2020 Candian MO

Part of Canadian Students Math Olympiad

Subcontests

(3)

1

Wierd problem in linear algebra

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.

1

2nd Geometry problem :) ... c'mon ppl let's start solving

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.

1

1st Problem [< MAB = 10°, ... ==> triangle ABC isosceles]

ABC

be a triangle, and

M

an interior point such that

\angle MAB=10^\circ

\angle MBA=20^\circ

\angle MAC=40^\circ

\angle MCA=30^\circ

. Prove that the triangle is isosceles.