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School Olympiads - International Mathematics Olympiad (IMO) previous year questions (PYQ)

251 previous year School Olympiads - International Mathematics Olympiad (IMO) questions, grouped by year, with answers and solutions.

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2025 (7 questions)

Q1.
A water tank is $\frac{3}{5}$ full. When 15 litres of water are drawn out, it becomes $\frac{3}{8}$ full. What is the total capacity of the water tank in litres?

(A)
{'key': 'A', 'text': '60 litres'}
(B)
{'key': 'B', 'text': '66.67 litres'}
(C)
{'key': 'C', 'text': '75 litres'}
(D)
{'key': 'D', 'text': '80 litres'}
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Q2.
A metallic sphere of radius $6\text{ cm}$ is melted and recast into small cones, each of radius $2\text{ cm}$ and height $3\text{ cm}$. How many such small cones are formed?

(A)
{'key': 'A', 'text': '54'}
(B)
{'key': 'B', 'text': '72'}
(C)
{'key': 'C', 'text': '108'}
(D)
{'key': 'D', 'text': '216'}
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2024 (129 questions)

Q1.

Olympiad Geometry: In a cyclic quadrilateral $ABCD$ in 'Number Theory: Primes, Divisibility Rules, Modular Arithmetic and Cryptarithms', the side lengths are $AB = 2, BC = 3, CD = 4, DA = 5$. By Ptolemy's Theorem, the product of diagonals $AC \cdot BD$ is:

(A)

23 ($AC \cdot BD = AB \cdot CD + BC \cdot DA = 2(4) + 3(5) = 8 + 15 = 23$) (Number Theory: Primes, Divisibility Rules, Modular Arithmetic and Cryptarithms)

(B)

20

(C)

26

(D)

30

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Q2.

Olympiad Geometry: In a cyclic quadrilateral $ABCD$ in 'Number Theory: Primes, Divisibility Rules, Modular Arithmetic and Cryptarithms', the side lengths are $AB = 2, BC = 3, CD = 4, DA = 5$. By Ptolemy's Theorem, the product of diagonals $AC \cdot BD$ is:

(A)

23 ($AC \cdot BD = AB \cdot CD + BC \cdot DA = 2(4) + 3(5) = 8 + 15 = 23$) (Number Theory: Primes, Divisibility Rules, Modular Arithmetic and Cryptarithms)

(B)

20

(C)

26

(D)

30

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2023 (21 questions)

Q1.
In an Olympiad competition problem on Coordinate Geometry & Trigonometric Ratios:
Find the numerical value of $X$ satisfying $$X = (16 \times 9) - 12 + 20$$.

(A)
{'key': 'A', 'text': '152'}
(B)
{'key': 'B', 'text': '156'}
(C)
{'key': 'C', 'text': '148'}
(D)
{'key': 'D', 'text': '161'}
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Q2.
A metallic sphere of radius $6\text{ cm}$ is melted and recast into small cones, each of radius $2\text{ cm}$ and height $3\text{ cm}$. How many such small cones are formed?

(A)
{'key': 'A', 'text': '54'}
(B)
{'key': 'B', 'text': '72'}
(C)
{'key': 'C', 'text': '108'}
(D)
{'key': 'D', 'text': '216'}
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2022 (21 questions)

Q1.
[Achievers Section - HOTS]
In an Olympiad competition problem on Mirror & Water Images, Paper Folding & Embedded Figures:
Find the numerical value of $X$ satisfying $$X = (39 \times 6) - 35 + 5$$.

(A)
{'key': 'A', 'text': '204'}
(B)
{'key': 'B', 'text': '208'}
(C)
{'key': 'C', 'text': '200'}
(D)
{'key': 'D', 'text': '213'}
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Q2.
In an Olympiad competition problem on Achievers Section: Multi-Step Algebra & Olympiad Geometry:
Find the numerical value of $X$ satisfying $$X = (43 \times 8) - 22 + 12$$.

(A)
{'key': 'A', 'text': '334'}
(B)
{'key': 'B', 'text': '338'}
(C)
{'key': 'C', 'text': '330'}
(D)
{'key': 'D', 'text': '343'}
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2021 (21 questions)

Q1.
In an Olympiad competition problem on Arithmetic & Geometric Progressions:
Find the numerical value of $X$ satisfying $$X = (42 \times 9) - 29 + 11$$.

(A)
{'key': 'A', 'text': '360'}
(B)
{'key': 'B', 'text': '364'}
(C)
{'key': 'C', 'text': '356'}
(D)
{'key': 'D', 'text': '369'}
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Q2.
[Achievers Section - 3 Marks]
Find the remainder when $3^{2024} + 5^{2024}$ is divided by $8$.

(A)
{'key': 'A', 'text': '0'}
(B)
{'key': 'B', 'text': '2'}
(C)
{'key': 'C', 'text': '4'}
(D)
{'key': 'D', 'text': '6'}
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2020 (15 questions)

Q1.
[Achievers Section - HOTS]
In an Olympiad competition problem on Mensuration (Surface Area & Volume):
Find the numerical value of $X$ satisfying $$X = (13 \times 4) - 20 + 17$$.

(A)
{'key': 'A', 'text': '49'}
(B)
{'key': 'B', 'text': '53'}
(C)
{'key': 'C', 'text': '45'}
(D)
{'key': 'D', 'text': '58'}
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Q2.
In an Olympiad competition problem on Number Theory, Advanced Algebra & Olympiad Geometry:
Find the numerical value of $X$ satisfying $$X = (40 \times 5) - 22 + 3$$.

(A)
{'key': 'A', 'text': '181'}
(B)
{'key': 'B', 'text': '185'}
(C)
{'key': 'C', 'text': '177'}
(D)
{'key': 'D', 'text': '190'}
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2019 (18 questions)

Q1.
[Achievers Section - HOTS]
In an Olympiad competition problem on Comparing Quantities, Profit-Loss & Simple Interest:
Find the numerical value of $X$ satisfying $$X = (42 \times 9) - 29 + 9$$.

(A)
{'key': 'A', 'text': '358'}
(B)
{'key': 'B', 'text': '362'}
(C)
{'key': 'C', 'text': '354'}
(D)
{'key': 'D', 'text': '367'}
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Q2.
In an Olympiad competition problem on Computation Operations & BODMAS Rules:
Find the numerical value of $X$ satisfying $$X = (26 \times 8) - 11 + 2$$.

(A)
{'key': 'A', 'text': '199'}
(B)
{'key': 'B', 'text': '203'}
(C)
{'key': 'C', 'text': '195'}
(D)
{'key': 'D', 'text': '208'}
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2018 (19 questions)

Q1.
In an Olympiad competition problem on Computation Operations & BODMAS Rules:
Find the numerical value of $X$ satisfying $$X = (17 \times 5) - 35 + 8$$.

(A)
{'key': 'A', 'text': '58'}
(B)
{'key': 'B', 'text': '62'}
(C)
{'key': 'C', 'text': '54'}
(D)
{'key': 'D', 'text': '67'}
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Q2.
In an Olympiad competition problem on Algebraic Expressions & Linear Equations:
Find the numerical value of $X$ satisfying $$X = (33 \times 7) - 32 + 8$$.

(A)
{'key': 'A', 'text': '207'}
(B)
{'key': 'B', 'text': '211'}
(C)
{'key': 'C', 'text': '203'}
(D)
{'key': 'D', 'text': '216'}
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