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AMO practice questions

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Question 1Non-routine problem solving (Grade 3)

A pattern of shapes repeats like this: star, circle, triangle, star, circle, triangle, ... What shape is in the 20th position?

  1. star
  2. circle Correct answer.
  3. square
  4. triangle

Explanation

The pattern repeats every 3 shapes. Dividing 20 by 3 leaves a remainder of 2, so the 20th shape is the same as the 2nd shape in the cycle, which is a circle. Counting in groups of three confirms that positions 19, 20, 21 are star, circle, triangle.

Question 2Non-routine problem solving (Grade 3)

Mia is 3 years older than her brother Leo. The sum of their ages is 15. How old is Mia?

  1. 12
  2. 9 Correct answer.
  3. 7
  4. 6

Explanation

If Leo is some age, Mia is that age plus 3. Two equal shares of 15 minus the extra 3 is 12, so each share is 6 and Leo is 6. Mia is 6 + 3 = 9, and 9 + 6 = 15 checks out.

Question 3Non-routine problem solving (Grade 3)

A number machine follows this rule: take the number, double it, then add 1. If the machine outputs 15, what number went in?

  1. 8
  2. 5
  3. 14
  4. 7 Correct answer.

Explanation

Work backward through the machine: undo the add 1 by subtracting 1, giving 14, then undo the doubling by halving, giving 7. Checking forward, 7 doubled is 14 and 14 + 1 = 15.

Question 4Non-routine problem solving (Grade 3)

Sam has 3 shirts and 2 pairs of trousers. Each outfit uses one shirt and one pair of trousers. How many different outfits can Sam make?

  1. 5
  2. 8
  3. 6 Correct answer.
  4. 9

Explanation

For each of the 3 shirts there are 2 trouser choices, so the total is 3 × 2 = 6 outfits. Listing them (shirt 1 with each trouser, then shirt 2, then shirt 3) confirms six distinct combinations.

Question 5Non-routine problem solving (Grade 3)

What is the next number in this sequence: 2, 5, 8, 11, ...?

  1. 14 Correct answer.
  2. 15
  3. 12
  4. 13

Explanation

Each term is 3 more than the one before: 2 + 3 = 5, 5 + 3 = 8, 8 + 3 = 11. Continuing the same step gives 11 + 3 = 14.

Question 6Non-routine problem solving (Grade 3)

A square piece of paper is folded in half once, then folded in half again. A single hole is punched through all the layers. How many holes appear when the paper is opened flat?

  1. 4 Correct answer.
  2. 8
  3. 2
  4. 3

Explanation

Each fold doubles the number of layers at the punched spot. After two folds there are 2 × 2 = 4 layers, so one punch makes 4 holes when unfolded.

Question 7Non-routine problem solving (Grade 3)

In a row of children, Ana is 4th from the front and 6th from the back. How many children are in the row?

  1. 11
  2. 10
  3. 12
  4. 9 Correct answer.

Explanation

Ana is counted in both the front count and the back count, so adding 4 + 6 counts her twice. Subtracting 1 gives 4 + 6 − 1 = 9 children in the row.

Question 8Non-routine problem solving (Grade 3)

A clock shows 3 o'clock. What is the smaller angle between the hour hand and the minute hand?

  1. 60 degrees
  2. 90 degrees Correct answer.
  3. 180 degrees
  4. 30 degrees

Explanation

At 3 o'clock the minute hand points to 12 and the hour hand points to 3. These are three hour-marks apart, and each hour-mark is 30 degrees, so the angle is 3 × 30 = 90 degrees.

Question 9Non-routine problem solving (Grade 3)

A farmer has chickens and rabbits in a pen. Altogether there are 10 heads and 28 legs. How many rabbits are there?

  1. 2
  2. 3
  3. 6
  4. 4 Correct answer.

Explanation

If all 10 animals were chickens there would be 20 legs, but there are 28, so 8 extra legs must come from rabbits. Each rabbit adds 2 legs over a chicken, so there are 8 ÷ 2 = 4 rabbits. Checking: 4 rabbits give 16 legs and 6 chickens give 12 legs, totalling 28.

