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Found: 4

Calculus Real numbers Integer and fractional parts. Archimedean property Problems Algebra Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other)

2.0

In a group of friends, each two people have exactly five common acquaintances. Prove that the number of pairs of friends is divisible by 3.

Problem #PRU-78705

Problems Number Theory Divisibility Divisibility of a number. General properties Methods Examples and counterexamples. Constructive proofs Calculus Real numbers Integer and fractional parts. Archimedean property

14–16 3.0

Does there exist a number \(h\) such that for any natural number \(n\) the number \(\lfloor h \times 2021^n\rfloor\) is not divisible by \(\lfloor h \times 2021^{n-1}\rfloor\)?

Problem #PRU-97790

Problems Methods Algebraic methods Counting in two ways Algebra Exponential functions and logarithms Exponential functions and logarithms (other) Calculus Real numbers Integer and fractional parts. Archimedean property

16–18 4.0

Prove that for every natural number \(n > 1\) the equality: \[\lfloor n^{1 / 2}\rfloor + \lfloor n^{1/ 3}\rfloor + \dots + \lfloor n^{1 / n}\rfloor = \lfloor \log_{2}n\rfloor + \lfloor \log_{3}n\rfloor + \dots + \lfloor \log_{n}n\rfloor\] is satisfied.

Problem #PRU-116627

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