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Problem #PRU-35257

Problems Algebra and arithmetic Polynomials Algebraic identities for polynomials Factoring polynomials

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

Problem #PRU-116627

Problems Algebra and arithmetic 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: \(\lfloor x\rfloor \times \{x\} < x - 1\).