CEMC Gauss 8 滑铁卢 Gauss 8 数学竞赛
📌 Course at a glance
· 10 units · key topics drilled across the official outline
· Next exam: 2026-05-13 · 2026-05-13
📝 Full chapter-by-chapter lecture walkthroughs are being written. This page is the course outline — every unit and topic that will be covered, with key concepts you can already start to recognise. Use it as a study map; the chapter notes drop in over the coming weeks.
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Course outline
Unit 1 · Number Theory — Primes, Divisors & Exponents 数论 — 质数、约数与指数
Exam weight: ≈12–16% (every paper has 2–3)
1.1 — Prime factorization & exponent sums 质因数分解与指数和
· Factor trees and unique factorization · Writing n as 2^a·3^b·5^c · Summing exponents a+b+c · Factorials and products in prime form
1.2 — Divisibility rules & digit problems 整除规则与数字题
· Tests for 2,3,4,5,6,8,9,10,11 · Digit-sum and alternating-sum reasoning · Finding a missing digit
1.3 — GCD & LCM — coincidences and constraints 最大公因数与最小公倍数
· GCD/LCM from prime factorization · GCD × LCM = product · Recurring-event LCM · Reverse use of the identity
1.4 — Counting divisors & largest-GCD problems 约数计数与最大 GCD 问题
· Divisor count (a₁+1)(a₂+1)… · Odd divisor count ⇔ perfect square · Largest possible GCD of a fixed-sum set
Unit 2 · Integers, Order of Operations, Powers & Roots 整数、运算顺序、幂与根
Exam weight: ≈10–14% (BC's flagship Grade-8 number unit)
2.1 — Integer operations & the number line 整数运算与数轴
· Sign rules for +,−,×,÷ · Subtracting a negative · Distance on the number line
2.2 — Order of operations with integers & exponents 含整数与指数的运算顺序
· BEDMAS with negatives · (−a)^n vs −a^n · Nested brackets and fraction bars
2.3 — Powers, perfect squares & cubes 幂、完全平方与立方
· Exponent meaning and laws · Recognizing squares/cubes · Units-digit patterns of powers
2.4 — Square roots & cube roots 平方根与立方根
· √ of a perfect square · Estimating a root between integers · Simplifying √(a²·b)
Unit 3 · Fractions, Decimals & Percent 分数、小数与百分比
Exam weight: ≈12–16%
3.1 — Fraction operations & comparisons 分数运算与比较
· × and ÷ of fractions · Order of operations with fractions · Comparing without a common denominator
3.2 — Percent & reverse percent 百分比与逆百分比
· Percent of a number · Percents <1 and >100 · Reverse: find the whole from a part
3.3 — Percent change & successive percents 百分比变化与连续百分比
· Increase/decrease as ×(1±r) · Successive changes multiply · +20% then −20% ≠ unchanged
3.4 — Complement & 'one-minus' reasoning 补集与反向求解
· 1 − P(not) · At-least problems · Discount/tax reversibility
Unit 4 · Ratios, Rates & Proportional Reasoning 比、率与比例推理
Exam weight: ≈10–12%
4.1 — Ratios & the 'multiple of the sum' trick 比与'和的倍数'技巧
· a:b notation and scaling · Total = (sum of parts) × k · Which totals are impossible
4.2 — Three-term ratios & shared quantities 三项比与共享量
· a:b:c chaining · Linking two ratios through a common term · Coin/value ratio problems
4.3 — Rates: speed, distance, time & average speed 率：速度、距离、时间与平均速率
· d = rt · Two-leg trips · Average speed = total distance / total time
4.4 — Proportion, mixtures & best buys 比例、混合与最优购买
· Setting a proportion · Concentration/mixture · Unit-price 'best buy' comparisons
Unit 5 · Algebra — Expressions, Equations & Systems 代数 — 表达式、方程与方程组
Exam weight: ≈12–16% (the Grade-8 algebra core)
5.1 — Expressions & substitution 表达式与代入
· Translate phrases to algebra · Evaluate by substitution · Like terms
5.2 — Two-step linear equations 两步一次方程
· Integer coefficients & solutions · Balance method · Solutions check
