After login you will be able to create your own lists of problems.

Problems

Sort

Age

Difficulty

Found: 4

Problem #PRU-109704

Problems Algebra and arithmetic 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

Prove that for any positive integer $n$ the inequality

is true.

Problem #PRU-110122

Problems Algebra and arithmetic Algebraic inequalities and systems of inequalities Algebraic inequalities (other) Calculus Functions of one variable. Continuity Monotonicity, boundedness

The functions $f (x) - x$ and $f (x^{2}) - x^{6}$ are defined for all positive $x$ and increase. Prove that the function

also increases for all positive $x$ .

Problem #PRU-61411

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

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.

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: $⌊ x ⌋ \times {x} < x - 1$ .