We solve problems

Problem #PRU-107992

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Algebra Sequences Algebraic methods Symmetry and involutions

14–16 5.0

For each pair of real numbers \(a\) and \(b\), consider the sequence of numbers \(p_n = \lfloor 2 \{an + b\}\rfloor\). Any \(k\) consecutive terms of this sequence will be called a word. Is it true that any ordered set of zeros and ones of length \(k\) is a word of the sequence given by some \(a\) and \(b\) for \(k = 4\); when \(k = 5\)?

Note: \(\lfloor c\rfloor\) is the integer part, \(\{c\}\) is the fractional part of the number \(c\).

Problem #PRU-109493

Problems Methods Algebraic methods Iterations Mathematical induction Mathematical induction (other) Processes and operations Calculus Real numbers Rational and irrational numbers Set theory and logic Theory of algotithms Theory of algorithms (other)

14–16 4.0

With a non-zero number, the following operations are allowed: \(x \rightarrow \frac{1+x}{x}\), \(x \rightarrow \frac{1-x}{x}\). Is it true that from every non-zero rational number one can obtain each rational number with the help of a finite number of such operations?

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

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 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-32132

Problems Methods Extremal principle Extremal principle (other) Set theory and logic Order relations Algebraic methods Proof by exhaustion

13–15 3.0

The order of books on a shelf is called wrong if no three adjacent books are arranged in order of height (either increasing or decreasing). How many wrong orders is it possible to construct from \(n\) books of different heights, if: a) \(n = 4\); b) \(n = 5\)?

Problem #PRU-76475

Problems Algebra and arithmetic Number theory. Divisibility Division with remainders. Arithmetic of remainders Division with remainder Calculus Functions of one variable. Continuity Periodicity and aperiodicity Probability and statistics Probability theory Probability theory (other) Methods Algebraic methods Proof by exhaustion

13–16 4.0

What has a greater value: \(300!\) or \(100^{300}\)?

Problem #PRU-98152

Problems Algebra and arithmetic Number theory. Divisibility Division with remainders. Arithmetic of remainders Division with remainder Calculus Functions of one variable. Continuity Periodicity and aperiodicity Probability and statistics Probability theory Probability theory (other) Methods Algebraic methods Proof by exhaustion

14–17 4.0

A numerical sequence is defined by the following conditions: \[a_1 = 1, \quad a_{n+1} = a_n + \lfloor \sqrt{a_n}\rfloor .\]

Prove that among the terms of this sequence there are an infinite number of complete squares.

Problem #PRU-100625

Problems Methods Algebraic methods Counting in two ways

Orcs and goblins, 40 creatures altogether, are standing in a rectangular formation of \(4\) rows and \(10\) columns. Is it possible that the total number of orcs in each row is \(7\), while the number of orcs in each column is the same?

Problem #PRU-100626

Problems Methods Algebraic methods Counting in two ways

There are some cannons in every fortress on the Cannon Island. The star marks the Grand Fortress, the capital, and the 10 circles mark 10 smaller fortresses. The total number of cannons located in all the fortresses along the east-west road is known to be \(130\). The total number of cannons along each of the other 3 roads is \(80\). Also it is known that there is a total of \(280\) cannons in all the fortresses. How many cannons are in the capital?

Problem #PRU-100627

Problems Methods Algebraic methods Counting in two ways

There are \(5\) directors of \(5\) banks sitting at the round table. Some of these banks have a negative balance (they owe more money that they have) and some have a positive balance (they have more money that they owe). It is known that for any 3 directors sitting next to each otehr, their 3 banks together have a positive balance. Does it mean that the \(5\) banks together have a positive balance?