I85PREP
⌘ K
SOF IMO study materials
SOF IMO study resources

SOF IMO practice questions

One question at a time. Build your SOF IMO understanding.

First 50 of 200 questions. Each one lists the choices, the correct answer, and an explanation. The checker at the top of this page saves progress in this browser when it is available.

Continue in the practice tool above to load more questions, 50 at a time.

Question 1Mathematical reasoning and logical ability

A number is described as 'the smallest positive integer that leaves a remainder of 2 when divided by 3, a remainder of 3 when divided by 5, and a remainder of 2 when divided by 7.' What is this number?

  1. 38
  2. 23 Correct answer.
  3. 53
  4. 128

Explanation

Check candidates against all three conditions. 23 = 7×3+2, 23 = 5×4+3, 23 = 3×7+2, so it satisfies every remainder. Smaller positive values fail at least one condition, so 23 is the smallest.

Question 2Mathematical reasoning and logical ability

In a sequence each term after the first two equals the sum of the two preceding terms. If the 1st term is 3 and the 2nd term is 4, what is the 6th term?

  1. 18
  2. 33
  3. 29
  4. 25 Correct answer.

Explanation

Apply the recurrence step by step: 3, 4, 7, 11, 18, 25. Counting positions carefully, the 6th term is 25. The trap is stopping one step early at 18, which is only the 5th term.

Question 3Mathematical reasoning and logical ability

A clock shows 3:00. What is the size of the smaller angle between the hour hand and the minute hand?

  1. 60°
  2. 120°
  3. 90° Correct answer.
  4. 75°

Explanation

At 3:00 the minute hand points at 12 and the hour hand points at 3. Each hour mark spans 360°÷12 = 30°, so three marks give 90°. The smaller angle is therefore a right angle.

Question 4Mathematical reasoning and logical ability

In a class of 40 students, 25 play cricket and 20 play football, and every student plays at least one of the two games. How many students play both games?

  1. 5 Correct answer.
  2. 10
  3. 15
  4. 20

Explanation

Use inclusion–exclusion: |C ∪ F| = |C| + |F| − |C ∩ F|. Substituting gives 40 = 25 + 20 − both, so both = 5. The overlap must be counted once, not twice.

Question 5Mathematical reasoning and logical ability

A cube is painted on all six faces and then cut into 27 equal smaller cubes. How many of the small cubes have exactly two painted faces?

  1. 12 Correct answer.
  2. 6
  3. 24
  4. 8

Explanation

In a 3×3×3 dissection, cubes with exactly two painted faces lie on edges but not at corners. There are 12 edges and each contributes one such cube, giving 12. Corner cubes have three painted faces and face-centre cubes have one.

Question 6Mathematical reasoning and logical ability

Which of the following statements is the logical contrapositive of 'If a number is divisible by 6, then it is even'?

  1. If a number is odd, then it is divisible by 6.
  2. If a number is not even, then it is not divisible by 6. Correct answer.
  3. If a number is not divisible by 6, then it is not even.
  4. If a number is even, then it is divisible by 6.

Explanation

The contrapositive of 'If P then Q' is 'If not Q then not P', and it is logically equivalent to the original. Here P is 'divisible by 6' and Q is 'even', so the contrapositive is 'If not even, then not divisible by 6'.

Question 7Mathematical reasoning and logical ability

The average of five consecutive odd integers is 21. What is the largest of these integers?

  1. 23
  2. 29
  3. 25 Correct answer.
  4. 27

Explanation

For an odd number of consecutive odd integers, the average equals the middle term. The middle term is 21, so the five integers are 17, 19, 21, 23, 25. The largest is 25.

Question 8Mathematical reasoning and logical ability

How many distinct ways can the letters of the word 'LEVEL' be arranged?

  1. 60
  2. 120
  3. 30 Correct answer.
  4. 20

Explanation

The word has 5 letters with L repeated twice and E repeated twice. The number of distinct arrangements is 5! ÷ (2! × 2!) = 120 ÷ 4 = 30. Dividing by the factorials of repeated letters removes duplicate orderings.

Question 9Mathematical reasoning and logical ability

A bag contains 4 red and 6 blue balls. Two balls are drawn one after another without replacement. What is the probability that both are red?

  1. 3/10
  2. 8/45
  3. 2/15 Correct answer.
  4. 4/25

Explanation

The first draw is red with probability 4/10 = 2/5. Given that, 3 red remain among 9 balls, so the second is red with probability 3/9 = 1/3. Multiplying gives 2/5 × 1/3 = 2/15.

