Maths Olympiad Questions for Class 10 with Solutions

Here’s something many students notice on their first olympiad paper: every chapter looks familiar, but the questions don’t. You know the formula, yet you can’t see where to use it. That gap is exactly what maths olympiad questions for class 10 are designed to test, and it closes only with practice.

This article gives you 20 solved questions from the class 10 syllabus, each with a short note on what makes it tricky, and five more for you to solve alone. Try each question first. Even five honest minutes of trying teaches you more than reading the answer straight away.

☰ Table of Contents

What Is a Maths Olympiad?

A maths olympiad tests how well you can reason, not how many formulas you can recall. Class 10 students in India usually hear about two exams. The International Mathematics Olympiad (IMO), conducted by the Science Olympiad Foundation (SOF), is open to school students. The Indian Olympiad Qualifier in Mathematics (IOQM) is the first step towards the national and international olympiads.

Why is it worth attempting?

It strengthens your basics, which helps in your board exam.
It builds speed and the habit of staying calm under time pressure.
It gives you a head start for NTSE, JEE and scholarship tests.
It looks good on your academic record.

What to Study

You don’t need to learn anything new. Most questions come from these chapters:

Real Numbers
Polynomials and Quadratic Equations
Pair of Linear Equations
Arithmetic Progressions
Triangles and Circles
Coordinate Geometry
Trigonometry and its Applications
Surface Areas and Volumes
Statistics and Probability

Papers also include logical reasoning, mathematical reasoning, everyday mathematics and a harder achievers section. Formats change from time to time, so confirm the latest pattern on the official website.

Maths Olympiad Questions for Class 10 with Answers

Real Numbers

Q1. Find the HCF and LCM of 96 and 404 using prime factorisation. Then check that HCF × LCM equals 96 × 404.

96 = 2⁵ × 3 and 404 = 2² × 101.
HCF = 2² = 4
LCM = 2⁵ × 3 × 101 = 9696
Check: 4 × 9696 = 38,784 = 96 × 404.

Watch out: HCF takes the smallest power of each common prime, LCM takes the biggest power of every prime.

Q2. Find the smallest number that leaves remainder 5 when divided by 12, 15 and 20.

The LCM of 12, 15 and 20 is 60. Any multiple of 60 leaves remainder 0, so add 5.
Answer: 65

Q3. How many zeros are there at the end of 100!?

Each zero needs a 2 and a 5. Twos are plentiful, so count the fives:
⌊100/5⌋ + ⌊100/25⌋ = 20 + 4 = 24
Answer: 24 zeros

Watch out: Many students stop at 20 and forget that 25, 50, 75 and 100 each carry an extra 5.

Polynomials and Quadratic Equations

Q4. If α and β are the zeros of 2x² − 7x + 3, find α² + β².

α + β = 7/2 and αβ = 3/2.
α² + β² = (α + β)² − 2αβ = 49/4 − 3 = 37/4

Q5. If x + 1/x = 3, find x³ + 1/x³.

Use a³ + b³ = (a + b)³ − 3ab(a + b) with a = x, b = 1/x. Here ab = 1.
x³ + 1/x³ = 27 − 9 = 18

Q6. For what value of k does 2x² + kx + 3 = 0 have equal roots?

Set the discriminant to zero: k² − 24 = 0, so k = ±2√6.

Q7. The product of two consecutive positive integers is 306. Find them.

Let them be n and n + 1. Then n² + n − 306 = 0, which gives (n + 18)(n − 17) = 0.
Since n is positive, the integers are 17 and 18.

Pair of Linear Equations

Q8. Solve 2x + 3y = 11 and 2x − 4y = −24. Then find m if y = mx + 3.

Subtract the equations: 7y = 35, so y = 5. Then 2x + 15 = 11, so x = −2.
Put both in y = mx + 3: 5 = −2m + 3, so m = −1.

Arithmetic Progressions

Q9. Find the sum of the first 20 terms of 3, 7, 11, 15, …

a = 3, d = 4, n = 20.
S₂₀ = 10 × [6 + 76] = 820

Q10. Which term of the AP 5, 11, 17, 23, … is 131?

5 + (n − 1)6 = 131, so n − 1 = 21 and n = 22.
It is the 22nd term.

Q11. Find the sum of all two-digit numbers divisible by 7.

The list is 14, 21, …, 98. From 98 = 14 + (n − 1)7, n = 13.
Sum = 13/2 × (14 + 98) = 728

Triangles

Q12. In triangle ABC, DE ∥ BC, with D on AB and E on AC. If AD = 4 cm, DB = 6 cm and AE = 6 cm, find EC.

By the Basic Proportionality Theorem, AD/DB = AE/EC.
4/6 = 6/EC, so EC = 9 cm.

Q13. The areas of two similar triangles are in the ratio 16 : 25. If the larger triangle’s perimeter is 50 cm, find the smaller one’s perimeter.

Areas are in the ratio of squared sides, so the sides are in the ratio 4 : 5. Perimeters share that ratio.
(4/5) × 50 = 40 cm

Q14. A right triangle has legs 5 cm and 12 cm. Find the altitude to the hypotenuse.

