Maths Olympiad Practice Test 1 — IMO-style (Class 9)

Q4. Riya walks 12 m towards the north. She turns right and walks 5 m, then turns right again and walks 20 m. Finally she turns left and walks 10 m. How far is she from her starting point, and in which direction?

Q5. In a row of 40 students, Anil is 14th from the left end and Bina is 9th from the right end. How many students are sitting between Anil and Bina?

Q6. Pointing to a girl in a photograph, Kiran said, “She is the daughter of the only son of my father’s father.” Kiran is a boy. How is the girl related to Kiran?

Q7. Find the odd one out: 203, 209, 221, 289

Q8. Find the missing number: 3 : 10 :: 7 : 50 :: 11 : ?

Q9. What is the angle between the hour hand and the minute hand of a clock at 4:36?

Q10. A mirror is placed along the dashed line MN, to the right of the figure. Which option shows the correct mirror image of the figure?

Q11. A square sheet of paper is folded as shown (Fold 1: left half over the right half; Fold 2: top half down over the bottom half). A triangular hole is then punched through all the layers. Which option shows the sheet when it is completely unfolded?

Q12. How many triangles are there in the figure below? (The figure is a square with both diagonals drawn and the mid-points of adjacent sides joined.)

Q13. Which option completes the pattern in the grid?

Q14. Statements: All pens are books. Some books are bags.
Conclusions: I. Some pens are bags. II. Some bags are books.
Which conclusion(s) follow from the statements?

Q15. Six friends P, Q, R, S, T and U sit in a row facing north. S sits at one end of the row. Exactly two people sit between S and R. T sits immediately to the right of R. U sits immediately to the left of S. Q does not sit at either end. Who sits third from the left end?

Q16. The value of (√5 + √3)/(√5 − √3) + (√5 − √3)/(√5 + √3) is

Q17. Simplify: (2ⁿ⁺² − 2ⁿ⁺¹) ÷ (2ⁿ⁺¹ + 2ⁿ)

Q18. Which fraction is equal to the recurring decimal 0.2454545… (the digits 45 repeat)?

Q19. When x³ − 3x² + 4x + k is divided by (x − 2), the remainder is 7. What is the value of k?

Q20. If (x + 1) and (x − 2) are both factors of x³ + ax² + bx − 2, then a + b equals

Q21. If x + 1/x = 5, then x³ + 1/x³ equals

Q22. Without cubing the numbers, find the value of (−12)³ + 7³ + 5³.

Q23. The line 3x + 4y = 24 meets the x-axis at P and the y-axis at Q. If O is the origin, what is the area of triangle OPQ?

Q24. For what value of k is x = 2, y = −1 a solution of the equation kx + 3y = k + 5?

Q25. Two parallel lines are cut by a transversal. Of the two interior angles on the same side of the transversal, one is 40° more than the other. The larger angle is

Q26. In triangle ABC, ∠A : ∠B : ∠C = 2 : 3 : 4. The bisectors of ∠B and ∠C meet at O. Find ∠BOC.

Q27. In parallelogram ABCD, ∠A = (3x + 10)° and ∠C = (5x − 30)°. What is the measure of ∠B?

Q28. The sides of a triangle are 8 cm, 10 cm and 12 cm. The mid-points of its sides are joined to form a new triangle. What is the perimeter of the new triangle?

Q29. A chord of a circle of radius 13 cm is at a distance of 5 cm from the centre. What is the length of the chord?

Q30. Arc AB of a circle subtends an angle of 130° at the centre O. C is a point on the major arc and D is a point on the minor arc. What is ∠ADB?

Q31. Using Heron’s formula, find the area of a triangle whose sides are 17 cm, 25 cm and 28 cm.

Q32. A cone has base radius 7 cm and height 24 cm. What is its curved surface area? (Take π = 22/7.)

Q33. The radius of a sphere is increased by 20%. By what percentage does its surface area increase?

Q34. The observations 11, 13, 15, x, x + 2, 23, 25, 27 are in ascending order and their median is 18. What is their mean?

Q35. A coin is tossed 400 times and a head appears 236 times. Based on this experiment, what is the probability of getting a tail?

Q36. A school bag is marked at ₹2,400. The shop gives a discount of 20% and then a further discount of 10% on the reduced price. How much does the bag cost?

Q37. A shopkeeper sells two mobile phones for ₹9,900 each. On one he gains 10% and on the other he loses 10%. What is his overall result?

Q38. Meera deposits ₹15,000 in a bank at 8% per annum compound interest, compounded yearly. How much interest does she earn in 2 years?

Q39. A train 240 m long passes completely through a platform 360 m long in 30 seconds. What is the speed of the train?

Q40. Arjun can paint a wall in 12 days and Sameer can paint it in 18 days. They work together for 4 days, after which Arjun leaves. In how many more days will Sameer finish the work?

Q41. 15 August 2026 is a Saturday. On which day of the week does 26 January 2027 fall?

Q42. A wall clock runs fast: in every real hour it gains 3 minutes. It is set to the correct time at 7:00 a.m. What is the correct time when this clock shows 11:12 a.m. the same morning?

Q43. A cuboidal water tank measures 2.5 m × 1.6 m × 1.2 m. A pipe fills it at 40 litres per minute. How long will it take to fill the empty tank?

Q44. The price of cooking gas rises by 25%. By what percentage must a family reduce its consumption so that its spending on gas does not change?

Q45. Grandmother divides ₹6,500 among her three grandchildren Anu, Bittu and Chinu in the ratio 1/2 : 1/3 : 1/4. How much does Bittu get?

Q46. What is the sum of all two-digit numbers which leave a remainder of 3 when divided by 7?

Q47. If x = 3 + 2√2, find the value of (x⁶ + 1)/x³.

Q48. The area of triangle ABC is 72 cm². D is the mid-point of BC, E is the mid-point of AD and F is the mid-point of BE. What is the area of triangle AEF?

Q49. A solid metal sphere of radius 6 cm is melted and drawn into a uniform wire of diameter 4 mm. What is the length of the wire?

Q50. The mean of 25 observations is 36. The mean of the first 13 observations is 32 and the mean of the last 13 observations is 40. What is the 13th observation?