MathDB

4

Part of 2006 Estonia National Olympiad

Problems(4)

Estonian Math Competitions 2005/2006

Source: Final Round Grade 9 Pro 3

7/30/2008
Triangle ABC ABC is isosceles with AC \equal{} BC and \angle{C} \equal{} 120^o. Points D D and E E are chosen on segment AB AB so that |AD| \equal{} |DE| \equal{} |EB|. Find the sizes of the angles of triangle CDE CDE.
trigonometrygeometry unsolvedgeometry
[x/3]+[2x/3]=x

Source: 2006 Estonia National Olympiad Final Round grade 10 p4

3/12/2020
Solve the equation [x3]+[2x3]=x\left[\frac{x}{3}\right]+\left [\frac{2x}{3}\right]=x
floor functionequationalgebra
Estonian Math Competitions 2005/2006

Source: Final Round Grade 11 Pro 4

7/30/2008
In a triangle ABC with circumcentre O and centroid M, lines OM and AM are perpendicular. Let AM intersect the circumcircle of ABC again at A′. Let lines BA′ and AC intersect at D and let lines CA′ and AB intersect at E. Prove that the circumcentre of triangle ADE lies on the circumcircle of ABC.
geometrycircumcirclegeometric transformationhomothetyratiogeometry unsolved
Estonian Math Competitions 2005/2006

Source: Final Round Grade 12 Pro 4

7/30/2008
Let O be the circumcentre of an acute triangle ABC and let A′, B′ and C′ be the circumcentres of triangles BCO, CAO and ABO, respectively. Prove that the area of triangle ABC does not exceed the area of triangle A′B′C′.
geometrytrigonometrygeometry unsolved