We solve problems

Problem #PRU-109704

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-32132

Problems Methods Extremal principle Extremal principle (other) Discrete Mathematics 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 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-88001

Problems Methods Algebraic methods Processes and operations Discrete Mathematics Algorithm Theory Theory of algorithms (other)

11–13 3.0

A traveller rents a room in an inn for a week and offers the innkeeper a chain of seven silver links as payment – one link per day, with the condition that they will be payed everyday. The innkeeper agrees, with the condition that the traveller can only cut one of the links. How did the traveller manage to pay the innkeeper?

Problem #PRU-88031

Problems Methods Algebraic methods Processes and operations Discrete Mathematics Algorithm Theory Theory of algorithms (other)

10–13 3.0

There are 6 locked suitcases and 6 keys for them. It is not known which keys are for which suitcase. What is the smallest number of attempts do you need in order to open all the suitcases? How many attempts would you need if there are 10 suitcases and keys instead of 6?

Problem #PRU-60299

Problems Number Theory Divisibility Divisibility of a number. General properties Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

13–15 4.0

From the set of numbers 1 to \(2n\), \(n + 1\) numbers are chosen. Prove that among the chosen numbers there are two, one of which is divisible by another.

Problem #PRU-60366

Problems Number Theory Numeral systems Decimal number system Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

13–14 4.0

Prove that amongst any 11 different decimal fractions of infinite length, there will be two whose digits in the same column – 10ths, 100s, 1000s, etc – coincide (are the same) an infinite number of times.

Problem #PRU-61535

Problems 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

Prove that the 13th day of the month is more likely to occur on a Friday than on other days of the week. It is assumed that we live in the Gregorian style calendar.

Problem #PRU-65167

Problems Discrete Mathematics Algorithm Theory Cooperative algorithms Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

13–15 4.0

A pack of 36 cards was placed in front of a psychic face down. He calls the suit of the top card, after which the card is opened, shown to him and put aside. After this, the psychic calls out the suit of the next card, etc. The task of the psychic is to guess the suit as many times as possible. However, the card backs are in fact asymmetrical, and the psychic can see in which of the two positions the top card lies. The deck is prepared by a bribed employee. The clerk knows the order of the cards in the deck, and although he cannot change it, he can prompt the psychic by having the card backs arranged in a way according to a specific arrangement. Can the psychic, with the help of such a clue, ensure the guessing of the suit of

a) more than half of the cards;

b) no less than 20 cards?