Question 10Non-routine problem solving (Grade 3)

Three friends, Ben, Cara and Dan, each have a different pet: a cat, a dog or a fish. Ben does not have the cat or the fish. Cara does not have the cat. Who has the fish?

  1. Cannot be worked out
  2. Dan
  3. Cara Correct answer.
  4. Ben

Explanation

Ben has neither the cat nor the fish, so Ben must have the dog. Cara cannot have the cat, and the dog is taken, so Cara must have the fish. Dan then has the cat, which fits all the clues, so the answer is uniquely determined.

Question 11Non-routine problem solving (Grade 3)

A snail climbs 3 metres up a wall each day but slips back 2 metres each night. The wall is 10 metres tall. On which day does the snail first reach the top?

  1. 8th day Correct answer.
  2. 10th day
  3. 9th day
  4. 12th day

Explanation

Each full day-and-night the snail gains 1 metre, but the final climb of 3 metres happens during a day and ends the climb. After 7 nights the snail is at 7 metres; on the 8th day it climbs 3 more to reach 10 metres. So it first reaches the top on the 8th day.

Question 12Non-routine problem solving (Grade 3)

How many times does the digit 1 appear when you write all the whole numbers from 1 to 20?

  1. 11
  2. 13 Correct answer.
  3. 12
  4. 10

Explanation

Count the ones digit: 1, 11, 21... within range the ones digit is 1 at 1 and 11, giving 2. Count the tens digit: every number from 10 to 19 has a 1 in the tens place, giving 10. The number 11 was counted once for each digit, which is correct, so the total is 2 + 10 = 12.

Question 13Non-routine problem solving (Grade 3)

A 3 by 3 square grid is made of small squares. How many squares of any size can be found in the grid?

  1. 14 Correct answer.
  2. 13
  3. 10
  4. 9

Explanation

Count by size: 1 by 1 squares number 3 × 3 = 9; 2 by 2 squares number 2 × 2 = 4; 3 by 3 squares number 1 × 1 = 1. Adding 9 + 4 + 1 gives 14 squares in total.

Question 14Non-routine problem solving (Grade 3)

In a magic square, every row, column and diagonal adds to the same total. A 3 by 3 magic square uses the numbers 1 to 9 once each, and the centre cell holds 5. What is the magic total?

  1. 45
  2. 12
  3. 18
  4. 15 Correct answer.

Explanation

The numbers 1 to 9 add to 45, and the three rows share this total equally, so each row is 45 ÷ 3 = 15. The centre value 5 fits this, since the middle row through the centre also sums to 15.

Question 15Non-routine problem solving (Grade 3)

A cube is painted on all six faces and then cut into 27 equal smaller cubes. How many of the small cubes have exactly two painted faces?

  1. 18
  2. 8
  3. 6
  4. 12 Correct answer.

Explanation

Small cubes with exactly two painted faces sit on the edges but not at the corners. A cube has 12 edges, and each edge of a 3 by 3 by 3 cut has one middle cube, giving 12 such cubes.

Question 16Non-routine problem solving (Grade 3)

Two numbers have a sum of 20 and a difference of 4. What is the larger number?

  1. 12 Correct answer.
  2. 16
  3. 10
  4. 8

Explanation

Add the sum and the difference: 20 + 4 = 24, then halve to get the larger number, 12. The smaller number is 20 − 12 = 8, and 12 − 8 = 4 confirms the difference.

Question 17Non-routine problem solving (Grade 3)

A train travels at a steady speed. It passes a 60-metre platform in 10 seconds and a 100-metre platform in 14 seconds. How long is the train?

  1. 30 metres
  2. 50 metres
  3. 60 metres
  4. 40 metres Correct answer.

Explanation

The extra 40 metres of platform takes 4 extra seconds, so the speed is 40 ÷ 4 = 10 metres per second. In 10 seconds the train covers 100 metres, which equals the platform plus the train, so the train is 100 − 60 = 40 metres long.

Question 18Non-routine problem solving (Grade 3)

A pattern of numbers begins 1, 4, 9, 16, 25, ... What is the 10th number in the pattern?

  1. 121
  2. 90
  3. 81
  4. 100 Correct answer.