Hypotenuse = 13 cm and area = 30 cm². Using the hypotenuse as base, ½ × 13 × h = 30.
h = 60/13 cm, about 4.62 cm.

Coordinate Geometry

Q15. Find the point on the x-axis equidistant from (2, −5) and (−2, 9).

Let the point be (x, 0) and equate squared distances:
(x − 2)² + 25 = (x + 2)² + 81
−4x + 29 = 4x + 85, so x = −7.
The point is (−7, 0).

Q16. Find the area of the triangle with vertices (1, 2), (4, 6) and (7, 2).

Area = ½ |1(6 − 2) + 4(2 − 2) + 7(2 − 6)| = ½ × 24 = 12 square units.
Check: base 6, height 4, so ½ × 6 × 4 = 12.

Trigonometry

Q17. Find sin²1° + sin²2° + sin²3° + … + sin²89°.

Pair the first term with the last: sin²1° + sin²89° = sin²1° + cos²1° = 1. This works for 44 pairs (1° to 44°), and sin²45° = 1/2 is left in the middle.
Total = 44.5

Watch out: Students often count 45 pairs. Remember that 45° pairs with itself.

Q18. Prove that (sin θ + cosec θ)² + (cos θ + sec θ)² = 7 + tan²θ + cot²θ.

Expand the left side:
sin²θ + 2 + cosec²θ + cos²θ + 2 + sec²θ
= 1 + 4 + (1 + cot²θ) + (1 + tan²θ)
= 7 + tan²θ + cot²θ. Proved.

Q19. From a point 30 m from the foot of a tower, the angle of elevation of the top is 60°. Find the height.

tan 60° = h/30, so h = 30√3 ≈ 51.96 m.

Circles

Q20. PA and PB are tangents from an external point P to a circle with centre O. If ∠APB = 80°, find ∠OAB.

The radius meets the tangent at 90°, so ∠OAP = ∠OBP = 90°. In quadrilateral OAPB, ∠AOB = 360° − 90° − 90° − 80° = 100°.
OA = OB, so ∠OAB = (180° − 100°)/2 = 40°.

Try These Yourself

Two concentric circles have radii 5 cm and 3 cm. Find the length of the chord of the larger circle that touches the smaller one. (Answer: 8 cm)
A cone has radius 7 cm and height 24 cm. Find its curved surface area. (Answer: 550 cm²)
A metal sphere of radius 3 cm is melted and recast into a cylinder of radius 1 cm. Find the cylinder’s height. (Answer: 36 cm)
Two dice are thrown together. What is the probability that the sum is a prime number? (Answer: 5/12)
The mean of five numbers is 27. After one number is removed, the mean of the other four is 25. Find the removed number. (Answer: 35)

A set like this makes good practice maths olympiad questions for 10th graders, especially right after you finish a chapter.

Where to Find Maths Olympiad Past Papers for Class 10

An actual paper shows you the true difficulty better than any guide. If you’re wondering where to find maths olympiad past papers for class 10, try these:

Official olympiad websites. The SOF site usually lists sample and earlier papers, but availability changes, so check the current page.
HBCSE (Homi Bhabha Centre for Science Education). Its site shares previous question papers for IOQM and later stages.
Your school or coaching centre. Teachers often keep old papers and mock tests.
Olympiad workbooks. These give chapter-wise questions with full solutions.

Much of this material is shared as a class 10 maths olympiad problems PDF, which is easy to print and solve offline. Use trusted sources only, since a wrong answer key can teach you a wrong method.

How to Practise Well

Reading solutions feels like progress, but solving is what builds skill.

Finish NCERT first. Olympiad problems are built on the same ideas.
Struggle for a while. Give each question 5 to 10 minutes before checking.
Keep a mistake notebook. Write down formulas, tricks and repeated errors.
Go chapter by chapter, then attempt timed papers.
Redo wrong questions after a few days. If you can solve them cold, you’ve learned them.

Using 10th class mathematics olympiad sample questions this way improves your speed and your confidence together.

Common Mistakes

Attempting hard problems before the basics are firm.
Reading word problems too fast and missing a condition.
Making small calculation slips.
Skipping the reasoning and everyday maths sections.
Guessing without first removing options that are clearly wrong.

Exam Day Tips

Begin with the questions you find easy.
In MCQs, cross out options you know are wrong.
Recheck signs and units.
If one question eats five minutes, move on and return later.
Stay calm. Panic costs more marks than difficulty.

Conclusion

There are no shortcuts here. Solve a few problems every day, learn from what you get wrong, and slowly move to full timed papers. Be patient with yourself. In a few weeks, maths olympiad questions for class 10 will feel far less intimidating, and your school exams will likely feel easier too.

Frequently Asked Questions

Are olympiad questions harder than board exam questions?

Usually yes. The syllabus overlaps, but you have to apply ideas in unfamiliar ways.

How much should I practise daily?

One to two hours of focused work, along with regular school study, is enough for most students.

Is NCERT enough?

It’s the foundation. Add a workbook and previous papers for extra practice.

Can I find maths olympiad questions for class 10 with answers online?

Yes. Official sample papers, workbooks and teacher-made worksheets often include answers. Study the method, not only the final answer.