EXERCISES IN ALGEBRA (INCLUDING TRIGONOMETRY)
I KAYWA˜NAN Algebra Contest Olympiad Algebra Book (Vol I): Polynomials & Trigonometry 1 1 Olympiad Algebra 101: Polynomials 3 11 Olympiad Algebra 001: Definitions & Corollaries of Pre–Algebra 4 111 Introduction to Olympiad Algebra 4 112 Essential Polynomial Theorems 5 113 In Search of Rational Roots 7 114 Vi`ete’s Formulas 10 12 Factorization 12 121 One–Variable Identities 12 122 Two–Variable Identities 12 123 Three–Variable Identities 14 13 Exercises in Expressions 17 131 Functional Expressions 17 132 Polynomial Expressions 21 14 Polynomial Division 23 141 Remainder of Polynomial Division 23 142 Greatest Common Factor of Polynomials 26 15 Advanced Polynomial Problems & Theorems 28 151 Symmetric Sums 28 152 Pool of Advanced Polynomial Theorems 31 16 Complex Numbers 38 161 Polar Representation of Complex Numbers 39 17 Number Theoretic Study of Polynomials 41 171 Essential Number Theoretic Theorems 41 172 Modular Arithmetic for Polynomials 44 18 Algebraic Systems 48 181 Polynomial Equations 48 182 Trigonometric Tricks for Solving Algebraic Systems 52 183 Irrational Equations 53 184 Reciprocal Equations 58 185 Equations Containing the Floor & Absolute Value Function 59 186 Arithmetic of Polynomial Roots 60 1861 Miscellaneous Treacheries on Polynomial Roots 60 1862 Calculating Sum of Powers of the Roots 64 1863 Forming Equations Given the Roots 65 1864 Common Roots of Equations 66 1865 Number Theoretic Wizardry on Polynomial Roots 67 187 Using Vi`ete’s Formulas to Solve Systems of Equations 67 188 Homogeneous Equations 68 189 Miscellaneous Systems of Equations 69 19 Nice Polynomial Problems 76 191 100 Nice Polynomial Problems 76 192 300 Ancient Polynomial Problems 85 2 Olympiad Algebra 201: Trigonometry 101, 201, 301, 401, and Beyond 121 21 Math Olympiad Trigonometry 101: Identities & Half-angled Trigonometry 122 211 Trigonometric Identities 122 212 Sum-to-Product and Product-to-Sum 126 213 Half-angled and Double-angled Trigonometry of Triangle 126 2131 Pool of Half– and Double–angled Identities in Triangle 127 214 Trigonometry of Geometrical Quantities 130 2141 Law of Sines, Law of Cosines, and Law of Tangents 130 2142 Trigonometry of Perimeter and Area 131 2143 Trigonometry of Heights 132 2144 Trigonometry of Internal and External Angle Bisectors 134 2145 Trigonometry of Inradius and Exradii 135 2146 Trigonometry of Distance Between Triangle Centers 137 2147 Trigonometry of Quadrilaterals 141 22 Math Olympiad Trigonometry 201: JEE, Roorkee, ISM 144 221 Agarwal’s and Pearson’s Problems 144 222 Rejaul Makshud’s Problems 146 23 Math Olympiad Trigonometry 301: Series, Graphing & Logarithms 157 231 Trigonometric Series 157 232 Advanced Trigonometry: Limits, Infinite Sums, & Roots 161 2321 Advanced Trigonometric Identities: Sums 161 2322 Advanced Trigonometric Identities: Products 163 2323 Trigonometric Limits and Infinite Sums 164 2324 Trigonometric Roots of Equations 164 2325 Various Trigonometric Inequalities 165 2326 Triangle Trigonometric Inequalities 165 233 Logarithms, Graphing and Nonroutine Problems 167 24 Math Olympiad Trigonometry 401 and Beyond: 50 Unsolved Math Olympiad Trigonometry Problems 169 25 Math Olympiad Trigonometry 501: Beyond Plane Euclidean Trigonometry (Egregia Introductio ad Monstruosam Trigonometriam) 175 251 63 Ancient Spherical Geometry Problems 192 II Hints & Select Solutions 201