MathDB

5

Part of 2000 IMO Shortlist

Problems(3)

nice [symmedians in a triangle, < ABM = < BAN]

Source: IMO Shortlist 2000, G5

11/14/2004
The tangents at BB and AA to the circumcircle of an acute angled triangle ABCABC meet the tangent at CC at TT and UU respectively. ATAT meets BCBC at PP, and QQ is the midpoint of APAP; BUBU meets CACA at RR, and SS is the midpoint of BRBR. Prove that ∠ABQ=∠BAS\angle ABQ=\angle BAS. Determine, in terms of ratios of side lengths, the triangles for which this angle is a maximum.
geometrycircumcircletrigonometryIMO Shortlist
Semiperimeter, inradius of triangle with int side lengths

Source: IMO Shortlist 2000, N5

8/10/2008
Prove that there exist infinitely many positive integers n n such that p \equal{} nr, where p p and r r are respectively the semiperimeter and the inradius of a triangle with integer side lengths.
Diophantine equationnumber theoryinradiusperimeterTriangleIMO Shortlist
Bound for sum of vertices of regions cut by rectangles

Source: IMO Shortlist 2000, C5

1/22/2005
In the plane we have nn rectangles with parallel sides. The sides of distinct rectangles lie on distinct lines. The boundaries of the rectangles cut the plane into connected regions. A region is nice if it has at least one of the vertices of the nn rectangles on the boundary. Prove that the sum of the numbers of the vertices of all nice regions is less than 40n40n. (There can be nonconvex regions as well as regions with more than one boundary curve.)
geometryrectanglecombinatoricsIntersectiongraph theoryIMO Shortlistdouble counting