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

14–16 4.0

A numeric set $M$ containing 2003 distinct numbers is such that for every two distinct elements $a, b$ in $M$ , the number $a^{2} + b \sqrt{2}$ is rational. Prove that for any $a$ in $M$ the number $q \sqrt{2}$ is rational.

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

14–16 3.0

Solve the inequality: $⌊ x ⌋ \times {x} < x - 1$ .