### Archive

Posts Tagged ‘AMM’

## Zero matrix product

Let $A,B$ be $n\times n$ real symmetric matrices such that $tr((A+B)^k)=tr(A^k)+tr(B^k)$ for every nonzero positive integer $k$. Show that $AB=0$.
AMM 11483

Categories: Algebra, Problem Solving Tags:

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

Let $a,b>1$ be integers such that for all $n>0$ we have $a^n-1|b^n-1$. Prove that $b$ is power of $a$ (there exists $k$ a positive integer such that $b=a^k$).
Marius Cavachi AMM 10674

## Permutation with distinct distances

For what values of $n \geq 1$ there exist permutations $(x_1,...,x_n)$ of the set $\{1,2,...,n\}$ such that all the differences $|x_k-k|$ are distinct, for $k=1..n$.

AMM 1989

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

Suppose $A$ is a $n\times n$ non-singular matrix having integer entries such that for any positive integer $k$ there exists a matrix $X$ having integer entries with $A=X^k$. Prove that $A=I_n$.
AMM11401 Marius Cavachi, Romania Read more…

## Thebault’s Theorem

Through the vertex $A$ of a triangle $ABC$, a straight line $AM$ is drawn, cutting the side $BC$ at $M$. $O$ and $I$ are the centres of the circumcircle and the incircle of the triangle $ABC$. Let $P$ be the centre of the circle which touches $MA, MC$ and the circumcircle, and let $Q$ be the centre of a further circle which touches $MA, MB$ and the circumcircle. Then $P,I$ and $Q$ are collinear.

Categories: Geometry, Problem Solving Tags:

## Preserving sum 3 cubes

Determine all functions $f:\mathbb{Z} \to \mathbb{Z}$ satisfying $f(x^3+y^3+z^3)=(f(x))^3+(f(y))^3+(f(z))^3$, for all integers $x,y,z$.
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 $\mathbb{Z} \times \mathbb{Z}$. ( i.e. we can find three points in $\mathbb{Z}\times \mathbb{Z}$ such that the initial triangle is congruent with the triangle formed by these three points )