3

Part of 2008 Hungary-Israel Binational

Problems(2)

Special partition of a rectangle in smaller rectangles.

Source: ISL 2007, C3 /

D

is partitioned in several (

\ge2

) rectangles with sides parallel to those of

D

. Given that any line parallel to one of the sides of

D

, and having common points with the interior of

D

, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with

D

's boundary.
Author: Kei Irie, Japan

geometryrectanglecombinatoricsdissectionIMO Shortlist

Hungary-Israel Binational 2008\6

Source: construction of an isosceles triangle

P and Q are 2 points in the area bounded by 2 rays, e and f, coming out from a point O. Describe how to construct, with a ruler and a compass only, an isosceles triangle ABC, such that his base AB is on the ray e, the point C is on the ray f, P is on AC, and Q on BC.

geometryconicshyperbolageometric transformationreflectionangle bisectorgeometry unsolved