Thailand International Mathematical Olympiad

Preliminary Round - Practice Set

Senior Secondary – Easy Level
Time Allowed: 90 minutes
Total Questions: 25
Marks per Question: 4
Total Score: 100
Instructions: Answers in simplest form; no calculators allowed.

Logical Thinking
General Skills Tested: Basic arithmetic reasoning, simple patterns, and spa-
tial understanding.
Questions and Solutions:
• Q1: Peter walks 15 km north, then 8 km south, and finally 3 km
north. How far is he from his starting point?
We need Peter’s final distance from his starting point after three moves.
Define the path:
- Start at 0 km.
- North 15 km: 0 + 15 = 15.
- South 8 km: 15 − 8 = 7.
- North 3 km: 7 + 3 = 10.
Calculate distance:
- Final position: 10 km north.
- Distance from start = |10 − 0| = 10.
Verify:
- Net displacement: 15 − 8 + 3 = 10.
- Straight-line path, no turns.
Final Answer: 10
• Q2: If the day after tomorrow is Wednesday, what day is it
today?
We need today’s day given the day after tomorrow is Wednesday.
Work backwards:
- Day after tomorrow = Wednesday.
- Tomorrow = Tuesday (one day before Wednesday).
- Today = Monday (one day before Tuesday).
Verify:

1
 - Today (Monday) + 1 = Tuesday.
- Tuesday + 1 = Wednesday, matches.
Final Answer: Monday

• Q3: Solve 2x + 5 = 13.

We need x such that 2x + 5 = 13.
Isolate x:
- Subtract 5: 2x + 5 − 5 = 13 − 5.
- 2x = 8.
8
- Divide by 2: x = 2 = 4.
Verify:
- 2 · 4 + 5 = 8 + 5 = 13.
- Equation holds.
Final Answer: 4
• Q4: A sequence starts 2, 5, 8, 11. What is the next number?
We need the next term in the sequence 2, 5, 8, 11.
Find the pattern:
- Differences: 5 − 2 = 3, 8 − 5 = 3, 11 − 8 = 3.
- Constant difference = 3.
Calculate next term:
- 11 + 3 = 14.
Verify:
- Sequence: 2, 5, 8, 11, 14.
- Differences: 3, 3, 3, 3, consistent.
Final Answer: 14
• Q5: If a1 = 3 and an+1 = an + 4 for n ≥ 1, find a4 .
We need the 4th term of the sequence starting with a1 = 3.
Compute terms:
- a2 = a1 + 4 = 3 + 4 = 7.
- a3 = a2 + 4 = 7 + 4 = 11.
- a4 = a3 + 4 = 11 + 4 = 15.
Verify:
- a1 = 3, a2 = 7, a3 = 11, a4 = 15.
- Differences: 4, 4, 4, matches rule.
Final Answer: 15

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Algebra
General Skills Tested: Simple equations, factoring, and exponent rules.
Questions and Solutions:
• Q6: Solve x − 7 = 12.
We need x such that x − 7 = 12.
Isolate x:
- Add 7: x − 7 + 7 = 12 + 7.
- x = 19.
Verify:
- 19 − 7 = 12.
- Holds true.
Final Answer: 19
• Q7: Factorize x2 + 5x + 6.
We need to factor x2 + 5x + 6.
Find factors:
- Seek a · b = 6, a + b = 5.
- Pairs: (1, 6), (2, 3), (-1, -6), (-2, -3).
- 2 + 3 = 5, 2 · 3 = 6.
Write factorization:
- x2 + 5x + 6 = (x + 2)(x + 3).
Verify:
- Expand: x · x + x · 3 + 2 · x + 2 · 3 = x2 + 3x + 2x + 6 = x2 + 5x + 6.
- Correct.
Final Answer: (x + 2)(x + 3)
• Q8: Simplify 42 · 43 .
We need to simplify 42 · 43 .
Use exponent rule:
- am · an = am+n .
- 42 · 43 = 42+3 = 45 .
Compute:
- 45 = 4 · 4 · 4 · 4 · 4.
- 16 · 4 = 64, 64 · 4 = 256, 256 · 4 = 1024.
Verify:
- 42 = 16, 43 = 64, 16 · 64 = 1024.
Final Answer: 1024

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 • Q9: If 2x + 3 = 5x − 6, find x.
We need x such that 2x + 3 = 5x − 6.
Solve:
- Subtract 2x: 3 = 5x − 2x − 6.
- 3 = 3x − 6.
- Add 6: 3 + 6 = 3x.
- 9 = 3x.
- Divide by 3: x = 3.
Verify:
- Left: 2 · 3 + 3 = 6 + 3 = 9.
- Right: 5 · 3 − 6 = 15 − 6 = 9.
- 9 = 9, true.
Final Answer: 3
4
• Q10: Simplify 12x
3x2 .
4
We need to simplify 12x
3x2 .
Divide coefficients and exponents:
12
- 3 = 4.
x4
- x2 = x4−2 = x2 .
Combine:
12x4
- 3x2 = 4x2 .
Verify:
- 4x2 · 3x2 = 12x2+2 = 12x4 .
- Matches numerator.
Final Answer: 4x2

Number Theory
General Skills Tested: Basic divisibility, remainders, and prime recognition.
Questions and Solutions:
• Q11: Find the remainder when 25 is divided by 7.
We need 25 mod 7.
Divide:
- 7 · 3 = 21.
- 25 − 21 = 4.
Verify:
- 25 = 7 · 3 + 4.

