MathDB

2

Part of 2005 Irish Math Olympiad

Problems(2)

centroid

Source: Ireland 2005

7/6/2009
Let D,E D,E and F F be points on the sides BC,CA BC,CA and AB AB respectively of a triangle ABC ABC, distinct from the vertices, such that AD,BE AD,BE and CF CF meet at a point G G. Prove that if the angles AGE,CGD,BGF AGE,CGD,BGF have equal area, then G G is the centroid of △ABC \triangle ABC.
geometryalgebrasystem of equationsgeometry proposed
game

Source: Ireland 2005

7/6/2009
Using the digits: 1,2,3,4,5, 1,2,3,4,5, players A A and B B compose a 2005 2005-digit number N N by selecting one digit at a time: A A selects the first digit, B B the second, A A the third and so on. Player A A wins if and only if N N is divisible by 9 9. Who will win if both players play as well as possible?
combinatorics unsolvedcombinatorics