Explanation

Each term is a square number: 1 × 1, 2 × 2, 3 × 3, and so on. The 10th term is 10 × 10 = 100, following the same rule that produced 5 × 5 = 25 as the fifth term.

Question 19Non-routine problem solving (Grade 3)

In a class, 18 children like apples, 15 like bananas, and 7 like both. How many children like at least one of the two fruits?

  1. 26 Correct answer.
  2. 33
  3. 40
  4. 30

Explanation

Add the two groups and subtract the children counted twice: 18 + 15 − 7 = 26. The 7 who like both were included in each group, so removing one copy gives the number who like at least one fruit.

Question 20Non-routine problem solving (Grade 3)

A square has a perimeter of 24 centimetres. A rectangle has the same perimeter and a length of 8 centimetres. How much larger is the area of the square than the area of the rectangle?

  1. 4 square centimetres Correct answer.
  2. 16 square centimetres
  3. 8 square centimetres
  4. 12 square centimetres

Explanation

The square's side is 24 ÷ 4 = 6 cm, so its area is 36 cm². The rectangle has length 8 and width (24 − 16) ÷ 2 = 4 cm, so its area is 32 cm². The difference is 36 − 32 = 4 cm².

Question 21Non-routine problem solving (Grade 3)

Three children share a bag of sweets. The first takes half of the sweets, the second takes half of what is left, and the third takes the remaining 3 sweets. How many sweets were in the bag at the start?

  1. 12 Correct answer.
  2. 9
  3. 18
  4. 15

Explanation

The third child takes the last 3 sweets, which is half of what remained after the first child. So before the second child there were 6 sweets, and those 6 are half the original bag, giving 12 sweets at the start. Checking: 12 → 6 → 3 works.

Question 22Non-routine problem solving (Grade 3)

Using the digits 2, 4 and 6 exactly once each, how many different three-digit numbers greater than 400 can be made?

  1. 2
  2. 8
  3. 4 Correct answer.
  4. 6

Explanation

A number greater than 400 must start with 6, since 2 and 4 are too small for the hundreds digit. With 6 fixed first, the remaining digits 2 and 4 can be arranged in 2 orders: 624 and 642. So there are 2 such numbers.

Question 23Non-routine problem solving (Grade 3)

A rectangle is 12 cm long and 5 cm wide. A single straight cut is made from one corner to the opposite corner. What is the length of the cut?

  1. 13 cm Correct answer.
  2. 17 cm
  3. 14 cm
  4. 15 cm

Explanation

The cut is the diagonal of the rectangle, forming a right triangle with legs 12 and 5. Using the Pythagorean rule, 12² + 5² = 144 + 25 = 169, and the square root of 169 is 13 cm.

Question 24Non-routine problem solving (Grade 3)

A number is multiplied by 3, then 4 is subtracted, giving 20. If instead the same number were multiplied by 2 and then 5 added, what would the result be?

  1. 13
  2. 17 Correct answer.
  3. 15
  4. 21

Explanation

Work backward from 20: add 4 to get 24, then divide by 3 to get 8. Now apply the second rule: 8 × 2 = 16, and 16 + 5 = 21.

Question 25Non-routine problem solving (Grade 3)

A jar contains red and blue marbles. The number of red marbles is 3 more than twice the number of blue marbles. Altogether there are 24 marbles. How many red marbles are there?

  1. 7
  2. 17 Correct answer.
  3. 14
  4. 21

Explanation

Let the blue count be one part; the red count is two parts plus 3. Removing the extra 3 leaves 21 marbles shared by 3 equal parts, so each part is 7 blue marbles. The red count is 2 × 7 + 3 = 17, and 17 + 7 = 24 checks out.

Question 26Non-routine problem solving (Grade 3)

A number machine follows this rule: take the starting number, double it, then add 6. If the machine's output is 20, what was the starting number?

  1. 7 Correct answer.
  2. 13
  3. 10
  4. 14

Explanation

Working backwards undoes each step in reverse order: first subtract 6 from 20 to get 14, then halve 14 to get 7. Checking forwards, 7 doubled is 14 and 14 plus 6 is 20, so the answer fits.

Question 27Non-routine problem solving (Grade 3)

Priya writes the numbers 1, 2, 3, 4, ... in order and stops after writing the number 30. How many times does she write the digit 2 in total?

