References. Springer International Publishing Switzerland 2015 E. Grigorieva, Methods of Solving Nonstandard Problems, DOI /

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1 References 1. Graham, et al.: Concrete Mathematics a Foundation for Computer Science. Addison-Wesley (1988) 2. Embry, M., Schell, J., Pelham, J.: Calculus and Linear Algebra. W.B. Sounders Company (1972) 3. Grigorieva E.: Methods of Solving Complex Geometry Problems. Birkhauser (2013) 4. Lidsky, B., Ovsyannikov, L., Tulaikov, A., Shabunin, M.: Problems in Elementary Mathematics. MIR Publisher, Moscow (1973) 5. Grigorieva, E.: Complex Math Problems and How to Solve Them. TWU Press. Library of Congress, TX u / (2001) 6. Grigoriev, E. (ed.): Problems of the Moscow State University Entrance Exams, pp MAX-Press. (In Russian) (2002) 7. Grigoriev, E. (ed.): Problems of the Moscow State University Entrance Exams, pp. 1 92, MAX-Press. (in Russian) (2000) 8. Grigoriev, E. (ed.): Olympiads and Problems of the Moscow State University Entrance Exams. MAX-Press. (in Russian) (2008) 9. Eves, H.: An Introduction to the History of Mathematics. Thomson Brooks/Cole, New York, NY (1990) 10. Alfutova, N., Ustinov, A.: Algebra and Number Theory. MGU, Moscow (2009) (in Russian) 11. Suprun, V.: Selected Mathematics Problems. Librocom, Moscow (1998) (in Russian) 12. Chirsky, V.: Problems of Elementary Mathematics. MGU, Moscow (1998) (in Russian) 13. Budak, B., Zolotareva, N., Fedotov, M.: Algebra for High Schools. Moscow State University, Moscow (2013) (in Russian) Springer International Publishing Switzerland 2015 E. Grigorieva, Methods of Solving Nonstandard Problems, DOI /

2 324 References Contest Problems for Further Reading Shklarsky, D.O., Chentzov, N.N., Yaglom, I.M.: The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics. Dover (1993) Williams K. Hardy, K.S.: The Red Book of Mathematics Problems. (Undergraduate William Lowell Putnam competition). Dover (1996) Past USAMO tests. The USA Mathematical Olympiads from the late 70s and early 80s can be found at. Art of Problem Solving. Andreescu, T., Feng, Z.L.: Olympiad books. USA and International Mathematical Olympiads (MAA Problem Books Series)

3 Index A Abel, 137 Ancient Babylonians, 59, 110, 112 Arithmetic and geometric means, 25, 26, 30, 33, 37, 120, 239, , 253, 312, 315 Auxiliary argument, 6, 26, 45, 159, 190, 218, 221 B Bernoulli s inequality, 239, 251, 255, 316 Bezout s Theorem, 51, 52, 61 Binomial coefficients, 92, 93, 95 97, 151 Biquadratic equation, Bounded above, 12, 30, 44 Bounded below, 12, 44, 45 Bounded function, 6, 12 15, 30, 45, 222, 233, 240, 252, 275, 316 Boundedness, 1, 14, 30, 43, 44, 47, 159, 218, 219, , 261, 265, 274, 275, 279, 311 C Cardano s formula, 49, 113, Cauchy-Bunyakovsky (CB) inequality, 148, 239, , 314, 318 Complementary, 162, 168, 197, 206, 236, 275 Completing the square, 27, 31, 32, 38, 42, 66, 76, 88, 195, 222, 240, 243, 274, 278, 287, 290, 316, 319 Complex conjugate polynomials, 61, 81 Composition, 10, 11, 40, 287 Cubic equations, 49, 101, , 145, 146, 152, 155, Cyclic quadrilateral, 78 D Decreasing, 9 12, 16, 18, 26, 31, 34, 36, 40, 74, 115, 120, 121, 237, 241, 242 Decreasing functions, 9 12, 16, 18, 34, 36, 38 41, 59, 74, 120, 121, 241 Degree of the remainder polynomial, 50 De Moivre s Theorem, 204, 208, 209 D/4 formula, 63, 72, 105, 170, 314 The difference of squares, 68, 69, 80 85, 115, 132, 136, 139, 150, 183, 199, 252, 318 Differential equations, Discontinuity, 1 12 Discriminant, 43, 56, 57, 62, 64, 66, 67, 73, 75 77, 79, 80, 105, 118, 119, , 133, 134, 136, 144, 146, 170, 220, 243, 261, 265, 277, 314 Domain, 1 6, 10 12, 14, 15, 19 21, 31, 33, 34, 36, 38, 40, 41, 45, 46, 103, 157, 159, 165, 166, 222, , 241, 285, 286, 292, 310 Domain of the function, 4, 5, 14, 15, 38, 41, 231 E Euler s formula, 129, 205, 208, 209 Euler s method, 49, Even function, 3, 19 20, 37, 41, 208 Exponential functions, 12, 20, 32, 34, 99, 260 Springer International Publishing Switzerland 2015 E. Grigorieva, Methods of Solving Nonstandard Problems, DOI /