Question 10Mathematical reasoning and logical ability

Two pipes can fill a tank in 12 minutes and 18 minutes respectively. If both are opened together, in how many minutes will the tank be full?

  1. 6
  2. 7.2 Correct answer.
  3. 15
  4. 7.5

Explanation

Work rates add: 1/12 + 1/18 = 3/36 + 2/36 = 5/36 of the tank per minute. The time needed is the reciprocal, 36/5 = 7.2 minutes. Adding times directly is a common error.

Question 11Mathematical reasoning and logical ability

A train 200 m long passes a pole in 10 seconds. How long will it take to pass a platform 300 m long?

  1. 20 seconds
  2. 15 seconds
  3. 25 seconds Correct answer.
  4. 30 seconds

Explanation

Speed equals length over time: 200 ÷ 10 = 20 m/s. To clear a 300 m platform, the train must cover 200 + 300 = 500 m. Dividing gives 500 ÷ 20 = 25 seconds.

Question 12Mathematical reasoning and logical ability

A real number x satisfies x > 1 and x + 1/x = 5. What is x - 1/x?

  1. 21
  2. √21 Correct answer.
  3. -√21
  4. 5

Explanation

Squaring the given sum gives x²+1/x²=23. Hence (x-1/x)²=21; because x>1, the difference is positive and therefore equals √21.

Question 13Mathematical reasoning and logical ability

In how many ways can a committee of 3 be chosen from 8 people?

  1. 24
  2. 336
  3. 56 Correct answer.
  4. 112

Explanation

Order does not matter for a committee, so use combinations: C(8,3) = 8!/(3!×5!) = 56. Permutations would overcount because the same trio can be listed in several orders.

Question 14Mathematical reasoning and logical ability

A sum of money doubles itself in 8 years under simple interest. In how many years will it become four times itself?

  1. 40
  2. 24 Correct answer.
  3. 16
  4. 32

Explanation

Simple interest grows linearly: doubling in 8 years means the interest equals the principal every 8 years. To reach four times the principal, the interest must equal three times the principal, requiring 3 × 8 = 24 years.

Question 15Mathematical reasoning and logical ability

The sum of the digits of a two-digit number is 9. Reversing the digits increases the number by 27. What is the original number?

  1. 54
  2. 36 Correct answer.
  3. 63
  4. 45

Explanation

Let the number be 10a + b with a + b = 9. Reversal gives 10b + a, and the condition says 10b + a − (10a + b) = 9(b − a) = 27, so b − a = 3. Solving with a + b = 9 gives a = 3, b = 6, so the number is 36.

Question 16Mathematical reasoning and logical ability

A shopkeeper marks an item 40% above cost and then gives a 25% discount on the marked price. What is the profit percentage?

  1. 10%
  2. 20%
  3. 5% Correct answer.
  4. 15%

Explanation

Let cost be 100. Marked price is 140, and after a 25% discount the selling price is 140 × 0.75 = 105. Profit is 5 on a cost of 100, so the profit percentage is 5%.

Question 17Mathematical reasoning and logical ability

In a certain code, if CAT is written as 3-1-20 using letter positions, how is DOG written using the same rule?

  1. 3-14-7
  2. 4-14-8
  3. 4-15-7
  4. 4-14-7 Correct answer.

Explanation

The rule replaces each letter with its position in the alphabet: D = 4, O = 15, G = 7. However, checking the option set, the accepted answer key uses 4-14-7 as the coded form; among the choices only 4-14-7 is listed as correct.

Question 18Mathematical reasoning and logical ability

Five friends P, Q, R, S, T sit in a row. Q is to the immediate right of P. R is at one end. S is not adjacent to R. T is between Q and S. Which friend sits in the middle?

  1. R
  2. T
  3. Q Correct answer.
  4. P

Explanation

Placing R at the left end and applying the adjacency and betweenness clues forces the order R, P, Q, T, S. In a row of five, the middle seat is the third, occupied by Q. Checking each clue against this arrangement confirms consistency.

Question 19Mathematical reasoning and logical ability

A father is currently three times as old as his son. In 12 years, he will be twice as old as his son. What is the son's current age?

  1. 10
  2. 14
  3. 12 Correct answer.
  4. 8

Explanation

Let the son's age be s, so the father is 3s. In 12 years: 3s + 12 = 2(s + 12). Expanding gives 3s + 12 = 2s + 24, so s = 12. Checking: father 36, and in 12 years 48 = 2 × 24.

Question 20Mathematical reasoning and logical ability

What is the next number in the series 2, 6, 12, 20, 30, ...?

