Grade 8 · Mathematics · Olympiad

IMO Quick Revision Guide

High-yield concepts, trusted formulas, reasoning reminders and exam-day checks. A compact reference before a timed mock—not a replacement for learning how each formula works.

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How to use this sheet: Review one topic at a time, recall the key rule, then solve one question without looking at the explanation. This is an independent SOF-style preparation resource, not an official SOF publication.

1. Number Systems & Rapid Number Sense

These tools reduce calculation time and catch impossible options.

Rational-number operations: For fractions, use a common denominator to add or subtract. To divide by a non-zero fraction, multiply by its reciprocal.
Zero and negative powers: For non-zero a: a⁰ = 1; a⁻ⁿ = 1/aⁿ. Use positive exponent bases when simplifying.
Power rules: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0); (aᵐ)ⁿ = aᵐⁿ; (ab)ⁿ = aⁿbⁿ.
Units-digit cycles: Powers of 2 repeat 2, 4, 8, 6. For 2⁴⁵, 45 mod 4 = 1, giving units digit 2.
Square/cube clues: A perfect square cannot end in 2, 3, 7 or 8. Perfect squares leave remainder 0 or 1 on division by 3. These are necessary tests, not proof.
Useful divisibility tests: By 3 or 9: digit sum divisible by 3 or 9. By 4: last two digits divisible by 4. By 11: alternating digit sum multiple of 11.
Recurring fraction shortcuts: 12.5% = 1/8; 25% = 1/4; 37.5% = 3/8; 87.5% = 7/8. One-third is exactly 33⅓%, not 33.33%.

2. Algebra, Identities & Factorisation

Identify structure before expanding everything.

Fundamental identities: (a+b)² = a²+2ab+b²; (a−b)² = a²−2ab+b²; a²−b² = (a−b)(a+b).
Binomial products: (x+a)(x+b) = x²+(a+b)x+ab. For (ax+b)(cx+d), use acx²+(ad+bc)x+bd.
Common-factor method: 6x²+9x = 3x(2x+3). Always check for a common factor before any other technique.
Regrouping and difference of squares: ax+ay+bx+by = (a+b)(x+y); 4a²−(3b−1)² = (2a−3b+1)(2a+3b−1).
Linear equations: For ax+b=c (a ≠ 0), x=(c−b)/a. Apply the same operation to both sides and verify by substitution.
Forming word-problem equations: Define the unknown and units first. Translate ‘twice a number less 5’ carefully: 2x−5; a percentage increase of p% multiplies by (1+p/100).
Check algebra by substitution: Test the alleged identity with more than one valid set of values; one successful test does not prove an identity. Never choose a value that makes a denominator zero.

3. Squares, Cubes & Roots

Factor powers into pairs or triples when finding roots.

Perfect square vs. perfect cube: pair or triple prime factors

Right triangle: a² + b² = c² (only for a right angle)

Squares and cubes: (ab)²=a²b²; (ab)³=a³b³. To square a two-digit number near a multiple of 10, use (a±b)².
Prime-factor method: In a perfect square, every prime exponent is even; in a perfect cube, every prime exponent is a multiple of 3.
Square roots of decimals: √0.0016 = 0.04. Check by squaring; decimal-place shortcuts only work under suitable representations and cannot replace verification.
Estimation: 15² = 225 and 16² = 256, so √240 lies between 15 and 16.
Pythagoras (extension): In a right triangle, c²=a²+b². Remember 3-4-5, 5-12-13, 8-15-17; apply only when a right angle is established.

4. Comparing Quantities & Proportions

Differentiate growth, discount and proportional relationships before calculating.

Percentage change: % change = (new−original)/original ×100. Reverse percentage: original = final ÷ (1±p/100).
Profit/loss and discount: Profit% = (SP−CP)/CP ×100; Discount% = (MP−SP)/MP ×100. CP = cost price, SP = selling price, MP = marked price.
Successive discounts: Discounts a% and b% combine to a+b−ab/100%, not simply a+b%.
Simple and compound interest: SI = PRT/100; for annual compounding, A=P(1+r/100)ⁿ and CI=A−P. For half-yearly compounding use r/2 and 2n periods.
Direct proportion: y=kx, so y/x stays constant (x≠0). Example: cost and quantity at a fixed unit price.
Inverse proportion: xy=k, so x₁y₁=x₂y₂ when conditions remain unchanged. Example: workers and time at equal productivity.
Speed, distance and time: Speed=distance/time; convert units before substituting. 1 m/s = 3.6 km/h. Average speed = total distance ÷ total elapsed time, including stops.

