Maths Olympiad Level 2 Solutions for Classes 3 to 12

Maths Olympiad Level 2 Solutions for Classes 3–12

When a Maths Olympiad question looks difficult, a clear solution can make the next problem easier. The key is to understand which idea to use, why each step works and how to check your final answer.

Practise with 24 Maths Olympiad Level 2 questions and worked solutions across four class groups. You’ll learn methods for number puzzles, fractions, algebra, geometry, probability and senior-level mathematics, with explanations you can discuss with a teacher or parent.

Choose the topics that suit your class from the School Connect International Maths Olympiad resources and the Maths Olympiad syllabus. For class-specific participation and later-round requirements, refer to the current examination rules.

Learn mathematical methods with step-by-step solutions

Choose Maths Olympiad Problems for Your Class

Use these learning stages to find questions that are challenging but approachable. School syllabi and exam stages can vary, so the Maths Olympiad syllabus will help you select the topics most useful to you.

Classes	Skills to develop	Useful solution method
3–5	Patterns, number puzzles, ratios, perimeter, simple counting	Draw or list the information, then calculate
6–8	Fractions, equations, percentages, factors and sequences	Write an equation and check the result
9–10	Quadratic equations, coordinate geometry, similarity, probability	Organise a multi-step solution with a clear formula or diagram
11–12	Functions, combinations, derivatives, integrals and matrices	Identify conditions and justify each transformation

How to Write Better Maths Olympiad Solutions

State what is given: list the numbers and conditions in the question.
Choose a method: draw a diagram, write an equation or look for a pattern.
Show your working: keep each calculation easy to follow.
Check the answer: substitute it into the question or use an estimate.
Try a new example: change the numbers and apply the same idea again.

24 Maths Olympiad Level 2 Questions with Step-by-Step Answers

Try each problem before revealing the explanation with your eyes. Read the steps carefully, then see whether you can solve a similar question without help.

Classes 3–5: Number Puzzles and Everyday Maths

Question 1. What comes next in the sequence 2, 5, 11, 23, __?
A. 47 | B. 46 | C. 45 | D. 48
Answer: A. 47. Each number is double the previous number plus 1. The next is 23 × 2 + 1 = 47.
Question 2. A two-digit number has digits that total 12. Its tens digit is 2 more than its ones digit. What is the number?
A. 66 | B. 75 | C. 84 | D. 93
Answer: B. 75. Let the ones digit be u and the tens digit be u+2. Then 2u+2=12, so u=5 and the tens digit is 7. The number is 75.
Question 3. Six boxes contain eight counters each. Eleven counters are removed. How many remain?
A. 35 | B. 36 | C. 37 | D. 38
Answer: C. 37. Six boxes hold 6 × 8 = 48 counters. Subtract 11 to get 37.
Question 4. A rectangle has perimeter 54 cm and length 16 cm. Find its width.
A. 9 cm | B. 10 cm | C. 12 cm | D. 11 cm
Answer: D. 11 cm. Perimeter is 2(length+width). Divide 54 by 2 to get 27; width = 27−16=11 cm.
Question 5. What is the sum 1 + 2 + 3 + … + 15?
A. 120 | B. 115 | C. 105 | D. 125
Answer: A. 120. Pair the first and last: 1+15=16, 2+14=16, and so on, with 8 in the middle. The sum is 15×16/2=120.
Question 6. Red and blue counters are in the ratio 3:4. If there are 49 counters altogether, how many are red?
A. 18 | B. 21 | C. 24 | D. 28
Answer: B. 21. There are 3+4=7 parts. Each part equals 49÷7=7 counters, so red counters=3×7=21.

Classes 6–8: Algebra, Fractions and Number Reasoning

Question 7. Solve 4(2x−3)=5x+15.
A. 7 | B. 8 | C. 9 | D. 10
Answer: C. 9. Expand to 8x−12=5x+15. Then 3x=27, so x=9.
Question 8. Calculate 3/4−2/5.
A. 1/20 | B. 13/20 | C. 11/20 | D. 7/20
Answer: D. 7/20. Use denominator 20: 3/4=15/20 and 2/5=8/20. The difference is 7/20.
Question 9. An item costs 600 units. A 15% discount is applied, followed by 10% tax on the discounted price. Find the final price.
A. 561 | B. 550 | C. 540 | D. 570
Answer: A. 561. The discounted price is 600×0.85=510. Add tax: 510×1.10=561.
Question 10. The mean of 5, 9, 13, 19 and x is 12. Find x.
A. 12 | B. 14 | C. 13 | D. 15
Answer: B. 14. Five numbers with mean 12 total 60. The known four total 46, leaving x=60−46=14.
Question 11. What is the units digit of 3 raised to the power 2026?
A. 1 | B. 3 | C. 9 | D. 7
Answer: C. 9. The units digits cycle 3, 9, 7, 1. Since 2026 leaves remainder 2 when divided by 4, the units digit is 9.
Question 12. What is the least common multiple of 12 and 18?
A. 24 | B. 30 | C. 72 | D. 36
Answer: D. 36. Multiples of 12 are 12,24,36 and multiples of 18 are 18,36. The first shared multiple is 36.

