MathDB

5

Part of 2011 Dutch IMO TST

Problems(2)

DF divides the line segment EG into two equal parts

Source: Dutch IMO TST1 2011 p5

8/6/2019
Let ABCABC be a triangle with ∣AB∣>∣BC∣|AB|> |BC|. Let DD be the midpoint of ACAC. Let EE be the intersection of the angular bisector of ∠ABC\angle ABC and the line ACAC. Let FF be the point on BEBE such that CFCF is perpendicular to BEBE. Finally, let GG be the intersection of CFCF and BDBD. Prove that DFDF divides the line segment EGEG into two equal parts.
bisects segmentangle bisectorgeometry
a red, b blue and c green points such that a+b+c = 10

Source: Dutch IMO TST2 2011 p5

1/10/2020
Find all triples (a,b,c)(a, b, c) of positive integers with a+b+c=10a+b+c = 10 such that there are aa red, bb blue and cc green points (all different) in the plane satisfying the following properties: ∙\bullet for each red point and each blue point we consider the distance between these two points, the sum of these distances is 3737, ∙\bullet for each green point and each red point we consider the distance between these two points, the sum of these distances is 3030, ∙\bullet for each blue point and each green point we consider the distance between these two points, the sum of these distances is 11.
combinatoricsColoring