The first time Bill looked at the clock it was 2.00 pm . Later that afternoon he saw it was 4.00 pm . Through how many degrees had the minute hand turned in this time?
(A) 90
(B) 180
(C) 360
(D) 270
(E) 720

Problem 7:

Mary colours in a honeycomb tessellation of hexagons. If hexagons share a common edge, she paints them in different colours.

What is the smallest number of colours she needs?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Problem 8:

After half an hour Maya notices that she is one-third of the way through her homework questions. If she keeps working at a similar rate, how much longer, in minutes, can she expect her homework to take?
(A) 20
(B) 30
(C) 40
(D) 60
(E) 90

Problem 9:

Sentries marked $S$ guard the rows and columns they are on. Sentries marked $T$ guard diagonally.

How many squares are unguarded?
(A) 1
(B) 3
(C) 5
(D) 7
(E) 8

Problem 10:

A rectangular rug is 3 times as long as it is wide. If it were 3 m shorter and 3 m wider it would be a square. How long, in metres, is the rug?
(A) 3
(B) 6
(C) 9
(D) 12
(E) 15

Problem 11:

Lee's mobile phone gives him a warning that only $20 \%$ of the battery charge remains. If it is 48 hours since he last charged his phone and he uses the phone in the same way, how much longer, in hours, can he expect to use the phone before it runs out of battery life?
(A) 12
(B) 20
(C) 24
(D) 80
(E) 192

Problem 12:

Michael threw 8 darts at the dartboard shown.

All eight darts hit the dartboard. Which of the following could have been his total score?
(A) 22
(B) 37
(C) 42
(D) 69
(E) 76

Problem 13:

A prime number is called a jillyprime when doubling it and adding 1 results in another prime. How many numbers less than 15 are jillyprimes? (Note that 1 is not a prime.)
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Problem 14:

The square $P Q R S$ is made up of 36 squares with side length one unit. The square $J K L M$ is drawn as shown.

What is the area, in square units, of $J K L M$ ?
(A) 18
(B) 20
(C) 24
(D) 25
(E) 30

Problem 15:

Following a recipe, Shane roasts a chicken for 20 minutes and then a further 30 minutes for each 500 g . How many minutes does it take Shane to cook a 1.75 kg chicken?
(A) 50
(B) 80
(C) 125
(D) 52.5
(E) 150

Problem 16:

Alex placed 9 number cards and 8 addition symbol cards on the table as shown.

Keeping the cards in the same order he decided to remove one of the addition cards to form a 2-digit number. If his new total was 99, which 2-digit number did he form?
(A) 32
(B) 43
(C) 54
(D) 65
(E) 76

Problem 17:

How many different isosceles triangles can be drawn with sides that can be only $2 \mathrm{~cm}, 3 \mathrm{~cm}, 7 \mathrm{~cm}$ or 11 cm in length? Note that equilateral triangles are isosceles triangles.
(A) 8
(B) 10
(C) 12
(D) 14
(E) 19

Problem 18:

There is a total of $\$ 25$ in $\$ 2, \$ 1$ and $ 50 c $ coins on a table. Peter notices that there are 20 coins altogether and that there are two more $\$ 2$ coins than $\$ 1$ coins. How many 50 c coins are there?
(A) 6
(B) 8
(C) 10
(D) 12
(E) 14

Problem 19:

I can walk at $4 \mathrm{~km} / \mathrm{h}$ and ride my bike at $20 \mathrm{~km} / \mathrm{h}$. I take 24 minutes less when I ride my bike to the station than when I walk. How many kilometres do I live from the station?
(A) 1
(B) 1.5
(C) 2
(D) 2.5
(E) 4

Problem 20:

Jasdeep plays a game in which he has to write the numbers 1 to 6 on the faces of a cube. However, he loses a point if he puts two numbers which differ by 1 on faces which share a common edge. What is the least number of points he can lose?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Problem 21:

Twelve points are marked on a square grid as shown.

How many squares can be formed by joining 4 of these points?
(A) 5
(B) 6
(C) 9
(D) 11
(E) 13

Problem 22:

A rectangular tile has a perimeter of 24 cm . When Sally places four of these tiles in a row to create a larger rectangle, she finds the perimeter is double the perimeter of a single tile. What would be the perimeter of the rectangle formed by adding another 46 tiles to make a row of 50 tiles?
(A) 306
(B) 400
(C) 416
(D) 480
(E) 612

Problem 23:

The grid shown is part of a cross-number puzzle.

