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Found: 3

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-116817

Problems Methods Algebraic methods Counting in two ways Discrete Mathematics Set theory and logic Mathematical logic Mathematical logic (other)

13–14 3.0

There is a group of 5 people: Alex, Beatrice, Victor, Gregory and Deborah. Each of them has one of the following codenames: V, W, X, Y, Z. We know that:

Alex is 1 year older than V,

Beatrice is 2 years older than W,

Victor is 3 years older than X,

Gregory is 4 years older than Y.

Who is older and by how much: Deborah or Z?

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.