What is the IMO pattern for Class 5?

50 multiple-choice questions for 60 marks in one hour: Logical Reasoning (15), Mathematical Reasoning (20), Everyday Mathematics (10), and 5 Achievers questions worth 3 marks each. There is no negative marking.

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No. Every question is written by us in the published pattern, so it is safe to print and share. We are not connected with the Science Olympiad Foundation, which runs the IMO.

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Sit it in one go with a 60-minute timer, then mark it with the answers at the end. Go back over the questions that were missed: the working shows the method for each.

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IMO Sample Paper Class 5 with Answers (Free)

Section	Questions	Marks
Logical Reasoning	15	15
Mathematical Reasoning	20	20
Everyday Mathematics	10	10
Achievers Section	5	15
Total	50	60

Scholar Tests · Sample paper

Maths olympiad sample paper, Class 5

50 questions · 60 marks · 60 minutes · no negative marking

Name ______________________ Date ____________

Section 1: Logical Reasoning

Questions 1 to 15, 1 mark each. Circle the letter of the best answer.

1.
The arrows in this grid follow a rule. Which arrow belongs where the question mark is?
1. A
2. B
3. C
4. D
2.
Which word does not belong with the others?
1. Aglider
2. Bbarge
3. Cyacht
4. Dcanoe
3.
Three views of the same cube are shown below. Each of its six faces carries a different number from 1 to 6. What is the sum of all the numbers on the faces next to (touching) the face numbered 1?
1. A19
2. B18
3. C11
4. D20
4.
21 January is a Sunday. What day of the week is 2 March? (January has 31 days.)
1. ASunday
2. BFriday
3. CSaturday
4. DThursday
5.
In a scholarship examination, Austin ranked 31st and Aisha ranked 35th. Among the girls, Aisha ranked 21st, and among the boys, Austin ranked 12th. Austin ranked 10th counting up from the bottom. Among the girls, Aisha ranked 3rd from the bottom. How many boys sat the examination?
1. A40
2. B23
3. C17
4. D5
6.
Each row of pictures follows the same rule from left to right. Which picture belongs where the question mark is?
1. A
2. B
3. C
4. D
7.
Five friends ran a race. Ethan did not finish first. James finished immediately ahead of Tom. Aaron finished fifth. Priya finished immediately ahead of Aaron. Who finished third?
1. APriya
2. BTom
3. CJames
4. DEthan
8.
In a certain month, the 8th falls on a Friday. On what day of the week does the 23rd of the same month fall?
1. ASaturday
2. BThursday
3. CSunday
4. DFriday
9.
Helen and Trevor have two children, Anna and Paul. Paul has two children, Zoe and Ethan.

Anna is Ethan's ______.
Which word fits the gap?
1. Auncle
2. Bcousin
3. Caunt
4. Dnephew
10.
Which figure is the odd one out?
1. A
2. B
3. C
4. D
11.
The first shape changes into the second in a certain way. Which shape completes the second pair in the same way?
1. A
2. B
3. C
4. D
12.
How many triangles of any size are there in this figure?
1. A20
2. B34
3. C10
4. D35
13.
Look at the grid below. The same rule works in every row. What number should replace the question mark?
1. A39
2. B80
3. C41
4. D40
14.
The dashed line is a mirror. The square marked P is reflected in the mirror. Which square does its reflection land in?
1. AD2
2. BD7
3. CE6
4. DE7
15.
If WIN is written in code as SEJ, how is TIN written in the same code?
1. AXMR
2. BPEJ
3. CNIT
4. DODI

Section 2: Mathematical Reasoning

Questions 16 to 35, 1 mark each. Circle the letter of the best answer.

