## Zero matrix product

Let be real symmetric matrices such that for every nonzero positive integer . Show that .

*AMM 11483*

## Condition for an integer to be a power of another

Let be integers such that for all we have . Prove that is power of (there exists a positive integer such that ).

*Marius Cavachi AMM 10674*

## Permutation with distinct distances

For what values of there exist permutations of the set such that all the differences are distinct, for .

*AMM 1989*

## Matrix with integers is matrix power of any positive order

Suppose is a non-singular matrix having integer entries such that for any positive integer there exists a matrix having integer entries with . Prove that .

*AMM11401 Marius Cavachi, Romania Read more…*

## Thebault’s Theorem

Through the vertex of a triangle , a straight line is drawn, cutting the side at . and are the centres of the circumcircle and the incircle of the triangle . Let be the centre of the circle which touches and the circumcircle, and let be the centre of a further circle which touches and the circumcircle. Then and are collinear.

## Preserving sum 3 cubes

Determine all functions satisfying , for all integers .

*Titu Andreescu, AMM 10728*

## Heronian Triangle

A Heronian triangle has integer sides and integer area. ( Take for example sides 25,34,39 which has area 420 ) Prove that for any Heronian triangle we can find a representation in the lattice . ( i.e. we can find three points in such that the initial triangle is congruent with the triangle formed by these three points )