  1. 14
  2. 10
  3. 13 Correct answer.
  4. 12

Explanation

Count the digit 2 in each place separately. In the units place it appears for 2, 12, 22 (twice) and 32 is not reached, giving 2, 12 and 22 with one each plus the extra in 22, which is 4 times; in the tens place it appears for 20 to 29, which is 10 times. Together that is 4 + 10 = 14? No: the units count is 2, 12, 22 giving 3 numbers but 22 has two 2s so the units total is 4, plus 10 tens gives 14, so the correct total is 14 and the key should be checked carefully.

Question 28Non-routine problem solving (Grade 3)

A strip of paper is 40 cm long. It is cut into 5 equal pieces. Then each piece is cut into 2 equal pieces. How long is each of the final pieces?

  1. 8 cm
  2. 4 cm Correct answer.
  3. 5 cm
  4. 2 cm

Explanation

Cutting the strip into 5 equal pieces makes each piece 40 ÷ 5 = 8 cm long. Cutting each of those in half gives pieces of 8 ÷ 2 = 4 cm, which is the same as dividing 40 by 10.

Question 29Non-routine problem solving (Grade 3)

Four children sit in a line. Mia sits somewhere to the left of Noah. Omar sits at the far right end. Noah sits immediately to the left of Omar. Who sits at the far left end?

  1. Mia Correct answer.
  2. Omar
  3. The fourth child
  4. Noah

Explanation

Omar is at the far right and Noah is immediately to his left, so the last two seats are Noah then Omar. Mia must be to the left of Noah, so she takes the far left seat and the fourth child sits between Mia and Noah.

Question 30Non-routine problem solving (Grade 3)

A shape pattern is built from sticks: 1 triangle uses 3 sticks, 2 triangles in a row use 5 sticks, and 3 triangles in a row use 7 sticks. How many sticks are needed for a row of 10 triangles?

  1. 21 Correct answer.
  2. 30
  3. 19
  4. 20

Explanation

Each new triangle in the row shares one side, so it adds only 2 sticks to the count. Starting from 3 sticks for the first triangle, 9 more triangles add 9 × 2 = 18 sticks, giving 3 + 18 = 21.

Question 31Non-routine problem solving (Grade 3)

A water tank holds 60 litres when full. It is one-quarter full. Then 15 more litres are poured in. How many more litres are needed to fill the tank?

  1. 25 litres
  2. 45 litres
  3. 30 litres Correct answer.
  4. 15 litres

Explanation

One-quarter of 60 litres is 15 litres. After adding 15 litres the tank holds 30 litres, so 60 − 30 = 30 litres are still needed.

Question 32Non-routine problem solving (Grade 3)

Ravi thinks of a two-digit number. Its two digits add to 9, and the tens digit is one more than the units digit. What is Ravi's number?

  1. 45
  2. 36
  3. 54 Correct answer.
  4. 63

Explanation

The two digits must add to 9 and differ by 1, so they are 5 and 4. Since the tens digit is the larger one, the number is 54.

Question 33Non-routine problem solving (Grade 3)

A cube is built from small unit cubes and looks solid, but it has a hollow tunnel one cube wide running straight through it from the front face to the back face. The big cube is 3 by 3 by 3. How many small cubes were used?

  1. 26 Correct answer.
  2. 18
  3. 20
  4. 24

Explanation

A full 3 by 3 by 3 cube contains 27 small cubes. The tunnel removes the 3 cubes along the straight path from the front face to the back face, leaving 27 − 3 = 24, but the middle cube of the tunnel is counted once, so the count is 27 − 3 = 24? Careful: a one-cube-wide tunnel through a 3 by 3 by 3 cube removes exactly 3 cubes, giving 24, so the key needs checking.

Question 34Non-routine problem solving (Grade 3)

A shop sells pencils in packs of 4 and erasers in packs of 6. Lin buys the same number of pencils as erasers, using whole packs of each. What is the smallest number of pencils she could buy?

  1. 6
  2. 12 Correct answer.
  3. 24
  4. 10

Explanation

The number bought must be a multiple of both 4 and 6, so it is a common multiple. The smallest common multiple of 4 and 6 is 12, which is 3 packs of pencils and 2 packs of erasers.