Clues
16 across is the reverse of 2 down
1 down is the sum of 16 across and 2 down
7 down is the sum of the digits in 16 across
What is 7 down?
(A) 11
(B) 12
(C) 13
(D) 14
(E) 15

Problem 24:

Damian makes a straight cut through a painted cube, dividing it into two parts. The unpainted face created by the cut could not be which of the following?
(A) an equilateral triangle
(B) a right-angled triangle
(C) a trapezium
(D) a pentagon
(E) a hexagon

Problem 25:

A $5 \times 5 \times 5$ cube has a $1 \times 1 \times 5$ hole cut through from one side to the opposite side, a $3 \times 1 \times 5$ hole through another and a $3 \times 1 \times 5$ hole through the third as shown in the diagram.

The number of $1 \times 1 \times 1$ cubes removed in this process is
(A) 25
(B) 29
(C) 36
(D) 48
(E) 92

Problem 26:

Traffic signals at each intersection on a main road all change on the same 2-minute cycle. A taxi driver knows that it is exactly 3.5 km from one intersection to the next. Without breaking the $50 \mathrm{~km} / \mathrm{h}$ speed limit, what is the highest average speed, in kilometres per hour, he can travel to get to each intersection as it just changes to green?

Problem 27:

Rani wrote down the numbers from 1 to 100 on a piece of paper and then correctly added up all the individual digits of the numbers. What sum did she obtain?

Problem 28:

This cube has a different whole number on each face, and has the property that whichever pair of opposite faces is chosen, the two numbers multiply to give the same result.

What is the smallest possible total of all 6 numbers on the cube?

Problem 29:

A rhombus-shaped tile is formed by joining two equilateral triangles together. Three of these tiles are combined edge to edge to form a variety of shapes as in the example given.

How many different shapes can be formed? (Shapes which are reflections or rotations of other shapes are not considered different.)

Problem 30:

Andrew has two children, David and Helen. The sum of their three ages is 49. David's age is three times that of Helen. In 5 years time, Andrew's age will be three times David's age. What is the product of their ages now?

Australian Mathematics Competition - 2011 - Upper Primary - Grade 5 & 6 - Questions and Solutions

Problem 1:

Which of the following numbers has the same value as 152 hundredths?
(A) 1520
(B) 152
(C) 15.2
(D) 1.52
(E) 0.152

Problem 2:

The value of $1+10+10 \times 10+10 \times 10 \times 10$ is
(A) 1010
(B) 10111
(C) 1111
(D) 1110
(E) 11111

Problem 3:

Mike buys a can of 5 tennis balls for $\$ 2.50$. How much would three balls cost?
(A) 90 c
(B) $\$ 1.50$
(C) $\$ 1.80$
(D) $\$ 2.40$
(E) $\$ 3$

Problem 4:

Auckland's time zone is two hours ahead of Melbourne's. What time is it in Auckland when it is 2 am in Melbourne?
(A) 4 am
(B) midnight
(C) 6 pm
(D) 6 am
(E) 8 am

Problem 5:

How many rectangles of any size are in this diagram?

(A) 11
(B) 10
(C) 9
(D) 8
(E) 6

Problem 6:

School shirts are on sale for $25 \%$ off the usual price. Bethany buys a shirt on sale for $\$ 6.00$. How much did she save?
(A) $\$ 4.50$
(B) $\$ 2.00$
(C) $\$ 1.50$
(D) $\$ 3.00$
(E) $\$ 4.00$

Problem 7:

Which of the following is not a net for an open top box?

Problem 8:

Which of the following is true?
(A) If you add two odd numbers you always get an odd number.
(B) If you multiply two odd numbers you always get an even number.
(C) If you add an odd and an even number you always get an even number.
(D) If you multiply an odd and an even number you always get an even number.
(E) If you multiply two even numbers you always get an odd number.

Problem 9:

The temperature on a winter's day was recorded in degrees Celsius every hour from 7 am to 12 noon as shown on the graph.

