Problems

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

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

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.

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.

In a row there are 2023 numbers. The first number is 1. It is known that each number, except the first and the last, is equal to the sum of two neighboring ones. Find the last number.