4
 - 4 < 7, remainder correct.
Final Answer: 4
• Q12: What is the smallest prime number greater than 10?
We need the smallest prime > 10.
Test numbers:
- 11: Not divisible by 2, 3 (1+1=2), 5 (not 0 or 5), prime.
- 10: 2 · 5, not prime.
Verify:
- 11 is prime, next is 13.
Final Answer: 11
• Q13: How many positive factors does 12 have?
We need the number of positive factors of 12.
Factorize:
- 12 = 22 · 3.
Count factors:
- (2 + 1)(1 + 1) = 3 · 2 = 6.
List:
- 1, 2, 3, 4, 6, 12.
Verify:
- 6 factors, matches calculation.
Final Answer: 6

• Q14: Find the units digit of 34 .

We need the units digit of 34 .
Compute:
- 31 = 3.
- 32 = 9.
- 33 = 27, units 7.
- 34 = 27 · 3 = 81, units 1.
Verify:
- 9 · 9 = 81, units 1.
Final Answer: 1
• Q15: If x is an integer and x ≡ 2 (mod 5), what is the smallest
positive x?
We need the smallest positive integer x such that x ≡ 2 (mod 5).
Solve:

5
 - x = 5k + 2, k ≥ 0.
- k = 0: x = 2.
Verify:
- 2 mod 5 = 2.
- 2 < 7 (next is 7).
Final Answer: 2

Geometry
General Skills Tested: Basic area, perimeter, and angle properties.
Questions and Solutions:
• Q16: Find the area of a rectangle with length 8 and width 5.
We need the area of a rectangle with dimensions 8 and 5.
Formula:
- Area = length · width.
Compute:
- 8 · 5 = 40.
Verify:
- Units irrelevant, 40 correct.
Final Answer: 40
• Q17: What is the perimeter of a square with side length 6?
We need the perimeter of a square with side 6.
Formula:
- Perimeter = 4 · side.
Compute:
- 4 · 6 = 24.
Verify:
- 6 + 6 + 6 + 6 = 24.
Final Answer: 24
• Q18: In a triangle, two angles are 50° and 60°. What is the third
angle?
We need the third angle of a triangle with angles 50° and 60°.
Use angle sum:
- 50◦ + 60◦ + x = 180◦ .
- 110◦ + x = 180◦ .
- x = 180◦ − 110◦ = 70◦ .

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 Verify:
- 50 + 60 + 70 = 180.
Final Answer: 70

• Q19: Find the area of a triangle with base 10 and height 4.

We need the area of a triangle with base 10 and height 4.
Formula:
1
- Area = 2 · base · height.
Compute:
1
- 2 · 10 · 4 = 5 · 4 = 20.
Verify:
- Simple triangle, calculation holds.
Final Answer: 20

• Q20: What is the circumference of a circle with radius 7? (Use

π = 22
7 ).
We need the circumference of a circle with radius 7, π = 22
7 .
Formula:
- Circumference = 2πr.
Compute:
22
- 2· 7 · 7.
- 2 · 22 = 44.
Verify:
44
- 7 · 7 = 44.
Final Answer: 44

Combinatorics
General Skills Tested: Simple counting, permutations, and probabilities.
Questions and Solutions:
• Q21: How many ways can you arrange 3 different books on a
shelf ?
We need the number of arrangements of 3 distinct books.
Use permutations:
- 3! = 3 · 2 · 1 = 6.
Verify:
- ABC, ACB, BAC, BCA, CAB, CBA = 6.
Final Answer: 6

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• Q22: A bag has 4 red and 5 blue balls. Find the probability of
picking a red ball.
We need the probability of picking a red ball from 4 red and 5 blue.
Total balls:
- 4 + 5 = 9.
Favorable:
- Red = 4.
Probability:
- 49 .
Verify:
- Simplified, 4 and 9 coprime.
4
Final Answer: 9

• Q23: How many 2-digit numbers can be formed using digits 1,

2, 3 (repetition allowed)?
We need the number of 2-digit numbers using 1, 2, 3 with repetition.
Choices:
- Tens: 3 (1, 2, 3).
- Units: 3 (1, 2, 3).
- Total: 3 · 3 = 9.
Verify:
- 11, 12, 13, 21, 22, 23, 31, 32, 33.
Final Answer: 9
• Q24: In how many ways can you choose 2 students from 5?
We need the number of ways to choose 2 students from 5.
Use combinations:
- 52 = 5·4


2·1 = 10.
Verify:
- Pairs: (1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,4), (3,5), (4,5).
- 10 pairs.
Final Answer: 10
• Q25: A coin is flipped 3 times. How many outcomes have exactly
2 heads?
We need the number of outcomes with exactly 2 heads in 3 coin flips.
Use binomial:
- 32 = 3·2


2·1 = 3.
List:

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- HHT, HTH, THH.
Verify:
- 3 outcomes, matches.
Final Answer: 3