MathDB

1

Part of 1996 Vietnam Team Selection Test

Problems(2)

n pairwise disjoint triangles

Source: Vietnam TST 1996 for the 37th IMO, problem 1

6/26/2005
In the plane we are given 3⋅n3 \cdot n points (n>n>1) no three collinear, and the distance between any two of them is ≤1\leq 1. Prove that we can construct nn pairwise disjoint triangles such that: The vertex set of these triangles are exactly the given 3n points and the sum of the area of these triangles <1/2< 1/2.
geometryrectanglecombinatorial geometryarea of a trianglecombinatorics unsolvedcombinatorics
d does not depend on the order of the axes of symmetry

Source: Vietnam TST 1996 for the 37th IMO, problem 4

6/26/2005
Given 3 non-collinear points A,B,CA,B,C. For each point MM in the plane (ABCABC) let M1M_1 be the point symmetric to MM with respect to ABAB, M2M_2 be the point symmetric to M1M_1 with respect to BCBC and M′M' be the point symmetric to M2M_2 with respect to ACAC. Find all points MM such that MM′MM' obtains its minimum. Let this minimum value be dd. Prove that dd does not depend on the order of the axes of symmetry we chose (we have 3 available axes, that is BCBC, CACA, ABAB. In the first part the order of axes we chose ABAB, BCBC, CACA, and the second part of the problem states that the value dd doesn't depend on this order).
symmetrygeometry unsolvedgeometry