Russian state exam, grade 9

Introduction

1. 0 What mathematics is

I Numbers

1. 1 Counting
2. 2 Zero and minus
3. 3 Primes
4. 4 Divisibility
5. 5 Fractions
6. 6 Root two

II Algebra

1. 7 Letters
2. 8 Equations
3. 9 Functions and graphs
4. 10 Quadratic equations
5. 11 Systems and inequalities
6. 12 Powers and logarithms
7. 13 Sequences
8. 14 Polynomials
9. 15 Complex numbers

III Geometry

1. 16 Euclid's Elements
2. 17 The triangle
3. 18 Pythagoras' theorem
4. 19 The circle and π
5. 20 Area and volume
6. 21 Trigonometry
7. 22 Vectors
8. 23 Symmetry
9. 24 Conic sections

IV Calculus

1. 25 Limits
2. 26 The derivative
3. 27 Extrema
4. 28 The integral
5. 29 The number e
6. 30 Series
7. 31 Differential equations
8. 32 Several variables
9. 33 Fourier series
10. 34 Complex analysis

V Linear algebra

1. 35 Matrices
2. 36 Gaussian elimination
3. 37 Vector spaces
4. 38 Eigenvectors
5. 39 Least squares and SVD

VI Structures

1. 40 Groups
2. 41 Remainders and ciphers
3. 42 Rings and fields
4. 43 Galois theory
5. 44 Zeta and curves

VII Chance & data

1. 45 Combinatorics
2. 46 Graphs
3. 47 Probability
4. 48 Random variables
5. 49 Statistics
6. 50 Markov and information

VIII Foundations

1. 51 Logic and sets
2. 52 Infinities
3. 53 Rigorous analysis
4. 54 Measure and integral
5. 55 Functional analysis
6. 56 Gödel and Turing

IX Horizons

1. 57 Topology
2. 58 Non-Euclidean geometries
3. 59 Chaos and fractals
4. 60 The frontier