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Problems Algebra Algebraic inequalities and systems of inequalities Algebraic inequalities (other) Methods Algebraic methods Counting in two ways Calculus Real numbers Integer and fractional parts. Archimedean property

13–15 4.0

Prove that for any positive integer $n$ the inequality

is true.

Problem #PRU-61411

Problems Algebra Algebraic inequalities and systems of inequalities Algebraic inequalities (other)

13–15 3.0

Let $p$ and $q$ be positive numbers where $1 / p + 1 / q = 1$ . Prove that $a_{1} b_{1} + a_{2} b_{2} + \dots + a_{n} b_{n} \leq (a_{1}^{p} + \dots a_{n}^{p})^{1 / p} (b_{1}^{q} + \dots + b_{n}^{q})^{1 / q}$ The values of the variables are considered positive.