AMC Y5-6

AMC Y5-6 Cheat Sheet

1Number and place value

Place value and rounding.
Factors, multiples and prime numbers.
Divisibility tests and remainders.
Order of operations.

AverageAverage = total valuenumber of values

Factor pairsfactor × matching factor = target number

Remaindernumber = divisor × quotient + remainder

Divisibility3 or 9: check digit sum. 4: check last two digits. 6: divisible by 2 and 3.

2Fractions, ratio and percent

Equivalent fractions and common denominators.
Fraction of a quantity.
Ratio sharing.
Benchmark percentages.

Fraction of an amountnumeratordenominator of amount = numerator × amountdenominator

Ratio sharingone share = total amounttotal ratio parts

Percent of amountpercent of amount = percent100 × amount

Benchmarks25% = one quarter. 50% = one half. 75% = three quarters.

3Geometry and measurement

Perimeter and area.
Angles on lines and around a point.
Symmetry and visual geometry.
Units and scale.

RectangleArea = length × widthPerimeter = 2 × (length + width)

TriangleArea = base × height2

SquareArea = side²Perimeter = 4 × side

AnglesStraight line = 180°. Around a point = 360°. Triangle = 180°.

4Patterns, counting and data

Number patterns and repeating cycles.
Tables, tree diagrams and organised lists.
Simple probability and data reading.
Hidden-operation word problems.

Term patternnext term = current term + repeated change

Product ruletotal choices = choice 1 × choice 2 × choice 3

Probabilityprobability = wanted outcomesall possible outcomes

Unit rateunit rate = total quantitynumber of units

5High-frequency triggers

Patternwrite term numbers, compare changes, then test whether the cycle repeats

Word problemcircle the final ask: total, difference, remainder, number of ways or final value

Diagramredraw neatly, label every known length, then split into simple shapes

Countingmake a table or tree; count each case exactly once

6Study plan and tips

More than 30 hours
Test foundation, practise advanced patterns, then complete one full mock exam.

10-30 hours
Do a mock exam first, find weak topics, then practise the related topic lesson deliberately.

Less than 10 hours
Memorise the formula map, redo mistakes, and complete one mixed timed set.

Tips
Use the AceAchievers Foundation lesson first, then move into advanced AMC question types.

AMC Y7-8

AMC Y7-8 Cheat Sheet

1Algebra and equations

Expanding and factorising.
Substitution into expressions.
Linear equations.
Number patterns and sequences.

Expand(a + b)(c + d) = a × c + a × d + b × c + b × d

Difference of squaresfirst² - second² = (first - second)(first + second)

Linear equationsame operation on both sides keeps equality true

Arithmetic sequenceterm number n = first term + (n - 1) × common difference

2Ratio, percent and rates

Ratio sharing.
Percentage increase and decrease.
Reverse percentages.
Speed, distance and time.

Ratio shareshare = part ratiototal ratio parts × total amount

Percent changenew value = original value × (1 + change rate)

Reverse percentoriginal value = new valuepercent multiplier

Ratespeed = distancetime

3Geometry and trigonometry

Pythagoras' theorem.
Similar triangles.
Angle reasoning.
Circle area and circumference.

Pythagorashypotenuse² = side one² + side two²

Triangle areaarea = base × height2

Circlecircumference = 2 × π × radiusarea = π × radius²

Angle factsStraight line = 180°. Around a point = 360°. Triangle = 180°.

4Counting and probability

Product rule and organised cases.
Ordered vs unordered counting.
Complements.
Probability experiments.

Product ruletotal outcomes = first choice × second choice × third choice

Pairsnumber of pairs from n objects = n × (n - 1)2

Probabilityprobability = wanted outcomesall possible outcomes

Complementprobability wanted = 1 - probability not wanted

5High-frequency triggers

Q21-Q30 geometryadd angle facts, then search for equal triangles, area split or Pythagoras

Countingdecide ordered or unordered before multiplying choices

Remainderstry parity, divisibility tests, last-digit cycles or repeating patterns

Word algebradefine one variable, then translate one sentence at a time

6Study plan and tips

More than 30 hours
Foundation diagnostic, advanced topic drills, full mock exam, then targeted repair.

10-30 hours
Start with a mock exam, rank weak topics, then practise the related course modules deliberately.

Less than 10 hours
Formula map, redo mistakes, then practise Q21-Q25 before attempting Q26-Q30.

Tips
Use the AceAchievers Foundation lesson first if algebra, geometry or counting basics are unstable.

AMC Y9-10

AMC Y9-10 Cheat Sheet

1Algebra and equations

Expansion and factorisation.
Quadratics and substitutions.
Indices and scientific notation.
Simultaneous equations.

