Problems

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?

Prove that if the irreducible rational fraction $p/q$ is a root of the polynomial $P(x)$ with integer coefficients, then $P(x)=(qx-p)Q(x)$, where the polynomial $Q(x)$ also has integer coefficients.

There are 8 glasses of water on the table. You are allowed to take any two of the glasses and make them have equal volumes of water (by pouring some water from one glass into the other). Prove that, by using such operations, you can eventually get all the glasses to contain equal volumes of water.

A function $f$ is given, defined on the set of real numbers and taking real values. It is known that for any $x$ and $y$ such that $x>y$, the inequality $(f(x){)}^2\le f(y)$ is true. Prove that the set of values generated by the function is contained in the interval $[0,1]$.

Author: I.I. Bogdanov

Peter wants to write down all of the possible sequences of 100 natural numbers, in each of which there is at least one 3, and any two neighbouring terms differ by no more than 1. How many sequences will he have to write out?

Author: I.I. Bogdanov

Peter wants to write down all of the possible sequences of 100 natural numbers, in each of which there is at least one 4 or 5, and any two neighbouring terms differ by no more than 2. How many sequences will he have to write out?

Author: G. Zhukov

The square trinomial $f(x)=a{x}^2+bx+c$ that does not have roots is such that the coefficient $b$ is rational, and among the numbers $c$ and $f(c)$ there is exactly one irrational.

Can the discriminant of the trinomial $f(x)$ be rational?

At what value of $k$ is the quantity $A_k=({19}^k+{66}^k)/k!$ at its maximum? You are given a number $x$ that is greater than 1. Is the following inequality necessarily fulfilled $\lfloor \sqrt{\sqrt{x}}\rfloor =\lfloor \sqrt{\sqrt{x}}\rfloor$?

Using mathematical induction prove that $$1+2+3+\cdots +n=\frac{n(n+1)}{2}.$$

There are $n$ lines on a plane, and all the lines intersect at exactly one point. Prove that the lines divide the plane into $2n$ parts.