We solve problems

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: $a_{1} = 1, a_{n + 1} = a_{n} + ⌊ \sqrt{a_{n}} ⌋ .$

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

Problem #PRU-61535

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

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-64494

Problems Methods Algebraic methods Proof by exhaustion Set theory and logic Mathematical logic Puzzles

10–11 2.0

Specify any solution of the puzzle: $2014 + Y E S = B E A R$ .

Problem #PRU-65269

Problems Algebra and arithmetic Mean values Methods Algebraic methods Proof by exhaustion Extremal principle Sorting in increasing (decreasing) order Probability and statistics Statistics

13–16 3.0

In the first term of the year Daniel received five grades in mathematics with each of them being on a scale of 1 to 5, and the most common grade among them was a 5 . In this case it turned out that the median of all his grades was 4, and the arithmetic mean was 3.8. What grades could Daniel have?

Problem #PRU-65309

Problems Probability and statistics Probability theory Continuous distribution Algebra and arithmetic Mean values Methods Algebraic methods Proof by exhaustion

15–16 4.0

The point $O$ is randomly chosen on piece of square paper. Then the square is folded in such a way that each vertex is overlaid on the point $O$ . The figure shows one of the possible folding schemes. Find the mathematical expectation of the number of sides of the polygon that appears.

Problem #PRU-78629

Problems Methods Pigeonhole principle Pigeonhole principle (other) Algebraic methods Proof by exhaustion

12–14 3.0

Is it possible to place the numbers $1, 2, \dots 12$ around a circle so that the difference between any two adjacent numbers is 3, 4, or 5?

Problem #PRU-79505

Problems Methods Pigeonhole principle Algebraic methods Proof by exhaustion

12–14 3.0

In March 2015 a teacher ran 11 sessions of a maths club. Prove that if no sessions were run on Saturdays or Sundays there must have been three days in a row where a session of the club did not take place. The 1st March 2015 was a Sunday.

Problem #PRU-87992

Problems Algebra and arithmetic Number systems Decimal number system Methods Algebraic methods Proof by exhaustion

10–12 2.0

One three-digit number consists of different digits that are in ascending order, and in its name all words begin with the same letter. The other three-digit number, on the contrary, consists of identical digits, but in its name all words begin with different letters. What are these numbers?