Perfect square(first + second)² = first² + 2 × first × second + second²

Difference of squaresfirst² - second² = (first - second)(first + second)

Indicessame base multiplication: add powerssame base division: subtract powers

Quadratic rootssum of roots = - coefficient of xcoefficient of x²product of roots = constant termcoefficient of x²

2Relationships and graphs

Gradient and intercept.
Linear relationships.
Quadratic relationships.
Coordinate geometry.

Gradientgradient = change in vertical valuechange in horizontal value

Line formvertical value = gradient × horizontal value + vertical intercept

Midpointmidpoint = (x one + x two2, y one + y two2)

Distancedistance = √((change in x)² + (change in y)²)

3Geometry and trigonometry

Pythagoras' theorem.
Similar figures.
Right-angle trigonometry.
Circle and angle facts.

Pythagorashypotenuse² = shorter side² + other shorter side²

Trigonometrysine θ = opposite sidehypotenusecosine θ = adjacent sidehypotenuse

Tangenttangent θ = opposite sideadjacent side

Similarityequal angles → proportional sidesarea scale = side scale²

4Counting, statistics and probability

Combinations and arrangements.
Complement counting.
Box plots and spread.
Expected value intuition.

Combinationsways to choose r from n = n factorialr factorial × (n - r) factorial

Probabilityprobability = wanted outcomesall possible outcomes

Complementprobability wanted = 1 - probability not wanted

Meanmean = sum of data valuesnumber of data values

5High-frequency triggers

Algebra structurefactor, substitute, or create a symmetric expression

Graph questionidentify gradient, intercept, scale and whether the relationship is linear

Geometry ratiolook for similar triangles, parallel lines, area ratios and angle chasing

Probabilitydefine sample space first; use complement when direct counting is messy

6Study plan and tips

More than 30 hours
Foundation diagnostic, advanced algebra/geometry/counting, full mock, then AIMO bridge.

10-30 hours
Do one mock exam, identify weak topics, then practise related topic courses deliberately.

Less than 10 hours
Memorise formulas, practise Q21-Q25, and review every missed mock question by topic.

Tips
Use AceAchievers Foundation lessons if algebra, graphing, geometry or probability formulas are not fluent.

AIMO

AIMO Cheat Sheet

1Number theory lemmas

Divisibility and greatest common divisor.
Modular arithmetic.
Parity and invariants.
Integer bounds.

Divisibilityif divisor divides first number and second number, it divides any integer combination of them

Remainderssame remainder means difference is divisible by the divisor

Greatest common divisorgreatest common divisor × least common multiple = product of the two numbers

Parityodd + odd = even; odd × odd = odd; even × any integer = even

2Algebra and inequalities

Factorisation and substitution.
Sequences and recurrence.
Inequality proof.
Working backwards from target form.

Difference of squaresfirst² - second² = (first - second)(first + second)

Square non-negativeany real expression squared is greater than or equal to zero

AM-GMfor positive values, value one + value two ≥ 2√(value one × value two)

Telescopingwrite terms so middle parts cancel after addition

3Geometry proof toolkit

Similarity and ratio chasing.
Circle angle facts.
Area comparison.
Auxiliary lines.

Similarityequal corresponding angles → proportional corresponding sides

Area ratioarea scale factor = side scale factor²

Tangentradius to tangent point is perpendicular to tangent line

Circle angleangle at centre = 2 × angle at circumference on same arc

4Combinatorics and invariants

Counting by cases.
Complement counting.
Pigeonhole principle.
Invariants and monovariants.

Product ruletotal arrangements = choices at step one × choices at step two × ...

Combinationchoose r objects from n = n factorialr factorial × (n - r) factorial

Pigeonholeplacing n objects into k boxes forces one box to contain at least ceiling(nk)

Invarianttrack what does not change after each allowed operation

5Proof frame

Definename variables, constraints and what must be proved

Observestate the lemma, invariant, congruence or geometry relationship

Proveshow why the method works; do not jump from pattern to conclusion

Concludecheck edge cases and answer in the exact language of the question

6Study plan and tips

More than 30 hours
Test foundation, complete advanced proof modules, then do an AIMO mock and rewrite solutions.

10-30 hours
Do a mock exam first, identify weak domains, then practise the matching topic course deliberately.

Less than 10 hours
Review this lemma map, attempt one mock, and rewrite two complete solutions slowly.

Tips
Use AceAchievers Foundation lessons if algebra, geometry or number theory facts are not automatic.

JSO Y7-8

Junior Science Olympiad Cheat Sheet

The Ace Pathway: From Acceleration to Competition Excellence

Younger JSO students do not need more random facts. They need school science connected into mechanisms, data and transfer.

