Mathegoras

Arithmetic808 problems · 8 topics

$1,2,3$ Integers and signed numbers 57 $(\,)$ Order of operations 33 $a/b$ Fractions 127 $0.5$ Decimals and repeating decimals 82 $\%$ Percentages 81 $a:b$ Ratio and proportion 146 $2^3$ Powers, roots and surds 211 $7\times 8$ Mental maths and products 71

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Algebra1,881 problems · 13 topics

$ax\!=\!b$ Linear equations 161 $1/x$ Rational equations 53 $\sqrt{x}$ Radical equations 59 $x^2$ Quadratic equations 220 $p(x)$ Polynomials 264 $a^2\!\!-\!\!b^2$ Expressions and identities 284 $\{x,y\}$ Systems of equations 204 $|x|$ Absolute value 102 $a\le b$ Inequalities 100 $\tbinom{n}{k}$ Binomial theorem 107 $\max$ Maximum and minimum of expressions 257 $y\propto x$ Direct and inverse variation 33 $(G,\cdot)$ Groups and abstract algebra 37

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Functions590 problems · 9 topics

$f\circ g$ Evaluating, composing and inverting functions 88 $f(\!-\!x)$ Even, odd and periodic functions 70 $f(xy)$ Functional equations 81 $ax\!+\!b$ Linear functions 28 $y\!=\!x^2$ Quadratic functions and parabolas 35 Graphs, intersections and asymptotes 83 $2^x$ Exponential equations 72 $\log$ Logarithms 103 $e^{kt}$ Exponential growth and decay 30

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Geometry1,673 problems · 12 topics

$\angle$ Lines and angles 168 Triangles 155 $a^2\!\!+\!\!b^2$ Right triangles and Pythagoras 113 Quadrilaterals 217 Polygons 113 Area and perimeter 225 $\pi r^2$ Circles: area, arcs and sectors 196 Chords, tangents and inscribed angles 96 Medians, bisectors and triangle centres 156 $\sim$ Similarity and ratios 95 $1\!:\!100$ Scale drawings and real-world geometry 48 $A_{\max}$ Geometric maximum and minimum 91

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Solid geometry598 problems · 6 topics

Cubes and cuboids 95 Prisms and pyramids 61 Cylinders, cones and spheres 193 Composite solids and recasting 73 Polyhedra and sections 135 Capacity and water levels 41

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Coordinate geometry648 problems · 7 topics

$|PQ|$ Points, distance and midpoints 122 Area from coordinates 16 Straight lines 107 $x^2\!\!+\!\!y^2$ Circles in coordinates 105 Parabolas, ellipses and hyperbolas 205 $(r,\theta)$ Polar and parametric curves 46 Reflections, rotations and translations 47

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Trigonometry528 problems · 9 topics

$\pi\,\mathrm{rad}$ Angles and radians 37 $30^\circ$ Exact trigonometric values 53 $\cos^2$ Trigonometric identities 78 $\sin x$ Trigonometric equations 86 Trigonometric functions: period and extremes 56 $\sin^{-1}$ Inverse trigonometric functions 30 $\tan\theta$ Right triangles, heights and distances 67 $\cos C$ Sine and cosine rules 64 $\textstyle\sum\sin$ Trigonometric sums and products 57

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Vectors and matrices405 problems · 5 topics

$\vec u+\vec v$ Vectors in the plane 137 $\vec u\times\vec v$ Lines, planes and vectors in space 93 $\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)$ Matrices 69 $\det A$ Determinants 58 $\lambda$ Rank, eigenvalues and linear maps 48

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Complex numbers269 problems · 4 topics

$a\!+\!bi$ Complex arithmetic 94 $re^{i\theta}$ The complex plane 67 $z^n\!=\!1$ Roots of unity and complex roots 84 $\oint$ Complex analysis 24

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Sequences and series673 problems · 6 topics

$+d$ Arithmetic sequences 119 $\times r$ Geometric sequences 95 $a_{n+1}$ Recursive sequences 90 $\textstyle\sum$ Finite sums 128 $\textstyle\sum^\infty$ Infinite series 158 $1,4,9$ Number patterns 83

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Number theory1,455 problems · 13 topics

$a\mid b$ Divisibility 181 $d(n)$ Divisors 119 $2,3,5$ Primes and factorisation 94 $\gcd$ GCD and LCM 124 $7\div 3$ Remainders 101 $\equiv$ Congruences and modular arithmetic 127 $x\!\in\!\mathbb{Z}$ Diophantine equations 87 $n^2,n^3$ Perfect squares and cubes 125 $\overline{abc}$ Digits 220 $\dots 7$ Last digits 110 $101_2$ Number bases 78 $5!$ Factorials 36 $\lfloor x\rfloor$ Floor and fractional parts 53

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Combinatorics1,208 problems · 11 topics

$m\cdot n$ Counting principles 180 ${}^nP_k$ Permutations and arrangements 113 ${}^nC_k$ Combinations and selections 115 $\star\mid\star\star$ Distributing objects 59 $A\!\cup\!B$ Sets and Venn diagrams 111 $n\!+\!1$ Pigeonhole and extremal problems 177 Graphs and networks 83 Colourings and symmetry 30 Combinatorial geometry 184 Grids and chessboards 121 Paths, stairs and recursion 35

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Probability1,046 problems · 15 topics

$\Omega$ Events and basic probability 43 Dice 105 Coins 39 Cards 86 Drawing from bags 87 $|\Omega|$ Probability by counting 87 $P(n)$ Probability with numbers 97 $A\!\mid\!B$ Conditional probability and Bayes 103 $A\!\cap\!B$ Independent events 38 $p^k$ Binomial probability 64 $E[X]$ Expected value 45 Geometric probability 69 Random walks and Markov chains 41 $f_X$ Random variables and distributions 101 Games of chance 41

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Statistics473 problems · 6 topics

$\bar x$ Mean and averages 194 $\tilde x$ Median and mode 69 $\sigma$ Range, quartiles and spread 90 Normal distribution 24 $H_0$ Sampling and inference 64 Charts, tables and regression 32

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Calculus1,298 problems · 14 topics

$\lim$ Limits of functions 149 $\to\!L$ Limits of sequences 130 $f(a)$ Continuity 34 $f'(x)$ Derivatives 168 $f'\!=\!0$ Extrema and curve sketching 115 $f_{\max}$ Optimisation problems 117 $\dot V$ Related rates 44 Motion 44 $\textstyle\int_a^b$ Definite integrals 190 $\textstyle\int^\infty$ Improper integrals 43 Areas, volumes and arc length 109 $\partial$ Multivariable calculus 84 $y'\!=\!y$ Differential equations 22 $\approx$ Power series and approximation 49

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Word problems1,200 problems · 9 topics

Money and shopping 184 Profit, loss and discounts 60 Interest and growth 114 $d\!=\!vt$ Distance, speed and time 244 Work and rates 217 Mixtures and solutions 80 Age problems 90 Tests and scores 40 Everyday problems 171

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Puzzles and logic460 problems · 5 topics

Logic puzzles 147 $\square\!+\!\square$ Number puzzles 92 Processes, invariants and strategy 74 Tournaments and games 94 Clocks and calendars 53

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