The Red Book of Mathematical Problems

Kenneth S. Williams, Kenneth Hardy

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486151823
• Topic: Mathématiques
• Subtopic: Mathématiques dans l'éducation

In North America, the most prestigious competition in mathematics at the undergraduate level is the William Lowell Putnam Mathematical Competition. This volume is a handy compilation of 100 practice problems, hints, and solutions indispensable for students preparing for the Putnam and other undergraduate mathematical competitions. Indeed, it will be of use to anyone engaged in the posing and solving of mathematical problems.
Many of the problems in this book were suggested by ideas originating in a variety of sources, including Crux Mathematicorum, Mathematics Magazine, and the American Mathematical Monthly, as well as various mathematical competitions. This result is a rich selection of carefully chosen problems that will challenge and stimulate mathematical problem-solvers at varying degrees of proficiency.

THE SOLUTIONS

Some people think we are wrong, but only time will tell: given all the alternatives, we have the solution.

Lev Davydovich Bronstein Trotsky (1879-1940)

1. Let p denote an odd prime and set ω = exp(2πi/p). Evaluate the product

(1.0)

where r₁, . . . , r_{(p – 1)/2} denote the (p – 1)/2 quadratic residues modulo p and n₁, . . . , n_{(p – 1)/2} denote the (p – 1)/2 quadratic nonresidues modulo p.

Solution: We set q = (p – 1)/2 and

(1.1)

and for k = 0, 1, . . . , p – 1 let

(1.2)

If k is a quadratic residue (resp. nonresidue) (mod p) {kr_i : i = 1, 2, . . . , q} is a complete system of quadratic residues (resp. nonresidues) (mod p) and {kn_j : j = 1, 2, . . . , q} is a complete system of quadratic nonresidues (resp. residues) (mod p). Replacing r_i by kr_i and n_j by kn_j in (1.2), where 1 ≤ k < p – 1, we obtain

(1.3)

Next, we note that

(1.4)

as – 1 is a quadratic residue (mod p) for p ≡ 1 (mod 4) and – 1 is a quadratic nonresid...