Core knowledge map

Biology: cells, body systems, ecosystems and adaptation.
Chemistry: particles, mixtures, reactions and materials.
Physics: forces, energy, light, heat and electricity.
Earth science: rocks, weather, climate and systems.

Competition thinking

Skill	What to practise
Data reading	Graphs, tables, units and trends.
Mechanism	Explain why a result happens.
Experiment	Variables, controls and fair tests.
Transfer	Apply school concepts to unfamiliar contexts.

Weekly routine

Read one science news story and identify the school topic.
Practise one graph or data question.
Write one mechanism explanation in 4-5 sentences.
Review one past-style multi-topic question.

Watch for these traps

Confusing correlation with causation.
Ignoring units on graphs.
Memorising definitions without knowing examples.
Writing vague explanations instead of cause-effect chains.

Soft next stepUse AceAchievers Science Signals to connect real science news to school science and JSO-style thinking.

JSO Y9-10

Junior Science Olympiad Advanced Cheat Sheet

The Ace Pathway: From Acceleration to Competition Excellence

Year 9-10 JSO preparation should bridge school science into senior-style reasoning: mechanisms, models, uncertainty and experimental design.

Advanced focus map

Biology: respiration, genetics, ecosystems and feedback systems.
Chemistry: reactions, bonding, acids, bases and conservation.
Physics: motion, energy, circuits, waves and forces.
Earth science: climate systems, plate tectonics and cycles.

Question style upgrade

From	To
Recall a fact	Use the fact in a new system.
Read a graph	Explain trend, anomaly and limitation.
Name a variable	Design the fair comparison.
Explain one cause	Trace a chain of mechanisms.

Data checklist

Identify axes, units and scale.
Describe the trend before explaining it.
Separate observation from inference.
Check whether the evidence supports the claim.
Write limitations clearly.

Bridge to ASO

Choose one discipline to deepen each term.
Practise past-style ASO snippets for exposure.
Build a concept map, not just notes.
Use science news to practise transfer and evaluation.

Soft next stepFor competition science, build an Ace pathway: accelerated concepts, science signal practice, then JSO/ASO exam readiness.

Parent Guide

AMC vs AIMO vs Olympiad: What Is the Right Sequence?

A decision matrix for families deciding where a high-achieving maths student should focus next.

Many parents ask whether Year 7 should prepare for AMC or AIMO. The better question is: what evidence shows the student is ready for the next layer?

Quick comparison

Path	Best for	Main signal
AMC	Y3-12 broad competition exposure	Non-routine problem solving
AIMO	Y7-10 high performers	Depth, stamina, written reasoning
AMO pathway	Elite invited students	Proof-based olympiad mathematics

Readiness signs

Move to AIMO if the student can explain solutions, not just guess answers.
Stay with AMC if accuracy and timing still fluctuate.
Add olympiad-style work when the student enjoys one hard problem for 30 minutes.
Do not rush proof writing before problem representation is strong.

Parent decision rule

Can my child solve Q21-Q25 reliably?
Can my child attempt Q26-Q30 without panic?
Can my child write why a method works?
Does my child enjoy deep problems enough to sustain training?

Common mistake

Treating AIMO as a status jump rather than a readiness jump. AIMO is powerful when a student is ready for depth; otherwise, it can create frustration without strengthening fundamentals.

Soft next stepUse the AMC Junior or AIMO AceAchievers pathway based on the student's current evidence, not just year level.

Y3-Y12 Roadmap

Australian Maths, Science and Coding Competition Roadmap

A shareable planning map for families building a multi-year competition pathway.

The strongest students do not do every competition. They choose the right exposure at the right time, then prepare with depth.

Competition pathway by stage

Y3-4Maths: KSF, AMC Middle PrimaryFocus: curiosity, pattern spotting, confidence.

Y5-6Maths: AMC Upper Primary, CAT, BebrasFocus: diagrams, logic, counting, careful reading.

Y7-8Maths: AMC Junior, APSMO Y7-8, CAT. Science: Big Science, JSO Y7-8.Focus: non-routine questions and science transfer.

Y9-10Maths: AMC Intermediate, AIMO, CAT. Science: JSO Y9-10, ASO exposure.Focus: written reasoning, mechanisms, data.

Y11-12Maths: AMO pathway. Science: ASO. Coding: AIO.Focus: discipline depth and high-performance pathway.

How to use this roadmap

Choose one broad benchmark and one stretch pathway per year.
Do not overload the calendar; use competitions to guide training.
Review after each sitting: topic gaps, thinking gaps, stamina gaps.

Shareable rule

If a competition does not change what your child practises next, it is only a calendar event. The value comes from turning results and preparation into a better learning pathway.

Soft next stepDownload the grade-specific AMC or JSO sheet, then choose one clear preparation focus for the next 8-10 weeks.