5. Geometry, Angles & Mensuration

Mark known lengths/angles, convert units, and separate perimeter, area, surface area and volume.

Quadrilateral family: properties depend on the shape

Exterior angles: one at each vertex sums to 360°

Circle: diameter d = 2r; area = πr²

Cylinder: radius r, height h; volume = πr²h

Polygon angles: Interior-angle sum of n-sided polygon = (n−2)×180°. Sum of one exterior angle at each vertex of a convex polygon = 360°.
Regular polygon angles: One exterior angle=360°/n; one interior angle=180°−360°/n.
Quadrilaterals: Parallelogram: opposite sides/angles equal and diagonals bisect. Rhombus: four equal sides, perpendicular diagonals. Rectangle: four right angles and equal diagonals. Square: both sets.
Triangles: Triangle angle sum=180°. Exterior angle equals sum of two non-adjacent interior angles.
2D area: Rectangle=lb; triangle=½bh; parallelogram=bh; trapezium=½(a+b)h; rhombus=½d₁d₂; circle=πr².
Circumference and perimeter: Circle circumference=2πr; perimeter of polygon=sum of side lengths.
3D volume: Cuboid=lbh; cube=a³; cylinder=πr²h. Surface areas: cuboid=2(lb+bh+hl), cube=6a², cylinder curved=2πrh and total=2πr(r+h).
Composite figures: Split into simple shapes or subtract cut-outs. Keep area in square units and volume in cubic units; never mix radii and diameters.

6. Data Handling, Probability & Logical Reasoning

Interpret the question—not just the diagram or formula.

Pie chart: sector angle = (part / total) × 360°

Directions: update your facing after each turn

Mean, median and mode: Mean=sum of observations/count; median=middle observation after sorting (or mean of two middle values); mode=most frequent value.
Range: Maximum−minimum. For a frequency table, mean = Σ(f×x)/Σf.
Pie charts: Component angle=(component ÷ total)×360°. Confirm the total includes every category.
Simple probability: For equally likely outcomes only: P(event)=favourable outcomes/total outcomes. Always 0≤P≤1; P(not A)=1−P(A).
Logical sequences: Check differences, alternating sub-sequences, squares/cubes and positional rules. A pattern consistent with a few terms may not be unique.
Direction and ranking: Draw compass directions; track turns from the CURRENT direction. Position from opposite end = total−position+1.
Venn/set logic: ‘All A are B’ does not mean ‘all B are A’. Test claims with an example or a small diagram.
Information sufficiency: Check whether conditions determine one unique answer, not merely a plausible answer.

7. Exam-Day Quick Checks

A flexible plan beats a rigid rule, particularly for multi-step Achievers questions.

Read the command word: Underline mentally NOT, INCORRECT, EXCEPT, least, greatest, approximately and units.
Work with options intelligently: Estimate to eliminate unreasonable values. Substitute options only when quicker than direct solving; confirm the surviving choice meets ALL conditions.
Time management: First collect reliable quick marks, then prioritise approachable high-value Achievers and medium questions. Leave time for the final review; don't guarantee an arbitrary 12–15 minutes to one section.
Don't confuse radius and diameter: If r=7 cm, diameter is 14 cm. If side lengths are in mixed units, convert before computing.
Avoid unproved shortcuts: A single substituted example can disprove an identity but cannot establish that it always holds. Estimate π≈3 only if options are far apart; use exact π when required.
OMR accuracy: Transfer responses in small batches; check number → bubble → number. Follow the actual exam's instructions for writing materials and answer changes.
Guessing policy: Read the current paper's marking instructions; only use a no-penalty guessing strategy when it is explicitly permitted.

How to use it: Pick three weak topics, recall the rules without looking, solve one fresh question on each, then revisit any mistakes. For full preparation, follow the Grade 8 lessons and timed Olympiad mock papers.