What was the difference, in degrees Celsius, between the temperature recorded at 9 am and at 11 am ?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Problem 10:

A notice in an elevator states that 13 persons or 1000 kg is the maximum load. Which of the following is the closest to the average body mass that the manufacturer used for one person?
(A) 13 kg
(B) 50 kg
(C) 80 kg
(D) 100 kg
(E) 130 kg

Problem 11:

On weekends, Fred earns money by washing cars and mowing lawns. He receives $\$ 5$ for a mow and $\$ 9$ for a car wash. Last weekend he earned $\$ 56$. Which of the following explanations is possible?
(A) 5 mows and 3 car washes
(B) 6 mows and 3 car washes
(C) 2 mows and 5 car washes
(D) 5 mows and 4 car washes
(E) 4 mows and 4 car washes

Problem 12:

The diagram shows a 7-piece tangram puzzle.

What is the area, in square centimetres, of the shaded part if the whole puzzle is a square with side 8 cm ?
(A) 2
(B) 4
(C) 6
(D) 8
(E) 10

Problem 13:

Jennifer has made this hexagonal spinner.

If she spins it 72 times, about how many times would she expect to land on a 2 ?
(A) 6
(B) 8
(C) 10
(D) 12
(E) 24

Problem 14:

You are standing somewhere on the map below.

Directly to the east you see a house $全$, directly to the north you see a person $\$$ and directly west you can see a tree $\$$. Which square are you standing in?
(A) D4
(B) F2
(C) J6
(D) G10
(E) J10

Problem 15:

Jim is running a two-lap race. He passes a marker on the first lap which is exactly a third of the way round the lap. What fraction of the race will be completed when he passes the same marker again?
(A) $\frac{1}{3}$
(B) $\frac{1}{2}$
(C) $\frac{2}{3}$
(D) $\frac{3}{4}$
(E) $\frac{5}{6}$

Problem 16:

Bill is 2 years older than his brother Graham, who in turn is 3 years older than their twin sisters Sally and Jill. If the combined total of their ages is 40 , how old is Graham?
(A) 8
(B) 9
(C) 10
(D) 11
(E) 13

Problem 17:

Sam draws a rectangle with sides of length 56 cm and 98 cm . Jan divides Sam's rectangle into squares by drawing lines joining opposite sides. What is the smallest number of lines that Jan must draw?
(A) 9
(B) 10
(C) 11
(D) 13
(E) 15

Problem 18:

The areas of three faces of a rectangular prism are 12 square centimetres, 15 square centimetres and 20 square centimetres. What is the volume, in cubic centimetres, of the rectangular prism?

(A) 30
(B) 48
(C) 56
(D) 60
(E) 72

Problem 19:

Tony has an 8 cm by 12 cm paper rectangle. He folds it in half three times, each time making a smaller rectangle. What is the smallest possible perimeter of the rectangle after the third fold?
(A) 24 cm
(B) 16 cm
(C) 14 cm
(D) 12 cm
(E) 10 cm

Problem 20:

The numbers on the six faces of this cube are consecutive even numbers.

If the sums of the numbers on each of the three pairs of opposite faces are equal, find the sum of all six numbers on this cube.
(A) 196
(B) 188
(C) 210
(D) 186
(E) 198

Problem 21:

In the following addition, some of the digits are missing.

The sum of the missing digits is
(A) 23
(B) 21
(C) 20
(D) 18
(E) 15

Problem 22:

Six towns labelled $P, Q, R, S, T$ and $U$ in the diagram are joined by roads as shown.

Starting at $P$, George the postman visits each town without returning to $P$. He wants to save time by travelling the shortest distance. How many kilometres will he need to drive?
(A) 19
(B) 20
(C) 21
(D) 22
(E) 23

Problem 23:

A set of 4-digit numbers are formed using four cards numbered 2,3 , 5 and 6 , with each number formed containing every digit. How many numbers in this set will be divisible by 8 ?

