2

Part of 1988 Bundeswettbewerb Mathematik

Problems(2)

heights and inner radius of a triangle (BWM1988_1_2)

Source: Bundeswettbewerb Mathematik 1988, first stage, second question

h_a

h_b

h_c

be the heights and

r

the inradius of a triangle. Prove that the triangle is equilateral if and only if

h_a + h_b + h_c = 9r

geometryinradiusgeometry solved

3k points divide a circle into k arcs of lengths 1, 2 and 3

Source: 1988 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p2

A circle is somehow divided by

3k

k

arcs of lengths

1, 2

3

each. Prove that two of these points are always diametrically opposite.

geometrycombinatoricscombinatorial geometry