MathDB

1

Part of 2014 International Zhautykov Olympiad

Problems(2)

International Zhautykov olympiad 2014 problem 1

Source:

1/14/2014
Points MM, NN, KK lie on the sides BCBC, CACA, ABAB of a triangle ABCABC, respectively, and are different from its vertices. The triangle MNKMNK is called beautiful if ∠BAC=∠KMN\angle BAC=\angle KMN and ∠ABC=∠KNM\angle ABC=\angle KNM. If in the triangle ABCABC there are two beautiful triangles with a common vertex, prove that the triangle ABCABC is right-angled.
Proposed by Nairi M. Sedrakyan, Armenia
geometrycircumcirclegeometry proposedeasy
Polynom

Source:

1/15/2014
Does there exist a polynomial P(x)P(x) with integral coefficients such that P(1+3)=2+3P(1+\sqrt 3) = 2+\sqrt 3 and P(3+5)=3+5P(3+\sqrt 5) = 3+\sqrt 5 ?
Proposed by Alexander S. Golovanov, Russia
algebrapolynomialcalculusintegrationalgebra proposed