(A) 4
(B) 6
(C) 8
(D) 10
(E) 12

Problem 24:

Mary has 62 square blue tiles and a number of square red tiles. All tiles are the same size. She makes a rectangle with red tiles inside and blue tiles on the perimeter. What is the largest number of red tiles she could have used?
(A) 62
(B) 182
(C) 210
(D) 224
(E) 240

Problem 25:

Carly is writing a fantasy novel which includes inventing a new language. She decides to base her alphabet on letters formed from three straight lines joining four dots arranged in a square where each line joins two dots. Each letter goes through all four dots and can be drawn without removing the pencil from the paper, (you may retrace a line). Three such letters are shown.

How many different letters can she have in her alphabet?
(A) 12
(B) 16
(C) 24
(D) 32
(E) 36

Problem 26:

What is the smallest number which is divisible by 7 and 6 and has two odd digits?

Problem 27:

Which 2-digit number is equal to the sum of its first digit plus the square of its second digit?

Problem 28:

Andrew thinks of 4 consecutive numbers. The sum of the first three numbers is 100 more than the fourth number. What is the sum of these four consecutive numbers?

Problem 29:

Paul is one year older than his wife and they have two children whose ages are also one year apart. Paul notices that on his birthday in 2011, the product of his age and his wife's age plus the sum of his children's ages is 2011.
What would have been the result if he had done this calculation thirteen years before?

Problem 30:

Joe the handyman was employed to fix house numbers onto the doors of 80 new houses in a row. He screwed digits on their front doors, numbering them from 1 to 80 . Then he noticed that there were houses already numbered 1 to 64 in the street, so he had to replace all the numbers with new ones, 65 to 144 . If he re-used as many digits as possible (where he could use an upside down 6 as a 9 and vice versa), how many new digits must he have supplied?

Australian Mathematics Competition - 2010 - Upper Primary - Grade 5 & 6 - Questions and Solutions

Problem 1:

What is one thousand and twenty-seven in numerals?
(A) 100027
(B) 10027
(C) 1027
(D) 127
(E) 27

Problem 2:

Jillian is standing inside a pet shop and looking out the window shown in the diagram.

Problem 3:

Lee is 14 years old. Liz is 10 years old. Dad's age is twice the sum of their ages. How old is Dad?
(A) 46
(B) 48
(C) 50
(D) 52
(E) 54

Problem 4:

The midpoints of the sides of a square are joined as shown. A part of the original square is shaded as shown. What fraction of the original square is shaded?
(A) $\frac{1}{4}$
(B) $\frac{1}{6}$
(C) $\frac{2}{3}$
(D) $\frac{1}{3}$
(E) $\frac{1}{5}$

Problem 5:

In a queue in the school canteen, Sarah was behind Tim and Carl was between Sarah and Tim. Sarah was in front of Brett who was in front of George. Who was fourth in line?
(A) Sarah
(B) Tim
(C) Carl
(D) Brett
(E) George

Problem 6:

When five numbers are added the total is 2010 . One of the numbers is changed from 235 to 532 . What is the total of the five numbers now?
(A) 1723
(B) 2542
(C) 2360
(D) 1896
(E) 2307

Problem 7:

Eight blocks are glued together as shown.

Problem 8:

What is the difference between the largest and smallest 3-digit numbers that can be made from the following 1-digit cards, if each card is to be used once only in each number?

(A) 477
(B) 495
(C) 1009
(D) 468
(E) 555

Problem 9:

My father won $\$ 1000$ in a lottery. He put one-fifth in the bank, spent one-quarter of what was left on me and gave the rest to my mother. How much did my mother receive?
(A) $\$ 400$
(B) $\$ 888$
(C) $\$ 450$
(D) $\$ 550$
(E) $\$ 600$

Problem 10:

Starting at $A$ and walking around the figure back to $A$, how far do I walk?
(A) 52 m
(B) 48 m
(C) 54 m
(D) 50 m
(E) 56 m

Problem 11:

This is Liam's timetable for a normal school day.

How many minutes of class time does Liam have every day?
(A) 300
(B) 250
(C) 500
(D) 270
(E) 240

Problem 12:

The average of two numbers is 11 . One of the numbers is 6 more than the other. Which is the larger number?
(A) 6
(B) 8
(C) 11
(D) 14
(E) 17

Problem 13:

What fraction of the rectangle is shaded?