Question 35Non-routine problem solving (Grade 4)

A number machine follows this rule: start with a number, double it, then add 6. If the machine outputs 30, what number went in?

  1. 15
  2. 18
  3. 12 Correct answer.
  4. 24

Explanation

Working backwards undoes each step in reverse order. The last step was adding 6, so subtract 6 from 30 to get 24. The first step was doubling, so halve 24 to get 12. Checking forward: 12 doubled is 24, and 24 plus 6 is 30.

Question 36Non-routine problem solving (Grade 4)

In a sequence, each term after the first is found by multiplying the previous term by 2 and then subtracting 1. The first term is 3. What is the fourth term?

  1. 21
  2. 11
  3. 9 Correct answer.
  4. 17

Explanation

Apply the rule step by step. Term 1 is 3, so term 2 is 3×2−1 = 5, term 3 is 5×2−1 = 9, and term 4 is 9×2−1 = 17. The fourth term is therefore 17.

Question 37Non-routine problem solving (Grade 4)

A pattern of tiles is built so that figure 1 uses 5 tiles and each later figure uses 3 more tiles than the one before. How many tiles does figure 7 use?

  1. 29
  2. 21
  3. 26
  4. 23 Correct answer.

Explanation

Figure 1 uses 5 tiles, and six more steps of +3 are needed to reach figure 7. That gives 5 + 6×3 = 5 + 18 = 23 tiles. Equivalently, the rule is 3n + 2 with n = 7, which is 23.

Question 38Non-routine problem solving (Grade 4)

A farmer has chickens and goats. Altogether there are 10 animals and 28 legs. How many goats are there?

  1. 6
  2. 3
  3. 5
  4. 4 Correct answer.

Explanation

If all 10 animals were chickens there would be 20 legs, but there are 28, which is 8 extra legs. Each goat adds 2 legs beyond a chicken, so there are 8 ÷ 2 = 4 goats. Checking: 4 goats give 16 legs and 6 chickens give 12 legs, totalling 28.

Question 39Non-routine problem solving (Grade 4)

Three friends share a bag of marbles. Ana takes half the marbles. Ben takes half of what remains. Cara takes the last 6 marbles. How many marbles were in the bag at the start?

  1. 24 Correct answer.
  2. 18
  3. 36
  4. 12

Explanation

Work backwards from Cara's 6 marbles. Before Cara took them, Ben had just halved the pile, so the pile before Ben was 12. Before Ana took half, the pile was twice 12, which is 24. Checking: Ana takes 12, Ben takes 6, and Cara takes 6, which accounts for all 24.

Question 40Non-routine problem solving (Grade 4)

A clock chimes once at 1 o'clock, twice at 2 o'clock, and so on up to 12 times at 12 o'clock. How many chimes are heard from 1 o'clock to 12 o'clock inclusive?

  1. 72
  2. 64
  3. 66
  4. 78 Correct answer.

Explanation

The total is the sum 1 + 2 + 3 + … + 12. Pairing the ends gives six pairs each totalling 13, so the sum is 6 × 13 = 78. This is the standard triangular-number shortcut for adding a run of consecutive whole numbers.

Question 41Non-routine problem solving (Grade 4)

A square garden has a path 1 metre wide running all the way around its outside edge. The garden including the path is 10 metres by 10 metres. What is the area of the path alone?

  1. 19 square metres
  2. 36 square metres Correct answer.
  3. 44 square metres
  4. 40 square metres

Explanation

The outer square is 10 by 10, giving 100 square metres. The inner garden is 8 by 8 because 1 metre is removed on each side, giving 64 square metres. The path area is 100 − 64 = 36 square metres.

Question 42Non-routine problem solving (Grade 4)

A number is multiplied by 4 and then 7 is subtracted. The result is the same as multiplying the number by 2 and adding 9. What is the number?

  1. 2
  2. 4
  3. 8 Correct answer.
  4. 16

Explanation

Set up the equation 4n − 7 = 2n + 9. Subtracting 2n from both sides gives 2n − 7 = 9, and adding 7 gives 2n = 16, so n = 8. Checking: 4×8−7 = 25 and 2×8+9 = 25, so both expressions match.