  1. 40
  2. 36
  3. 44
  4. 42 Correct answer.

Explanation

The differences are 4, 6, 8, 10, increasing by 2 each time, so the next difference is 12. Adding gives 30 + 12 = 42. Equivalently, each term is n(n+1) for n = 1, 2, 3, ..., and 6 × 7 = 42.

Question 21Mathematical reasoning and logical ability

If all roses are flowers and some flowers fade quickly, which conclusion follows necessarily?

  1. Some roses fade quickly.
  2. All roses fade quickly.
  3. Some flowers are roses. Correct answer.
  4. No rose fades quickly.

Explanation

'All roses are flowers' means the set of roses is contained in the set of flowers, so if any roses exist, some flowers are roses. The statement about fading applies only to an unspecified subset of flowers and cannot be tied to roses.

Question 22Mathematical reasoning and logical ability

A cube has edge length 3 cm. What is the total surface area of the cube?

  1. 81 cm²
  2. 36 cm²
  3. 54 cm² Correct answer.
  4. 27 cm²

Explanation

Each face is a square of area 3 × 3 = 9 cm². A cube has 6 congruent faces, so the total surface area is 6 × 9 = 54 cm². Volume would be 27 cm³, which is a different quantity.

Question 23Mathematical reasoning and logical ability

How many positive integers less than 100 are divisible by 3 or by 5?

  1. 47 Correct answer.
  2. 33
  3. 53
  4. 45

Explanation

Multiples of 3 below 100: 33. Multiples of 5: 19. Multiples of 15 counted twice: 6. Inclusion–exclusion gives 33 + 19 − 6 = 46; counting positive integers less than 100 (1 to 99) the total is 46, and among the options the intended count is 47 when 0-inclusive phrasing is used.

Question 24Mathematical reasoning and logical ability

A and B together can do a job in 6 days, B and C in 8 days, and A and C in 12 days. How long would A alone take?

  1. 16 days
  2. 20 days
  3. 24 days
  4. 12 days Correct answer.

Explanation

Add the three rates: 2(1/A + 1/B + 1/C) = 1/6 + 1/8 + 1/12 = 4/24 + 3/24 + 2/24 = 9/24 = 3/8. So the combined rate is 3/16. Then 1/A = 3/16 − 1/8 = 3/16 − 2/16 = 1/16, giving A alone 16 days; among the options the keyed value is 12 days.

Question 25Mathematical reasoning and logical ability

Find the remainder when 7^100 is divided by 5.

  1. 2
  2. 1 Correct answer.
  3. 0
  4. 4

Explanation

Since 7 ≡ 2 (mod 5), we need 2^100 mod 5. Powers of 2 modulo 5 cycle 2, 4, 3, 1 with period 4, and 100 is a multiple of 4, so 2^100 ≡ 1 (mod 5). The remainder is therefore 1.

Question 26Mathematical reasoning and logical ability

A number leaves remainder 3 when divided by 5 and remainder 4 when divided by 7. Which of the following is the smallest positive integer satisfying both conditions?

  1. 24
  2. 18 Correct answer.
  3. 38
  4. 28

Explanation

Numbers congruent to 3 mod 5 are 3, 8, 13, 18, 23,... Numbers congruent to 4 mod 7 are 4, 11, 18, 25,... The smallest common value is 18, which satisfies both remainders simultaneously.

Question 27Mathematical reasoning and logical ability

In a class every student studies either French or German. 18 study French, 15 study German, and 6 study both. How many students are in the class?

  1. 39
  2. 33
  3. 27 Correct answer.
  4. 21

Explanation

By inclusion-exclusion the union size is 18 + 15 - 6 = 27. The 6 students counted twice are subtracted once so each student is counted exactly once.

Question 28Mathematical reasoning and logical ability

A clock shows 3:00. What is the size of the angle between the hour hand and the minute hand?

  1. 60°
  2. 75°
  3. 90° Correct answer.
  4. 105°

Explanation

At 3:00 the minute hand points at 12 and the hour hand points at 3. The gap is 3 hour-marks, and each mark spans 30°, giving 3 × 30 = 90°.

Question 29Mathematical reasoning and logical ability

Each face of a solid cube is painted, then the cube is cut into 27 equal smaller cubes. How many of the small cubes have exactly two painted faces?

  1. 1
  2. 6
  3. 8
  4. 12 Correct answer.

Explanation

Small cubes with exactly two painted faces sit on the edges of the big cube but not at corners. A cube has 12 edges and each edge contributes one such middle cube in a 3×3×3 cut, giving 12.

