Putnam and Beyond

By Titu Andreescu

Putnam and past takes the reader on a trip during the global of faculty arithmetic, targeting the most vital ideas and ends up in the theories of polynomials, linear algebra, actual research in a single and a number of other variables, differential equations, coordinate geometry, trigonometry, straightforward quantity conception, combinatorics, and chance. utilizing the W.L. Putnam Mathematical pageant for undergraduates as an inspiring image to construct a suitable math historical past for graduate experiences in natural or utilized arithmetic, the reader is eased into transitioning from problem-solving on the highschool point to the college and past, that's, to mathematical research.

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6. 2. three Counting recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. 2. four The Inclusion–Exclusion precept . . . . . . . . . . . . . . . . . . . . . . . . . . 6. three likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. three. 1 both most likely situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. three. 2 developing family between chances . . . . . . . . . . . . . . . . . . 6. three. three Geometric chances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 291 294 294 298 302 308 310 310 314 318 strategies tools of facts . . . . . . . . . . . . . . . . . . . . .

Three. 2. eleven Taylor and Fourier sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three. three Multivariable Differential and critical Calculus . . . . . . . . . . . . . . . . . . . . . three. three. 1 Partial Derivatives and Their functions . . . . . . . . . . . . . . . . . . . . three. three. 2 Multivariable Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three. three. three the numerous types of Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . three. four Equations with capabilities as Unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . three. four. 1 sensible Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three. four.

Xn be specified confident integers. 2. 2 Polynomials forty five 2. 1. 7 different Inequalities We finish with a bit for the inequalities aficionado. in the back of every one challenge hides a recognized inequality. 139. If x and y are optimistic actual numbers, convey that x y + y x > 1. one hundred forty. end up that for all a, b, c ≥ zero, (a five − a 2 + 3)(b5 − b2 + 3)(c5 − c2 + three) ≥ (a + b + c)3 . 141. think that each one the zeros of the polynomial P (x) = x n + a1 x n−1 + · · · + an are genuine and optimistic. express that if there exist 1 ≤ m < p ≤ n such that am = (−1)m mn and ap = (−1)p pn , then P (x) = (x − 1)n .

2. 2 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2. 1 A Warmup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2. 2 Viète’s kinfolk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2. three The spinoff of a Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2. four the site of the Zeros of a Polynomial . . . . . . . . . . . . . . . . . . . 2. 2. five Irreducible Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2. 6 Chebyshev Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 25 25 28 32 36 39 forty two forty five forty five forty five forty seven fifty two fifty four fifty six fifty eight viii three Contents 2.

2. 1. 2 x 2 ≥ zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 1. three The Cauchy–Schwarz Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 1. four The Triangle Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 1. five The mathematics Mean–Geometric suggest Inequality . . . . . . . . . . . . . 2. 1. 6 Sturm’s precept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 1. 7 different Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2. 1 A Warmup . . . .

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