## On permutation

Let be vectors of – dimensional Euclidean space such that . Prove that there exists a permutation of the integers such that

for each .

Solution

We define inductively. Set .Assume is defined for and also

(1)

Note that is true for . We choose in a way that is fulfilled for instead of . Set and . Assume that for all .  Then and in view of ones gets which is impossible. Hence , there is such that

(2)

Put . Then using and we have

which verifies for . Thus we define for every . Finally from we get

## Irreducible fraction

Find all positive integers such that it holds that

(1)

where stands for the period.

Solution

Well,

## Divisibility

Prove that the product of consecutive positive integers divides .

Solution

The number of different combinations of over is of course an integer and equals to

and the result follows.

Any questions?

## gcd and subgroup

Let be a non trivial subgroup and . Prove that

Solution

It is enough to prove it for the case since we can then reduce it by using the fact that

We have that for any integers and . But by Bezout’s lemma we can find such that

and hence .

## On a transcedental number

Prove that the number is transcedental.

Solution

Let . Then

which is algebraic.  Hence it follows from Weierstrass – Lindemann theorem  that , as algebraic, must be transcedental.

The exercise can also be found at mathematica.gr .