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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

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

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.

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