Problems

Does a continuous function that takes every real value exactly 3 times exist?

Prove that the function $\cos \sqrt{x}$ is not periodic.

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.

The functions $f$ and $g$ are defined on the entire number line and are reciprocal. It is known that $f$ is represented as a sum of a linear and a periodic function: $f(x)=kx+h(x)$, where $k$ is a number, and $h$ is a periodic function. Prove that $g$ is also represented in this form.

The function $f(x)$ is defined for all real numbers, and for any $x$ the equalities $f(x+2)=f(2-x)$ and $f(x+7)=f(7-x)$ are satisfied. Prove that $f(x)$ is a periodic function.

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

We consider a function $y=f(x)$ defined on the whole set of real numbers and satisfying $f(x+k)\times (1-f(x))=1+f(x)$ for some number $k\ne 0$. Prove that $f(x)$ is a periodic function.

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.