12 original numerical-answer problems
IOQM Topic-wise Practice Questions: Number Theory, Algebra, Geometry and Combinatorics
Use these original, independently written practice problems to check method selection across the four major Olympiad domains. Attempt each question before opening the worked solution, and record every answer in two-digit form.
Allow a serious attempt before viewing the solution. Write the main idea, calculate carefully, record the two-digit answer and then compare the reasoning—not only the number. These are original practice questions, not official IOQM questions or a prediction of the 2026 paper.
Questions 1–3
Number theory practice
Q1 · Congruence
Find the smallest positive integer n such that 7n leaves remainder 1 when divided by 20.
Show solution
Test the inverse of 7 modulo 20. Since 7×3=21≡1 (mod 20), the smallest positive n is 3. Answer: 03.
Q2 · GCD structure
Find gcd(2¹⁵−1, 2⁹−1).
Show solution
Use gcd(aᵐ−1,aⁿ−1)=agcd(m,n)−1. Here gcd(15,9)=3, so the value is 2³−1=7. Answer: 07.
Q3 · Divisors
How many positive divisors does 360 have?
Show solution
360=2³·3²·5. A divisor chooses exponents 0–3, 0–2 and 0–1, giving (3+1)(2+1)(1+1)=24. Answer: 24.
Questions 4–6
Algebra practice
Q4 · Symmetric expression
A nonzero real number x satisfies x+1/x=5. Find x²+1/x².
Show solution
Square the given relation: x²+2+1/x²=25. Therefore x²+1/x²=23. Answer: 23.
Q5 · Arithmetic progression
The first term of an arithmetic progression is 3 and the common difference is 2. Find the sum of its first 6 terms.
Show solution
The terms are 3,5,7,9,11,13. Pairing ends gives three pairs of 16, so the sum is 48. Answer: 48.
Q6 · Roots without solving
If r and s are the roots of t²−7t+10=0, find r²+s².
Show solution
By Vieta, r+s=7 and rs=10. Hence r²+s²=(r+s)²−2rs=49−20=29. Answer: 29.
Questions 7–9
Geometry practice
Q7 · Right-triangle area
A right triangle has perpendicular sides 6 and 8. Find its area.
Show solution
The perpendicular sides are base and height. Area=½×6×8=24. Answer: 24.
Q8 · Cyclic angles
ABCD is a cyclic quadrilateral and angle A is 112°. Find angle C in degrees.
Show solution
Opposite angles of a cyclic quadrilateral are supplementary, so C=180°−112°=68°. Answer: 68.
Q9 · Similarity and area
Two similar triangles have corresponding side ratio 3:5. If the smaller area is 27, find the larger area.
Show solution
Areas are in the square of the side ratio, 9:25. The larger area is 27×25/9=75. Answer: 75.
Questions 10–12
Combinatorics practice
Q10 · Handshakes
Ten people each shake hands once with every other person. How many handshakes occur?
Show solution
Each handshake is an unordered pair, so the count is C(10,2)=10×9/2=45. Answer: 45.
Q11 · Unordered selection
How many two-person teams can be selected from 8 students?
Show solution
Order does not matter, so choose 2 from 8: C(8,2)=8×7/2=28. Answer: 28.
Q12 · Inclusion–exclusion
How many positive integers not exceeding 100 are divisible by 4 or by 6?
Show solution
There are 25 multiples of 4 and 16 multiples of 6. Multiples of lcm(4,6)=12 were counted twice; there are 8. Thus 25+16−8=33. Answer: 33.
Second pass
Remove the topic labels and test recognition
On a later day, copy only the question statements into a mixed order. Without the section labels, write the first useful cue beside each:
- inverse modulo 20;
- exponent gcd identity or Euclidean algorithm;
- prime-exponent choices;
- square a symmetric relation;
- pair an arithmetic progression;
- Vieta before solving;
- perpendicular base and height;
- cyclic supplementary angles;
- area scales as side ratio squared;
- unordered pairs;
- unordered selection;
- subtract an overlap.
The ability to choose the method without a chapter heading is closer to the real examination demand.
Error log
Classify the earliest cause of every miss
| Error type | Next action |
|---|---|
| Concept not known | Study the smallest relevant lesson and solve a graded set |
| Method not recognised | Write the cue and compare a disguised variation |
| Execution | Slow the algebra or arithmetic and add a verification step |
| Condition missed | Rewrite the theorem with its domain or configuration |
| Format/transcription | Practise two-digit recording and the OMR check routine |
Progression
Move from original drills to official mixed papers
- Finish the twelve questions without open solutions.
- Reattempt every miss after two or three days.
- Use the detailed topic hubs for the weakest domain.
- Solve selected official previous-paper questions.
- Later take a timed official paper under the current pattern.
Original practice builds specific skills; official papers are essential for authentic style, balance and difficulty.
Questions answered
Frequently asked questions
Are these official IOQM questions?
No. They are original independent practice questions. Use the linked HBCSE archive for official previous papers.
Do the questions follow IOQM topics?
They practise number theory, algebra, geometry and combinatorics, the four major Mathematical Olympiad areas.
Should I look at the solution immediately?
No. Make a serious independent attempt, write the main idea and only then compare the worked reasoning.
Why are answers written as two digits?
It reinforces current IOQM OMR practice, where numerical answers are recorded in two-digit form.
Do all answers fit the two-digit practice format?
Yes. The results are between 00 and 99, matching the numerical-answer recording habit used in current IOQM instructions.
How should I review an incorrect answer?
Find the earliest cause—concept, recognition, condition, execution or transcription—and choose a specific reattempt or repair.
What should I solve after this set?
Use the four topic hubs for weak areas, then move into official previous-paper questions and later timed papers.
Can I use these questions for a mock test?
Use them as a short diagnostic or topic drill, not as a prediction or full replica of the IOQM 2026 paper.
Next step
Attempt first. Diagnose second. Reattempt later.
Use this set to identify the weakest reasoning step, then move into the relevant topic hub and official paper archive.
MENTOR PERSPECTIVE
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