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104 Number Theory Problems: From the Training of the USA IMO Team

Titu Andreescu Dorin Andrica Zuming Feng

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No detectada	2007	SpringerLink

Información

Tipo de recurso:

libros

ISBN impreso

978-0-8176-4527-4

ISBN electrónico

978-0-8176-4561-8

Editor responsable

Springer Nature

País de edición

Reino Unido

Fecha de publicación

2007

Información sobre derechos de publicación

Cobertura temática

Matemáticas

Tabla de contenidos

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doi: 10.1007/978-0-8176-4561-8

104 Number Theory Problems

Pp. No disponible

doi: 10.1007/978-0-8176-4561-8_1

Foundations of Number Theory

Back in elementary school, we learned four fundamental operations on numbers (integers), namely, addition (+), subtraction (−), multiplication (× or ·), and division $$ ( \div or/or\tfrac{{}} {c}) $$ . For any two integers a and b , their sum a + b , differences a − b and b − a , and product ab are all integers, while their quotients a ÷ b (or a/b or a/b ) and b ÷ a are not necessarily integers.

Palabras clave: Positive Integer; Number Theory; Prime Divisor; Diophantine Equation; Residue Class.

Pp. 1-74

doi: 10.1007/978-0-8176-4561-8_2

Introductory Problems

1. Let 1, 4, ... and 9, 16, ... be two arithmetic progressions. The set S is the union of the first 2004 terms of each sequence. How many distinct numbers are in S ?

Pp. 75-82

doi: 10.1007/978-0-8176-4561-8_3

Advanced Problems

1. Prove that the sum of the squares of 3, 4, 5, or 6 consecutive integers is not a perfect square. Give an example of 11 consecutive positive integers the sum of whose squares is a perfect square.

Palabras clave: Positive Integer; Prime Divisor; Winning Strategy; Balance Scale; Fermat Number.

Pp. 83-90

doi: 10.1007/978-0-8176-4561-8_4

Solutions to Introductory Problems

1. [AMC10B 2004] Let 1, 4, ... and 9, 16, ... be two arithmetic progressions. The set S is the union of the first 2004 terms of each sequence. How many distinct numbers are in S ?

Palabras clave: Positive Integer; Prime Divisor; Arithmetic Progression; Residue Class; Euclidean Algorithm.

Pp. 91-130

doi: 10.1007/978-0-8176-4561-8_5

Solutions to Advanced Problems

1. [MOSP 1998] Prove that the sum of the squares of 3, 4, 5, or 6 consecutive integers is not a perfect square. Give an example of 11 consecutive positive integers the sum of whose squares is a perfect square.

Palabras clave: Positive Integer; Induction Hypothesis; Nonnegative Integer; Prime Divisor; Winning Strategy.

Pp. 131-187