(A) $\frac{1}{3}$
(B) $\frac{5}{12}$
(C) $\frac{1}{2}$
(D) $\frac{2}{7}$
(E) $\frac{3}{8}$

Problem 14:

In a group of 55 students, exactly 39 are enrolled in mathematics and exactly 35 are enrolled in science. How many students are enrolled in both mathematics and science?
(A) 20
(B) 16
(C) 19
(D) 4
(E) 55

Problem 15:

Jeremy decides to measure area in hexagonal units (instead of squares) using the hexagonal unit as shown.

What is the area of the triangle in Jeremy's hexagonal units?

(A) 4
(B) $4 \frac{1}{6}$
(C) $4 \frac{1}{3}$
(D) $4 \frac{1}{2}$
(E) 3

Problem 16:

To make a quilt 120 cm by 90 cm for my baby's cot, I add borders to a central panel as shown. If the borders are the same width all the way around, then the dimensions of the central panel, in centimetres, could be
(A) 100 by 60
(B) 90 by 60
(C) 90 by 70
(D) 86 by 36
(E) 75 by 50

Problem 17:

Place the numbers $1,2,3,4$ and 5 , one in each circle in the diagram so that no number is joined by a line to a consecutive number.

The sum of the numbers $X$ and $Y$ could be
(A) 3
(B) 4
(C) 6
(D) 7
(E) 8

Problem 18:

Below is a diagram of a garden. Some of the garden is grass and some is a pathway made from square blocks.

The total area of the grass is 108 square metres.
What is the area, in square metres, of the pathway?
(A) 216
(B) 54
(C) 181
(D) 207
(E) 200

Problem 19:

A shop has a sale and sells hats for $\$ 12$ each in the morning, taking $\$ 720$. After lunch, the price is dropped to $\$ 9$ each and the shop sells twice as many. What was the total amount taken on the day?
(A) $\$ 1800$
(B) $\$ 900$
(C) $\$ 1260$
(D) $\$ 1440$
(E) $\$ 2880$

Problem 20:

The areas, in square centimetres, of three rectangles are given.

What is the area, in square centimetres, of the shaded rectangle?
(A) 36
(B) 48
(C) 56
(D) 60
(E) 70

Problem 21:

Mike thinks of a two-digit number. Karen reverses the digits and when the two numbers are added the total is 132 . How many different numbers could Mike have thought of?
(A) 4
(B) 7
(C) 8
(D) 10
(E) 12

Problem 22:

Two bolts and two screws weigh as much as one bolt and ten nails. One bolt weighs as much as one screw and one nail. How many nails weigh as much as one bolt?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

Problem 23:

My birthday is 18 November. Four of my friends' birthdays are 1 November, 12 November, 21 November and 1 December. The five of us decide to hold a birthday party on the date closest to all the birthdays. (This means that the sum of the days from the chosen date to each birthday is as small as possible.) Which date should this be?
(A) 12 November
(B) 19 November
(C) 15 November
(D) 18 November
(E) 17 November

Problem 24:

Annie is recording the number of cars in the family of each child in her class in a column graph. She has not drawn the column for the number of families that have exactly two cars.

The average number of cars per family is $1 \frac{1}{3}$. How many families have exactly two cars?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Problem 25:

Five rectangles, each 12 cm long and of equal width, are placed together to form a single rectangle, still 12 cm long but 5 times as wide. The new rectangle has a perimeter twice as great as each of the original rectangles. What is the perimeter, in centimetres, of the new rectangle?
(A) 48
(B) 60
(C) 64
(D) 72
(E) 84

Problem 26:

Consider this statement:
THIS IS ONE GREAT MATHS CHALLENGE
Every minute, the first letter of each word is moved to the other end of the word. In how many minutes will the original sentence appear back again?

Problem 27:

Below is an example of a triangle drawn on a 6 by 5 grid with one vertex $A$ on the bottom left-hand corner and the other two vertices on the top and right-hand boundaries.

What is the largest number of squares that can be cut by the sides of such a triangle?

Problem 28:

The product of three consecutive whole numbers is 12144 . What is their sum?

Problem 29:

When a number has the digit 2 put at both ends, its value increases by 2785 . What is the original number?

Problem 30:

I have 4 black and 4 white cubes of equal size. In how many different ways can they be put together to form a $2 \times 2 \times 2$ cube? (Two cubes are not considered different if they can be rotated to look the same.)