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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-98152

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

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-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-65269

Problems Algebra 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 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.