5.3 — Two-unknown systems in word problems 应用题中的二元方程组
· Set two equations · Add/subtract to eliminate · 'Not-A / not-B' clue decoding
5.4 — Cryptarithms & bounded integer search 字母算术与有界搜索
· Column-by-column carries · Digit/parity constraints · Enumerate the small side
Unit 6 · Geometry I — Angles & Polygons 几何 I — 角与多边形
Exam weight: ≈10–12%
6.1 — Angle sums, straight & vertical angles 角和、平角与对顶角
· Straight = 180°, vertical equal · Triangle = 180°, quadrilateral = 360°
6.2 — Parallel lines & transversals 平行线与截线
· Corresponding & alternate angles · Co-interior sum 180° · Building a chain
6.3 — Isosceles/equilateral chains & exterior angle 等腰/等边链与外角
· Base angles equal · Exterior = sum of remote interiors · Multi-step chasing
6.4 — Polygon angle sums & symmetry 多边形内角和与对称
· (n−2)·180° · Regular-polygon interior angle · Lines of symmetry
Unit 7 · Geometry II — Area, Pythagoras & 3D Solids 几何 II — 面积、勾股定理与立体
Exam weight: ≈12–16% (Grade-8: Pythagoras & 3D solids)
7.1 — Composite & shaded regions 复合与阴影区域
· Add/subtract sub-areas · Shaded = whole − unshaded · Symmetry shortcuts
7.2 — The Pythagorean theorem & its converse 勾股定理及其逆定理
· a²+b²=c² · Common triples (3-4-5, 5-12-13) · Converse: testing a right angle
7.3 — Surface area & volume of prisms & cylinders 棱柱与圆柱的表面积与体积
· Prism volume = base × height · Cylinder V = πr²h · Surface area from the net · Subtracting a drilled hole
7.4 — Nets, views & shortest path on a surface 展开图、视图与表面最短路径
· Net of a prism/box · Top/front/side views · Unfold, then straight-line distance
Unit 8 · Patterns, Sequences & Modular Reasoning 数列、规律与周期
Exam weight: ≈8–10%
8.1 — Arithmetic sequences: nth term & sum 等差数列：通项与求和
· Common difference · aₙ = a₁+(n−1)d · Sum = n(first+last)/2
8.2 — Recursive & figurate patterns 递推与图形数
· Fibonacci-style recursion · Compute the first few terms · Triangular/square figurate numbers
8.3 — Repeating patterns & modular position 周期与模定位
· Period p · Position = ((n−1) mod p)+1 · Off-by-one trap
8.4 — Calendar, cycles & remainders 日历、循环与余数
· Days as 0–6 (mod 7) · Wrap-around · Remainder reasoning (÷10 leaves 8 ⇒ ÷5 leaves 3)
Unit 9 · Counting, Probability & Statistics 计数、概率与统计
Exam weight: ≈12–16% (Part C leans on this)
9.1 — Systematic counting & casework 系统计数与分类讨论
· List in order · Order matters vs not · Disjoint cases, no double-count
9.2 — Complement counting & pigeonhole 补集计数与鸽巢原理
· Total − bad = good · n+1 items in n holes · Worst-case construction
9.3 — Probability of two independent events 两独立事件的概率
· P = favorable/total · Multiply along a branch · Add across branches · Complement for 'at least'
9.4 — Mean, median, mode, range & optimization 均值、中位数、众数、极差与最优化
· Mean=sum/count; inverse mean · Median of sorted list; mode · Range=max−min · Push values to extremes for 'smallest possible'
Unit 10 · Contest Strategy — The Gauss 8 Playbook 考试策略 — Gauss 8 战术手册
Exam weight: Meta — applies to all 25 questions
10.1 — The 2-point blank rule & expected value 留空2分规则与期望值
· Blank = 2 pts (max 10) · Guess vs blank by expected value · Lock Part-C blanks as a floor
10.2 — Pacing: A → B → C in 60 minutes 节奏：60 分钟内 A→B→C
· ~12 min Part A · ~25 min Part B · ~20 min Part C + 3 min review
10.3 — Calculator use & estimation 计算器使用与估算
· Useful for messy arithmetic & roots · Useless for casework/algebra · Estimate to eliminate choices
10.4 — Common Gauss 8 traps 常见 Gauss 8 陷阱
· Sign errors with negatives · Diagrams not to scale · Average speed ≠ mean of speeds · 'Different' integers must be distinct