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Problem #PRU-116627

Problems Algebra Algebraic inequalities and systems of inequalities Algebraic inequalities (other) Polynomials Algebraic identities for polynomials Factoring polynomials Calculus Real numbers Integer and fractional parts. Archimedean property

Solve the inequality: \(\lfloor x\rfloor \times \{x\} < x - 1\).

Problem #PRU-35257

Problems Algebra Polynomials Algebraic identities for polynomials Factoring polynomials

Prove that, if \(b=a-1\), then \[(a+b)(a^2 +b^2)(a^4 +b^4)\dotsb(a^{32} +b^{32})=a^{64} -b^{64}.\]