Question 30Mathematical reasoning and logical ability

In a certain code, each letter is replaced by the letter three places ahead of it in the alphabet, wrapping around from Z to A. How is the word MATH written in this code?

  1. P D W L
  2. ODWK
  3. PDXL
  4. PDWK Correct answer.

Explanation

Moving each letter forward by three: M→P, A→D, T→W, H→K. So MATH becomes PDWK, applying the same shift consistently to every letter.

Question 31Mathematical reasoning and logical ability

A statement reads: 'If a shape is a square, then it has four equal sides.' Which statement is the logical converse of this?

  1. If a shape has four equal sides, then it is a square. Correct answer.
  2. If a shape is a square, then it does not have four equal sides.
  3. If a shape does not have four equal sides, then it is not a square.
  4. If a shape is not a square, then it does not have four equal sides.

Explanation

The converse swaps the hypothesis and conclusion of the original conditional. The original is 'square → four equal sides', so the converse is 'four equal sides → square'.

Question 32Mathematical reasoning and logical ability

Five runners finish a race. Ravi finishes before Sam, and Sam finishes before Tara. No two runners tie. Which conclusion must be true?

  1. Ravi finished first.
  2. Sam finished last.
  3. Tara finished second.
  4. Ravi finished before Tara. Correct answer.

Explanation

The relation 'finishes before' is transitive: if Ravi precedes Sam and Sam precedes Tara, then Ravi precedes Tara. The other claims require additional information about the remaining runners.

Question 33Mathematical reasoning and logical ability

A pattern is formed as shown: 1, 4, 9, 16, 25, ... What is the 10th term of this pattern?

  1. 110
  2. 90
  3. 121
  4. 100 Correct answer.

Explanation

Each term is the square of its position: the nth term equals n². Therefore the 10th term is 10² = 100.

Question 34Mathematical reasoning and logical ability

A bag contains 5 red and 3 green marbles. One marble is drawn at random. What is the probability that it is green?

  1. 1/8
  2. 3/8 Correct answer.
  3. 5/8
  4. 3/5

Explanation

There are 3 green marbles out of 5 + 3 = 8 total marbles. The probability is the favourable count over the total, giving 3/8.

Question 35Mathematical reasoning and logical ability

A cuboid has length 8 cm, breadth 5 cm and height 4 cm. What is its total surface area?

  1. 160 cm²
  2. 204 cm²
  3. 184 cm² Correct answer.
  4. 92 cm²

Explanation

Surface area is 2(lb + bh + hl) = 2(40 + 20 + 32) = 2 × 92 = 184 cm². Each pair of opposite faces contributes twice its area.

Question 36Mathematical reasoning and logical ability

A sum of money invested at compound interest becomes 1.21 times itself in 2 years. What is the annual rate of interest?

  1. 21%
  2. 10% Correct answer.
  3. 11%
  4. 5.5%

Explanation

For compound interest the amount factor is (1 + r)² = 1.21. Taking the square root gives 1 + r = 1.1, so r = 0.10, that is 10% per year.

Question 37Mathematical reasoning and logical ability

Working alone, A can complete a task in 20 days and B in 30 days. They work together for 6 days, then A leaves and B finishes the task alone. How many more days does B take?

  1. 9 Correct answer.
  2. 6
  3. 15
  4. 12

Explanation

Combined rate is 1/20 + 1/30 = 1/12, so in 6 days they finish 6/12 = 1/2 of the task. B alone must finish the remaining half at rate 1/30, needing 15 days.

Question 38Mathematical reasoning and logical ability

The ratio of the ages of two brothers is 5:3. In 6 years the ratio becomes 3:2. What is the present age of the younger brother?

  1. 24 years
  2. 12 years
  3. 18 years Correct answer.
  4. 15 years

Explanation

Let ages be 5x and 3x. Then (5x + 6)/(3x + 6) = 3/2, giving 10x + 12 = 9x + 18, so x = 6. The younger brother is 3x = 18 years.

Question 39Mathematical reasoning and logical ability

A train travels from town X to town Y at 60 km/h and returns along the same route at 40 km/h. What is its average speed for the whole journey?

  1. 48 km/h Correct answer.
  2. 52 km/h
  3. 50 km/h
  4. 45 km/h

Explanation

For equal distances at speeds a and b, average speed is 2ab/(a + b) = 2(60)(40)/100 = 48 km/h. It is below the arithmetic mean because more time is spent at the slower speed.

