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?

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 4 or 5, and any two neighbouring terms differ by no more than 2. How many sequences will he have to write out?

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.

There are $n$ lines on a plane, no two lines are parallel, and no three lines cross at one point. Show that those lines dived the plane into $\frac{n(n+1)}{2}+1$ regions.

In a sequence 2, 6, 12, 20, 30, ... find the number

(a) in the 6th place

(b) in the 2016th place.

Using mathematical induction prove that $$1+3+5+\cdots +(2n-1)=n^2.$$

Circles and lines are drawn on the plane. They divide the plane into non-intersecting regions, see the picture below.

Show that it is possible to colour the regions with two colours in such a way that no two regions sharing some length of border are the same colour.

Numbers $1,2,\dots ,n$ are written on a whiteboard. In one go Louise is allowed to wipe out any two numbers $a$ and $b$, and write their sum $a+b$ instead. Louise enjoys erasing the numbers, and continues the procedure until only one number is left on the whiteboard.

What number is it? What if instead of $a+b$ she writes $a+b-1$?