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Found: 4

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

14–16 3.0

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

Problem #PRU-35257

Problems Algebra and arithmetic Polynomials Algebraic identities for polynomials Factoring polynomials

13–14 3.0

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-64622

Problems Algebra and arithmetic Polynomials Algebraic identities for polynomials Identical transformations Geometry Plane geometry Quadrilateral Inscribed quadrilaterals

13–15 3.0

The number \(x\) is such a number that exactly one of the four numbers \(a = x - \sqrt{2}\), \(b = x-1/x\), \(c = x + 1/x\), \(d = x^2 + 2\sqrt{2}\) is not an integer. Find all such \(x\).

Problem #PRU-64638

Problems Algebra and arithmetic Polynomials Algebraic identities for polynomials Identical transformations Calculus Real numbers Rational and irrational numbers

14–15 3.0

The numbers \(x\), \(y\) and \(z\) are such that all three numbers \(x + yz\), \(y + zx\) and \(z + xy\) are rational, and \(x^2 + y^2 = 1\). Prove that the number \(xyz^2\) is also rational.