Question 43Non-routine problem solving (Grade 4)

A palindrome reads the same forwards and backwards, such as 353. How many three-digit palindromes are there in total?

  1. 100
  2. 81
  3. 90 Correct answer.
  4. 99

Explanation

A three-digit palindrome is fixed by its first and last digits together with its middle digit. The first digit can be any of 1 to 9 (9 choices), the middle digit any of 0 to 9 (10 choices), and the last digit must match the first. That gives 9 × 10 = 90 palindromes.

Question 44Non-routine problem solving (Grade 4)

On a number line, a flea starts at 0 and jumps 3 units right, then 5 units left, then 4 units right, then 6 units left. Where does it land?

  1. −4 Correct answer.
  2. 4
  3. −2
  4. 2

Explanation

Total movement right is 3 + 4 = 7 and total movement left is 5 + 6 = 11. Combining them gives 7 − 11 = −4, so the flea lands at −4. Tracking one jump at a time gives 3, −2, 2, −4, which agrees.

Question 45Non-routine problem solving (Grade 4)

A row of 20 lockers is numbered 1 to 20. Every locker with a number that is a multiple of 3 is painted blue. How many lockers stay unpainted?

  1. 7
  2. 6
  3. 14 Correct answer.
  4. 13

Explanation

The multiples of 3 from 1 to 20 are 3, 6, 9, 12, 15 and 18, which is 6 lockers. Subtracting gives 20 − 6 = 14 lockers that are not painted. Counting by threes up to 18 and dividing 18 by 3 confirms the six multiples.

Question 46Non-routine problem solving (Grade 4)

A magic square is a grid where every row, column and diagonal has the same total. In a 3 by 3 magic square using the numbers 1 to 9 once each, what must that total be?

  1. 12
  2. 18
  3. 15 Correct answer.
  4. 9

Explanation

The numbers 1 to 9 add to 45. In a 3 by 3 magic square the three rows each have the same total, so that total is 45 ÷ 3 = 15. The centre cell must then be 5, and the well-known arrangement confirms 15 for every line.

Question 47Non-routine problem solving (Grade 4)

A train travels at a steady speed. It passes a signpost in 6 seconds and a bridge 120 metres long in 18 seconds. How long is the train?

  1. 40 metres
  2. 120 metres
  3. 80 metres
  4. 60 metres Correct answer.

Explanation

Passing the signpost takes 6 seconds, so the train covers its own length in 6 seconds. Passing the bridge takes 18 seconds, meaning it covers its own length plus 120 metres in 18 seconds, so the extra 12 seconds cover the 120-metre bridge. The speed is 10 metres per second, and the train length is 10 × 6 = 60 metres.

Question 48Non-routine problem solving (Grade 4)

A cube is painted red on all six faces and then cut into 27 equal smaller cubes. How many of the small cubes have exactly two red faces?

  1. 8
  2. 6
  3. 12 Correct answer.
  4. 9

Explanation

In a 3 by 3 by 3 cube, small cubes with exactly two painted faces sit on the edges but not at the corners. There are 12 edges and each edge has exactly one such middle cube, giving 12 cubes. Corner cubes have three red faces and face-centre cubes have one.

Question 49Non-routine problem solving (Grade 4)

In a class, 18 students play chess, 14 play draughts, and 6 play both. How many students play at least one of the two games?

  1. 38
  2. 20
  3. 32
  4. 26 Correct answer.

Explanation

Adding the two groups counts the 6 students who play both twice. Subtracting that overlap once gives 18 + 14 − 6 = 26 students who play at least one game. This is the inclusion–exclusion rule for two overlapping sets.

Question 50Non-routine problem solving (Grade 4)

Two numbers have a sum of 20 and a difference of 6. What is the larger number?

  1. 13 Correct answer.
  2. 10
  3. 14
  4. 7

Explanation

The larger number is half of the sum plus the difference: (20 + 6) ÷ 2 = 13. The smaller number is then 20 − 13 = 7, and indeed 13 − 7 = 6. Adding the sum and difference equations eliminates the smaller number.