Let $ABC$ be a triangle, and $M$ an interior point such that $\angle MAB=10^\circ$, $\angle MBA=20^\circ$, $\angle MAC=40^\circ$ and $\angle MCA=30^\circ$. Prove that the triangle is isosceles.

okay this one is from Prof. Mircea Lascu from Zalau, Romaniaand Prof. V. Cartoaje from Ploiesti, Romania. It goes like this: given being a triangle ABC for every point M inside we construct the points AM, BM, CM on the circumcircle of the triangle ABC such that A, AM, M are collinear and so on. Find the locus of these points M for which the area of the triangle AM BM CM is bigger than the area of the triangle ABC.

$S= \{1,4,8,9,16,...\} $is the set of perfect integer power. ( $S=\{ n^k| n, k \in Z, k \ge 2 \}$. )We arrange the elements in $S$ into an increasing sequence $\{a_i\}$ . Show that there are infinite many $n$, such that $9999|a_{n+1}-a_n$

If A,B are invertible and the set {A^k - B^k | k is a natural number} is finite , then there exists a natural number m such that A^m = B^m.