16.
The net below is folded to make a cube. What is the total of the number on the shaded square and the number on the face that ends up opposite it?
1. A23
2. B21
3. C22
4. D24
17.
Elena used a balance to compare the masses of some objects. The results are shown below. Which of these must be true?
1. AOne ball and one cone are lighter than two cubes.
2. BTwo cones are heavier than one cube.
3. COne cube is heavier than one ball.
4. DOne cube is heavier than one ball and one cone.
18.
In the diagram below, what is the size of the angle marked x?
1. A145°
2. B85°
3. C125°
4. D135°
19.
What number should replace the question mark? 7 × ? + 24 = 59
1. A5
2. B12
3. C35
4. D6
20.
For any two numbers, a ◆ b means (2 × a) − (3 × b). What is 5 ◆ 3?
1. A1
2. B7
3. C15
4. D8
21.
A sequence begins 10, 16, 22, 28, … and carries on in the same way. What is the 19th term?
1. A124
2. B119
3. C190
4. D118
22.
Jonah started with a jar of marbles. Jonah gave away 40% to a friend, 15% to a cousin and 10% to a neighbour, each share worked out from the number Jonah started with. 210 marbles were left. How many marbles were there at the start?
1. A467
2. B600
3. C35
4. D323
23.
Aaron is standing in square P. Ethan is standing in square Q. Which one of these squares is the same distance, in a straight line, from both of them?
1. AF7
2. BE8
3. CF4
4. DD1
24.
Grace stacks three ordinary dice one on top of another on a table. On any die, the dots on opposite faces add to 7. The face on the very top of the stack shows 3. How many dots are on the faces that cannot be seen — the face resting on the table, and the pairs of faces where one die meets the next?
1. A14
2. B17
3. C4
4. D18
25.
In the triangle below, what is the size of the angle marked x?
1. A132°
2. B84°
3. C48°
4. D42°
26.
Find the missing number marked with the asterisk (*) in the magic square below. (Every row, column and diagonal adds to the same total.)
1. A19
2. B15
3. C16
4. D17
27.
A solid square pyramid and a square prism stand on bases of the same size, and they are the same height. The pyramid has a volume of 64 cubic centimetres. The prism is 4 cm along each side of its base. What is the volume of the prism, in cubic centimetres?
1. A60
2. B384
3. C192
4. D32
28.
How many tenths are there altogether in 36.8?
1. A37
2. B8
3. C36
4. D368
29.
Elena writes down the digits 1, 2, 4, 5, 6 and 8. Using every digit exactly once, 3-digit numbers are made: one number from 3 of the digits and a second number from the other 3. What is the smallest difference between the two numbers?
1. A741
2. B35
3. C26
4. D44
30.
Which of these fractions is not equal to 19/22?
1. A114/132
2. B57/66
3. C38/22
4. D95/110
31.
Two crowds come to 72, and one is 54 more than the other. What is the bigger crowd?
1. A18
2. B9
3. C27
4. D63
32.
This solid rectangular prism has been built from identical cubes. Some of the cubes cannot be seen in the drawing. How many cubes cannot be seen?
1. A45
2. B36
3. C3
4. D16
33.
Which of these is the largest?
1. A48%
2. B1/10
3. C0.84
4. D1/4
34.
Two whole numbers multiply together to make 2592, and one of them is 8 times the other. What is the smaller number?
1. A18
2. B19
3. C144
4. D324
35.
Each letter of the alphabet stands for a number: A = 1, B = 2, C = 3, and so on to Z = 26. What is the value of P × D + O − I?
1. A72
2. B70
3. C160
4. D51

Section 3: Everyday Mathematics

Questions 36 to 45, 1 mark each. Circle the letter of the best answer.

36.
The same rule turns every number in the top row into the number directly below it. What number belongs in the cell marked with a question mark?
1. A46
2. B6
3. C33
4. D26
37.
A party bag is filled with one of each: there are five sandwiches and six drinks to choose from. How many different bags are possible?
1. A30
2. B29
3. C11
4. D31
38.
A square photograph has an area of 196 cm². What is the distance all the way round it?
1. A56 cm
2. B196 cm
3. C54 cm
4. D28 cm
39.
A box holds 8 yellow counters and 2 green counters. Two counters are taken out one after the other, and the first is not put back. What is the chance that both are yellow?
1. A16/25
2. B7/9
3. C28/45
4. D4/5
40.
A train journey begins at 11:46 am and takes 1 hour 55 minutes. What time does it finish?
1. A1:41 pm
2. B2:41 pm
3. C9:51 am
4. D2:21 pm
41.
Jonah wants to post some art supplies. The brush has a mass of 1400 grams, the paint tin has a mass of 1050 grams, the sketchpad has a mass of 450 grams and the easel has a mass of 150 grams. Which group has a total mass as close as possible to 1100 grams?
1. Aall 4 items
2. Bthe brush and sketchpad
3. Cthe paint tin, sketchpad and easel
4. Dthe brush, paint tin and sketchpad
42.
Chloe collected data on the vehicles in a car park and graphed the results. Which statement about the vehicles is not true?
1. AHalf of the buses are diesel.
2. BAltogether 51 vehicles were counted.
3. CThere were 13 more petrol than diesel altogether.
4. DThere were more cars than vans.
43.
This graph shows how far Grace and Dev walk for a given number of paces. Grace takes 240 paces to walk from school to the swimming pool. How many paces does Dev take to walk from school to the swimming pool?
1. A720
2. B300
3. C80
4. D240
44.
A box measures 12 cm long, 3 cm wide and 8 cm high. What is its volume?
1. A92 cm³
2. B23 cm³
3. C312 cm³
4. D288 cm³
45.
At 4:30 pm two trams leave the terminus together. After that, the first leaves every 8 minutes and the second every 6 minutes. At what time will they next leave together?
1. A4:38 pm
2. B4:54 pm
3. C4:53 pm
4. D4:44 pm

Section 4: Achievers Section

Questions 46 to 50, 3 marks each. Circle the letter of the best answer.

