Recall with conditions and examples
IOQM Formula Sheet: Theorems to Know—and How to Use Them
A formula list helps only when the conditions, meaning and typical problem signals are understood. Use this compact IOQM revision guide for number theory, algebra, geometry and combinatorics, then apply every item in problems.
For each result, recall its conditions, explain why it is true at your level, solve one direct example and one disguised application, then revisit it in a mixed set. This is a revision map—not an official formula booklet or a substitute for problem solving.
Number theory
Core facts for divisibility and integers
| Fact | Conditions / meaning |
|---|---|
| Division algorithm: a = bq + r | For b > 0, integers q,r exist uniquely with 0 ≤ r < b |
| gcd(a,b) = gcd(b, a mod b) | Basis of the Euclidean algorithm |
| gcd(a,b) · lcm(a,b) = |ab| | For nonzero integers; often stated for positive a,b |
| a ≡ b (mod m) | Means m divides a − b; addition and multiplication respect congruence |
| If gcd(a,m)=1, a has an inverse mod m | The coprimality condition is essential |
| Prime factorisation is unique | For integers greater than 1, apart from order |
Algebra
Identities, sequences and inequalities
| Result | Useful form / condition |
|---|---|
| Difference of squares | a² − b² = (a−b)(a+b) |
| Sum/difference of cubes | a³±b³ = (a±b)(a²∓ab+b²) |
| Quadratic roots | For ax²+bx+c=0, x = (−b±√(b²−4ac))/(2a), a≠0 |
| Vieta for ax²+bx+c | Root sum = −b/a, product = c/a |
| Arithmetic sequence sum | Sₙ = n(first+last)/2 |
| Geometric sum | 1+r+…+rⁿ = (rⁿ⁺¹−1)/(r−1), r≠1 |
| AM–GM for two numbers | (x+y)/2 ≥ √(xy) for x,y ≥ 0; equality when x=y |
| Factor theorem | x−a divides P(x) exactly when P(a)=0 |
In inequalities, always record the domain and equality case. In polynomial work, factorisation and root structure are often more informative than expanding everything.
Geometry
Theorems require a precise configuration
| Theorem / relation | What to check |
|---|---|
| Triangle angle sum = 180° | Euclidean triangle |
| Pythagoras: a²+b²=c² | c is the hypotenuse of a right triangle |
| Similarity | Corresponding angles and sides; area ratio is the square of the side ratio |
| Cyclic quadrilateral | Opposite angles are supplementary; converse can prove concyclicity |
| Equal chords | Equal chords subtend equal central angles in the same circle |
| Power of a point | Intersecting secants/chords or tangent relation from one point |
| Area: Δ = ½bh | Height is perpendicular to the chosen base |
| Heron: Δ=√(s(s−a)(s−b)(s−c)) | s=(a+b+c)/2 and valid triangle sides |
A formula should follow the diagram, not replace it. Mark equal angles, collinearity, cyclic points and correspondence before using a theorem.
Combinatorics
Counting formulas begin with a model
| Result | Meaning |
|---|---|
| n! | Arrangements of n distinct objects |
| P(n,r)=n!/(n−r)! | Ordered selection of r distinct objects from n |
| C(n,r)=n!/(r!(n−r)!) | Unordered selection; C(n,r)=C(n,n−r) |
| Pascal identity | C(n,r)=C(n−1,r)+C(n−1,r−1) |
| Two-set inclusion–exclusion | |A∪B|=|A|+|B|−|A∩B| |
| Pigeonhole threshold | More objects than boxes forces a shared box; general capacity must be stated |
Recognition cues
Link a problem signal to a question—not a formula
Remainders repeat
Ask whether congruences, cycles or pigeonhole can organise them.
Expression is symmetric
Ask whether sum/product, Vieta, AM–GM or a substitution reveals structure.
Four points and angles
Test whether a cyclic quadrilateral or similarity can be proved.
“At least two” guarantee
Define objects, boxes and capacity for pigeonhole.
Many overlapping cases
Try complementary counting or inclusion–exclusion.
Repeated legal moves
Search for parity, residue, colour balance or another invariant.
Conditions matter
Formula mistakes IOQM problems are designed to expose
- Cancelling in a congruence without checking coprimality.
- Applying AM–GM to values outside the required nonnegative domain.
- Using a cyclic-quadrilateral fact before proving the points are concyclic.
- Mixing corresponding sides in similar triangles.
- Using combinations when order matters, or permutations when it does not.
- Adding case counts that overlap.
- Forgetting equality, zero or degenerate cases.
- Expanding an expression that was meant to be factored.
Turn recall into skill
A four-step theorem revision cycle
- Recall: write the result and all conditions from memory.
- Explain: give a proof or intuitive justification appropriate to your level.
- Apply: solve one direct and one disguised problem.
- Retrieve later: meet the idea again inside a mixed set.
Mark a result “ready” only when you can recognise and use it, not when the line looks familiar.
Important limitation
No short sheet can contain the whole Olympiad
IOQM questions can require constructions, lemmas, transformations and combinations of basic facts that no formula list predicts. Use the four detailed topic hubs for learning order and common methods:
Questions answered
Frequently asked questions
Is there an official IOQM formula sheet?
This page is an independent revision guide, not an official formula booklet. Check official syllabus and examination instructions.
Are formulas enough for IOQM?
No. IOQM requires method selection, adaptations, constructions and reasoning that cannot be reduced to a short list.
Which geometry theorems should I revise?
Begin with triangles, similarity, circles, cyclic figures, areas and power relationships, always including theorem conditions.
What algebra formulas are most useful?
Factorisation identities, polynomial roots, sequences and inequalities are useful, but structural recognition matters more than expansion.
Should I memorise nPr and nCr?
Understand ordered versus unordered outcomes first; then the formulas become consequences rather than guesses.
How do I revise formulas effectively?
Recall conditions, explain the result, solve direct and disguised applications, then retrieve it later in mixed practice.
What is the most common formula error?
Using a correct result when its conditions are not satisfied, such as invalid modular cancellation or unproved concyclicity.
Can I print this page?
Yes for personal revision, but use it as a checklist and attach solved examples rather than reading it passively.
Next step
A theorem becomes useful when you recognise its conditions.
Revise the result, prove or explain it, apply it twice and then retrieve it from a mixed official paper.
MENTOR PERSPECTIVE
“A formula is useful only when the student understands its conditions and purpose.”Meet your IOQM mentor →