Problems

You are given 1002 different integers that are no greater than 2000. Prove that it is always possible to choose three of the given numbers so that the sum of two of them is equal to the third.

Will this still always be possible if we are given 1001 integers rather than 1002?

Prove that amongst any 11 different decimal fractions of infinite length, there will be two whose digits in the same column – 10ths, 100s, 1000s, etc – coincide (are the same) an infinite number of times.

Let $a$, $b$, $c$ be integers; where $a$ and $b$ are not equal to zero.

Prove that the equation $ax+by=c$ has integer solutions if and only if $c$ is divisible by $d=\mathrm{GCD}(a,b)$.

Prove that in a three-digit number, that is divisible by 37, you can always rearrange the numbers so that the new number will also be divisible by 37.

$N$ points are given, no three of which lie on one line. Each two of these points are connected by a segment, and each segment is coloured in one of the $k$ colours. Prove that if $N>\lfloor k!e\rfloor$, then among these points one can choose three such that all sides of the triangle formed by them will be colored in one colour.

Let it be known that all the roots of some equation $x^3+p{x}^2+qx+r=0$ are positive. What additional condition must be satisfied by its coefficients $p,q$ and $r$ in order for it to be possible to form a triangle from segments whose lengths are equal to these roots?

Prove that amongst any 7 different numbers it is always possible to choose two of them, $x$ and $y$, so that the following inequality was true: $$0<\frac{x-y}{1+xy}<\frac{1}{\sqrt{3}}.$$

The frog jumps over the vertices of the hexagon $ABCDEF$, each time moving to one of the neighbouring vertices.

a) How many ways can it get from $A$ to $C$ in $n$ jumps?

b) The same question, but on condition that it cannot jump to $D$?

c) Let the frog’s path begin at the vertex $A$, and at the vertex $D$ there is a mine. Every second it makes another jump. What is the probability that it will still be alive in $n$ seconds?

d)* What is the average life expectancy of such frogs?

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.

Find the largest number of colours in which you can paint the edges of a cube (each edge with one colour) so that for each pair of colours there are two adjacent edges coloured in these colours. Edges are considered to be adjacent if they have a common vertex.