46.
A league has 8 clubs. Every one of them sends one letter to every other one, and then the whole round is played a second time. How many letters are there altogether?
1. A56
2. B112
3. C128
4. D57
47.
Finn started with a jar of marbles. Finn gave away 50% to a friend, 30% to a cousin and 10% to a neighbour, each share worked out from the number Finn started with. 70 marbles were left. How many marbles were there at the start?
1. A700
2. B78
3. C7
4. D350
48.
The net below is folded to make a cube. What is the total of the number on the shaded square and the number on the face that ends up opposite it?
1. A32
2. B30
3. C31
4. D28
49.
A bag holds 7 red marbles and 7 blue marbles. Two marbles are taken out one after the other, and the first is not put back. What is the chance that both are red?
1. A6/13
2. B3/13
3. C1/2
4. D25/26
50.
Which of these fractions is not equal to 5/6?
1. A30/36
2. B10/12
3. C15/12
4. D15/18

Answers and working

Logical Reasoning

1.A. Along each row, from left to right, every arrow is the one before it turned a quarter turn anticlockwise, so the missing arrow points down and to the right.
2.A (glider). All the others are vehicles that travel on water; glider is one of the vehicles that travel through the air.
3.B (18). Faces seen together on a cube cannot be opposite each other, and the three views leave only one possibility: 1–2, 3–4, 5–6. So 1 is opposite 2, and the four faces touching it total 21 − 1 − 2 = 18.
4.B (Friday). From 21 January to the end of January is 10 days, plus 2 more into March = 12 days. 12 = 1 weeks and 5 days, so count 5 on from Sunday: Friday.
5.C (17). Austin is 31st from the top and 10th from the bottom, so everyone numbers 31 + 10 − 1 = 40. The same trick on the girls gives 21 + 3 − 1 = 23 girls. Boys = 40 − 23 = 17. (The other two ranks are true but not needed.)
6.C. Along every row the shaded part moves one quarter anticlockwise each step, so the missing picture is the square with the bottom left quarter shaded.
7.D (Ethan). The clues fix one order only: first James, second Tom, third Ethan, fourth Priya, fifth Aaron. So Ethan finished third.
8.A (Saturday). From the 8th to the 23rd is 15 days. 15 = 2 full weeks and 1 day, so count 1 on from Friday: Saturday.
9.C (aunt). Anna is Ethan's aunt. Paul is Ethan's parent. Paul and Anna are both children of Helen and Trevor, so Anna is Paul's sister. A parent's sister is an aunt, so Anna is Ethan's aunt and Ethan is Anna's nephew.
10.A. In every figure but one, the line starts at a corner. The odd one out is the triangle whose line starts at the middle of a side.
11.A. In the first pair the shape is turned a quarter turn clockwise, with the black and white parts swapped. Doing the same to the third shape gives the answer.
12.D (35). Counting size by size, from the smallest up to the biggest, finds 35 triangles.
13.D (40). In every row, the middle number is the average of the two on the outside. In the row with the question mark that gives 40.
14.D (E7). P is 3 squares from the mirror, so its reflection is the same distance on the other side. That is E7.
15.B (PEJ). Every letter moves 4 places back in the alphabet, so TIN becomes PEJ.