4 326 Index F Factorial, 92, 93 Ferrari, 135 Ferrari s Method, 49, Floor function, 20 Fractional part of number, 20 Function, 1, 49, 159, 241 Fundamental Theorem of Algebra, 49, 59, 144 G Geometric, 21, 26, 30, 33, 37, 69, 82, 83, 120, 165, 172, 181, 198, 201, 209, 239, 246, 247, , 306, 307 Global maximum, 15 Global minimum, 15, 16 Graphing calculator, 4, 130, 249, 250, 252, 259, 261, 277, 279, 287 H Heron s formula, 268, 272 Homogeneous polynomials, Horner s algorithm, 56, 59, 109 I Increasing function, 1, 9 12, 14 16, 18, 33, 34, 36 41, 45, 58, 59, 113, 115, 120, 121, 129, 147, 149, 152, 157, 165, 223 Inequality of Cauchy, 250, 252, 257, 258, 312 Intermediate Value Theorem (IVT), 7, 8, 157 Interval method, 34, 310 Inverse trigonometric functions, N Newton s Binomial theorem, 49, Newton s formula, 94 Number theory, 55, 64, 294, 299, 307 O Odd function, 19 20, 24, 37, 41, 151, 152, 165, 166, 179, 194, 205, 231, 232 One-to-one function, 2 4, 165 Onto, 2, 3, 24 P Pascal, Blaisé, 92 Pascal s triangle, 92, 96, 97, 158 Periodic function, 20 25, 129, 162, 223, 250 Polynomial function, 4, 6 8, 19, 20, 49, 57, 59 61, 107, 111, 114, 115, 118, 140, 141, 144, 150, 152, 155 Polynomial of minimal degree, 20, 142, 143 Primes, 49, 56, 57, 80, 81, 87 90, 143, 157, 297, 300, 309 Problem on work, 307 Problems about motion, 306 Problems in integers, 293 Production rates, 304, 307 Q Quadratic equation, 24, 41, 43, 48, 49, 57, 61 79, 81, 84, , 113, , 127, 128, 135, 136, 138, 140, 142, 148, 155, 170, 171, 185, 207, 226, 260, 261, 265, 277, 283, 293, 314 Quartic equation, 49, 135, , 150, 154 L Lagrange s Mean Value Theorem, 16 Law of cosines, 267, 269, 270 Local minimum, 8, 15, 123, 124, 134, 152 Logarithmic function, 30, 34, 45, 46, 222 M Minimal period, 20 23, 162 Mixture, 222, 303, 310 Monotonic behavior, 11, 40, 44 Monotonic function, 9, 41 R Range, 1, 2, 4, 6, 7, 21, 165, 166, 285, 292, 310 Rational zero Theorem, 53, 54, 56, 57, 85, 86, 101, 107, 109, 110, 113, 120, 121, 124, 129, 132, 133, 137, 148, 150, 152, 154, 155, 158 Relative maximum, 15, 16 Relative minimum, 15, 16 Resolvent, 137, 138, 140, 141, 154 Reversed Vieta Theorem, 64, 127 Rolle s Theorem, 16, 17, 49 50, 144, 145 Roots of a quadratic equation, 62, 64, 66, 67, 69, 81, 105, 127, 283

5 Index 327 S Sawtooth function, 21, 22 The sum and the difference of cubes, 87 Supplementary, 78, 161, 176, 207, 236 Symmetric polynomial equations, 101, Synthetic division, 53 57, 119, 124, 150, 152, 154 T Tartaglia, 124 TI 81-82, 283 Transcendental equation, 1, 30, 250 Triangle Bisector Theorem, 269 Trigonometric series, Trigonometric substitution, 212, 237, , 318 Trinomial square, 32, 66, 84, 104, 218 U Unbounded function, 6, Unit circle, , , 185, 187, 189, 192, 196, 198, 199, 234, 273 V Vertical asymptote, 4 6, 14 Vertical line test, 3 Vieta s Theorem, 49, 60, 61, 64 69, 78, 80, , 116, 117, , 145, 147, 153, 155, 157, 293 W Working with radicals, 103, 198