Problems

What has a greater value: $300!$ or ${100}^{300}$?

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

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

What time is it going to be in $2019$ hours from now?

What is a remainder of $1203\times 1203-1202\times 1205$ when divided by $12$?

Show that a perfect square can only have remainders 0 or 1 when divided by 4.

What is a remainder of $7780\times 7781\times 7782\times 7783$ when divided by $7$?

Tim had more hazelnuts than Tom. If Tim gave Tom as many hazelnuts as Tom already had, Tim and Tom would have the same number of hazelnuts. Instead, Tim gave Tom only a few hazelnuts (no more than five) and divided his remaining hazelnuts equally between $3$ squirrels. How many hazelnuts did Tim give to Tom?

Prove that $n^3-n$ is divisible by $24$ for any odd $n$.

Show that if numbers $a-b$ and $c-d$ are divisible by $11$, then $ac-bd$ and $ad-bc$ are also both divisible by $11$.

For how many pairs of numbers $x$ and $y$ between $1$ and $100$ is the expression $x^2+y^2$ divisible by $7$?