Mathematical Reasoning

16.D (24). 10 and 14 are the two squares at the ends of a straight line of three, and the ends of a line of three always fold to opposite faces. So 10 + 14 = 24.
17.B (Two cones are heavier than one cube.). The pictures say two balls are lighter than one cone; two cubes are heavier than one ball; one cube and one ball are lighter than two cones. Put together, they make it certain that two cones are heavier than one cube, whatever the objects weigh.
18.D (135°). The angles at O go all the way round, so they come to 360°. The ones printed on the diagram take up 85° + 140° = 225°, which leaves 360° − 225° = 135°, so x = 135°.
19.A (5). Undo the operations in reverse order: 7 × ▢ + 24 = 59 gives ▢ = 5.
20.A (1). Put 5 where a is and 3 where b is, keeping them in that order — this rule gives a different answer the other way round. That makes it (2 × 5) − (3 × 3), which comes to 1.
21.D (118). Each term is 6 more than the last, so the nth term is 10 + (n − 1) × 6. For n = 19: 10 + 18 × 6 = 118.
22.B (600). The shares add to 65%, so the 210 left over are the other 35%. One per cent is 210 ÷ 35 = 6, so the whole is 600.
23.C (F4). From F4 to P is 1 across and 3 up or down; to Q it is 1 across and 3 up or down. Both give the same straight-line distance, so F4 is the square.
24.D (18). Each of the two touching places hides a pair of opposite faces, which is 7 dots, so that is 14. The face on the table is opposite the 3, so it is 4. Altogether 18.
25.D (42°). The two ticked sides are the same length, so the two angles at the ends of the third side are equal. Between them they take up 180° − 96° = 84°, so x = 84° ÷ 2 = 42°.
26.D (17). The magic total is 3 × centre = 45. The diagonal through * gives 45 − 15 − 13 = 17.
27.C (192). A pyramid on the same base and height as a prism is a third of it, so the prism is three times the pyramid: 3 × 64 = 192 cubic centimetres.
28.D (368). Each whole one holds 10 tenths, so 36 wholes are 360 tenths. Add the 8 already there: 360 + 8 = 368.
29.C (26). Putting the two closest digits at the front and running the tails in opposite directions gives 512 and 486, and 512 − 486 = 26.
30.C (38/22). Multiplying the top and the bottom by the same number leaves a fraction unchanged. In 38/22 only the top has been multiplied (by 2), so it is not equal to 19/22.
31.D (63). Take the difference off the total first: 72 − 54 = 18. That is what TWO of the smaller one come to, so the smaller is 9 and the larger is 9 + 54 = 63.
32.D (16). Three faces are showing, so peel one layer off each of them. What is left is a 4 by 2 by 2 block entirely inside, which is 16 cubes.
33.C (0.84). As decimals: 48% = 0.48, 1/10 = 0.1, 0.84 = 0.84, 1/4 = 0.25. The largest is 0.84.
34.A (18). Call the smaller number n. The larger is 8n, so the two multiply to 8 × n × n = 2592. That makes n × n = 2592 ÷ 8 = 324, so n = 18 and the larger number is 144.
35.B (70). The letters stand for P = 16, D = 4, O = 15 and I = 9. Do the multiplying first, then the adding and subtracting, and the answer is 70.

Everyday Mathematics

36.D (26). The rule is: multiply by 5, then add 1. It works on every column — 5 × 1 + 1 = 6 — so the missing number is 5 × 5 + 1 = 26.
37.A (30). Every choice can go with every other choice, so the counts multiply, which gives 30.
38.A (56 cm). A square's sides are all equal, so each side is the number that multiplies by itself to make 196, which is 14 cm. Round all four sides is 4 × 14 = 56 cm.
39.C (28/45). The first draw is 8 out of 10. One yellow is now gone, so the second is 7 out of 9. Both must happen, so multiply: 8/10 × 7/9 = 28/45.
40.A (1:41 pm). Add the hours first: 11:46 am plus 1 hour is 12:46 pm. Then the 55 minutes. That takes the minutes past 60, so an hour carries, and it finishes at 1:41 pm.
41.C (the paint tin, sketchpad and easel). The paint tin, sketchpad and easel come to 1650 g, which is 550 g from 1100 g — closer than any of the others.
42.D (There were more cars than vans.). Checked against the columns, three of the four statements hold. "There were more cars than vans" does not. There were more vans than cars, not the other way round.
43.C (80). 240 paces puts Grace at 40 metres. Reading 40 metres back down Dev's line gives 80 paces.
44.D (288 cm³). Volume multiplies all three edges: 12 × 3 × 8 = 288 cm³.
45.B (4:54 pm). The smallest number that both 8 and 6 divide into is 24. So they meet again 24 minutes after 4:30 pm, which is 4:54 pm.

Achievers Section

46.A (56). Each of the 8 meets 7 others, which counts 8 × 7 = 56 — but that counts every pair twice, so halve it to get 28, then double it for the second round: 56.
47.A (700). The shares add to 90%, so the 70 left over are the other 10%. One per cent is 70 ÷ 10 = 7, so the whole is 700.
48.A (32). Two squares that are joined along an edge, or that meet at a corner, always end up side by side on the cube, so none of those can be opposite 17. Pairing all six squares up under that rule, and keeping the ends of every straight line of three together, leaves 17 with 15. So 17 + 15 = 32.
49.B (3/13). The first draw is 7 out of 14. One red is now gone, so the second is 6 out of 13. Both must happen, so multiply: 7/14 × 6/13 = 3/13.
50.C (15/12). Multiplying the top and the bottom by the same number leaves a fraction unchanged. In 15/12 the top has been multiplied by 3 but the bottom by 2, so it is not equal to 5/6.

Written by Scholar Tests in the published IMO pattern. Not an SOF paper; we are not connected with the Science Olympiad Foundation.

IMO sample paper for Class 5, with answers