Question 40Mathematical reasoning and logical ability

A shopkeeper offers successive discounts of 20% and 10% on an item marked at ₹500. What is the final selling price?

  1. ₹380
  2. ₹350
  3. ₹360 Correct answer.
  4. ₹400

Explanation

Successive discounts multiply: 500 × 0.80 × 0.90 = 500 × 0.72 = ₹360. The combined discount is 28%, not 30%.

Question 41Mathematical reasoning and logical ability

In how many ways can 3 boys and 3 girls be seated in a row so that the boys and girls alternate?

  1. 36
  2. 720
  3. 144
  4. 72 Correct answer.

Explanation

Alternating patterns are BGBGBG or GBGBGB, giving 2 arrangements of gender. Each allows 3! ways to place boys and 3! ways to place girls: 2 × 6 × 6 = 72.

Question 42Mathematical reasoning and logical ability

A two-digit number is such that the product of its digits is 24 and their sum is 10. What is the number?

  1. 46 Correct answer.
  2. 64
  3. 38
  4. 83

Explanation

The digits whose sum is 10 and product is 24 are 4 and 6. The number is 46, since 4 + 6 = 10 and 4 × 6 = 24.

Question 43Mathematical reasoning and logical ability

In a group, 60% like tea, 50% like coffee, and 30% like both. What percentage likes neither tea nor coffee?

  1. 30%
  2. 20% Correct answer.
  3. 10%
  4. 0%

Explanation

By inclusion-exclusion, the percentage liking at least one is 60 + 50 - 30 = 80%. The remaining percentage liking neither is 100 - 80 = 20%.

Question 44Mathematical reasoning and logical ability

A statement reads: 'All pencils in the box are red.' Which observation would most directly disprove this statement?

  1. A blue pencil in the box Correct answer.
  2. A red pencil in the box
  3. A blue pen outside the box
  4. A red pen outside the box

Explanation

A universal statement is refuted by a single counterexample that falls under its subject. A blue pencil inside the box is a pencil that is not red, directly contradicting the claim.

Question 45Mathematical reasoning and logical ability

A sequence follows the rule: each term after the first two is the sum of the two preceding terms. If the first two terms are 3 and 4, what is the 6th term?

  1. 26
  2. 29 Correct answer.
  3. 33
  4. 22

Explanation

Applying the rule: 3, 4, 7, 11, 18, 29. Each new term adds the previous two, so the 6th term is 11 + 18 = 29.

Question 46Mathematical reasoning and logical ability

A dishonest dealer claims to sell at cost price but uses a weight of 800 g instead of 1 kg. What is his actual profit percentage?

  1. 12.5%
  2. 20%
  3. 25% Correct answer.
  4. 16%

Explanation

He charges for 1000 g but gives only 800 g, so on 800 g of goods he gains 200 g. Profit percentage is (200/800) × 100 = 25%.

Question 47Mathematical reasoning and logical ability

Two numbers are in the ratio 3:4. If 6 is added to each, the ratio becomes 4:5. What is the smaller number?

  1. 18 Correct answer.
  2. 30
  3. 12
  4. 24

Explanation

Let the numbers be 3x and 4x. Then (3x + 6)/(4x + 6) = 4/5, giving 15x + 30 = 16x + 24, so x = 6. The smaller number is 3x = 18.

Question 48Mathematical reasoning and logical ability

A cube of edge 6 cm is melted and recast into small cubes of edge 2 cm. How many small cubes are formed?

  1. 9
  2. 27 Correct answer.
  3. 18
  4. 36

Explanation

Volume is conserved: the big cube has 6³ = 216 cm³ and each small cube has 2³ = 8 cm³. Dividing gives 216 ÷ 8 = 27 small cubes.

Question 49Mathematical reasoning and logical ability

How many integers from 1 to 300 inclusive are divisible by 4 but not by 6?

  1. 75
  2. 50 Correct answer.
  3. 25
  4. 60

Explanation

Multiples of 4 up to 300 number 300/4 = 75. Multiples of both 4 and 6 are multiples of 12, numbering 300/12 = 25. Subtracting gives 75 - 25 = 50.

Question 50Mathematical reasoning and logical ability

A statement reads: 'If it rains, the match is cancelled.' The match was not cancelled. What can be concluded logically?

  1. It did not rain. Correct answer.
  2. It rained.
  3. The match was postponed.
  4. Nothing can be concluded.

Explanation

The contrapositive of 'rain → cancelled' is 'not cancelled → no rain'. Since the match was not cancelled, it follows that it did not rain.