Posts Tagged ‘AMM’

Zero matrix product

June 21, 2010 Leave a comment

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

May 11, 2010 Leave a comment

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

April 26, 2010 Leave a comment

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

April 18, 2010 Leave a comment

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

March 31, 2010 Leave a comment

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.

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Categories: Geometry, Problem Solving Tags:

Preserving sum 3 cubes

March 31, 2010 Leave a comment

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.
Titu Andreescu, AMM 10728

Heronian Triangle

March 31, 2010 Leave a comment

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 )

Categories: Geometry, Problem Solving Tags:
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