Classes 9–10: Geometry, Equations and Probability

Question 13. Solve x²−11x+30=0.
A. 5 and 6 | B. 4 and 7 | C. 3 and 10 | D. −5 and −6
Answer: A. 5 and 6. Factor as (x−5)(x−6)=0. Hence x=5 or x=6.
Question 14. An arithmetic sequence starts 3, 8, 13, … . Find its 15th term.
A. 68 | B. 73 | C. 70 | D. 78
Answer: B. 73. The common difference is 5. The 15th term is 3 + 14×5 = 73.
Question 15. Two similar triangles have corresponding sides in the ratio 2:3. The smaller triangle has area 40 cm². Find the larger area.
A. 60 cm² | B. 80 cm² | C. 90 cm² | D. 120 cm²
Answer: C. 90 cm². Area scales with the square of the side ratio: 40×(3/2)²=40×9/4=90 cm².
Question 16. A bag has five red and three blue counters. Two are selected without replacement. What is the probability both are red?
A. 1/4 | B. 1/2 | C. 3/8 | D. 5/14
Answer: D. 5/14. The first red probability is 5/8; the second red is 4/7. Multiply: 5/8×4/7=20/56=5/14.
Question 17. Solve the simultaneous equations 2x+3y=16 and x−y=3. What is x?
A. 5 | B. 4 | C. 3 | D. 6
Answer: A. 5. From x−y=3, x=y+3. Substitute: 2(y+3)+3y=16, giving 5y=10, y=2 and x=5.
Question 18. Find the slope of the straight line through (1,4) and (5,16).
A. 2 | B. 3 | C. 4 | D. 5
Answer: B. 3. Slope is change in y divided by change in x: (16−4)/(5−1)=12/4=3.

Classes 11–12: Functions, Calculus and Counting

Question 19. If f(x)=2x²−3x+1 and g(x)=x+2, find f(g(1)).
A. 8 | B. 9 | C. 10 | D. 12
Answer: C. 10. First g(1)=3. Then f(3)=2×3²−3×3+1=18−9+1=10.
Question 20. How many ways can three students be selected from eight distinct students when order does not matter?
A. 48 | B. 336 | C. 64 | D. 56
Answer: D. 56. Choose 3 from 8: 8C3=(8×7×6)/(3×2×1)=56.
Question 21. If f(x)=x³−6x²+4x, find f'(2).
A. −8 | B. −12 | C. 0 | D. 8
Answer: A. −8. Differentiate: f'(x)=3x²−12x+4. At x=2, f'(2)=12−24+4=−8.
Question 22. Evaluate the definite integral of 3x²+2 from x=0 to x=2.
A. 8 | B. 12 | C. 10 | D. 14
Answer: B. 12. An antiderivative is x³+2x. At 2 this is 8+4=12; at 0 it is zero.
Question 23. Two fair dice are rolled. Given that their sum is even, what is the probability both numbers are odd?
A. 1/4 | B. 1/3 | C. 1/2 | D. 3/4
Answer: C. 1/2. There are 18 outcomes with an even sum: 9 even-even and 9 odd-odd. Hence P(both odd | even sum)=9/18=1/2.
Question 24. Find the determinant of the matrix [[4,3],[2,5]].
A. 11 | B. 12 | C. 17 | D. 14
Answer: D. 14. For a two-by-two matrix, determinant=ad−bc=4×5−3×2=20−6=14.

Common Maths Mistakes and How to Correct Them

Challenge	Helpful check	Example
Fractions with different denominators	Find a common denominator before adding or subtracting	3/4−2/5=15/20−8/20=7/20
Percentage changes in sequence	Apply each percentage to the latest value	600 after 15% off becomes 510; adding 10% tax gives 561
Confusing side scale and area scale	Square the side scale factor to compare areas	A 3:2 side factor gives a 9:4 area factor
Counting the same selection twice	Use combinations when order does not matter	Choosing three students from eight gives 56 groups
Forgetting restrictions in algebra	Check whether every step is valid	A denominator must be nonzero when dividing

Seven-Day Level 2 Solutions Practice Plan

Day	Activity	Learning goal
1	Select your class syllabus and solve four questions	Find the topics that need practice
2	Work on patterns, fractions or algebra	Become confident with calculations
3	Practise geometry and measurement problems	Use diagrams and formulas correctly
4	Attempt sequences, probability or counting questions	Improve systematic reasoning
5	Explain three worked answers to a classmate or teacher	Make your methods easier to remember
6	Solve a short mixed-topic set with a comfortable time limit	Build speed after accuracy
7	Review mistakes and retry difficult questions	Check that you understand each method

Frequently Asked Questions

How can I practise Maths Olympiad Level 2 solutions?

Choose a few questions suited to your class. Work independently, compare your steps with the answers and then solve a similar problem using the same method.

Which maths topics are helpful for Classes 3–5?

Number patterns, arithmetic, fractions, shapes, measurement and simple puzzles are useful starting points. Use drawings to help explain each solution.

What should students in Classes 6–8 practise?

Try multi-step equations, fractions, ratios, percentages, geometry and number reasoning. Check calculations by substituting your answer into the original problem.

How can Classes 9–12 prepare for challenging questions?

Build confidence with algebra, geometry, functions, probability and senior topics that match your class syllabus. Practise showing each mathematical step clearly.

How do I check which Olympiad exam stage applies to me?

Check your class, examination session and qualification details in the School Connect Maths Olympiad rules.

Where can I find Maths Olympiad question papers?

Explore the class-wise Maths practice library and the official maths resources for papers, problems and further learning.

Can teachers use the worked solutions in class?

Yes. Students can present different methods, discuss common mistakes and work through challenging problems together.

What should I do after getting a question wrong?

Find the step where your reasoning changed, read the explanation carefully and then try another question with similar numbers or concepts.

Continue learning: Choose four questions from the group most suited to your class, write out the complete solutions, and visit the Level 2 practice guide for more mathematical challenges.