Problems

A continuous function $f$ has the following properties:

1. $f$ is defined on the entire number line;

2. $f$ at each point has a derivative (and thus the graph of f at each point has a unique tangent);

3. the graph of the function $f$ does not contain points for which one of the coordinates is rational and the other is irrational.

Does it follow that the graph of $f$ is a straight line?

Peter has 28 classmates. Each 2 out of these 28 have a different number of friends in the class. How many friends does Peter have?

To each pair of numbers $x$ and $y$ some number $x\ast y$ is placed in correspondence. Find $1993\ast 1935$ if it is known that for any three numbers $x,y,z$, the following identities hold: $x\ast x=0$ and $x\ast (y\ast z)=(x\ast y)+z$.

For each pair of real numbers $a$ and $b$, consider the sequence of numbers $p_n=\lfloor 2\{an+b\}\rfloor$. Any $k$ consecutive terms of this sequence will be called a word. Is it true that any ordered set of zeros and ones of length $k$ is a word of the sequence given by some $a$ and $b$ for $k=4$; when $k=5$?

Note: $\lfloor c\rfloor$ is the integer part, $\{c\}$ is the fractional part of the number $c$.

$x_1$ is the real root of the equation $x^2+ax+b=0$, $x_2$ is the real root of the equation $x^2-ax-b=0$.

Prove that the equation $x^2+2ax+2b=0$ has a real root, enclosed between $x_1$ and $x_2$. ($a$ and $b$ are real numbers).

In the number $a=0.12457\dots$ the $n$th digit after the decimal point is equal to the digit to the left of the decimal point in the number. Prove that $\alpha$ is an irrational number.

Solve the equation: $(x^3-2)(2^{\sin x}-1)+(2^{x^3}-4)\sin x=0$.

With a non-zero number, the following operations are allowed: $x\to \frac{1+x}{x}$, $x\to \frac{1-x}{x}$. Is it true that from every non-zero rational number one can obtain each rational number with the help of a finite number of such operations?

Find all functions $f(x)$ defined for all positive $x$, taking positive values and satisfying the equality $f(x^y)=f(x{)}^f(y)$ for any positive $x$ and $y$.

Seven triangular pyramids stand on the table. For any three of them, there is a horizontal plane that intersects them along triangles of equal area. Prove that there is a plane intersecting all seven pyramids along triangles of equal area.