MathDB

2

Part of 2003 Vietnam Team Selection Test

Problems(2)

incircle with center I of triangle ABC touches the side BC

Source: Vietnam TST 2003 for the 44th IMO, problem 2

6/26/2005
Given a triangle ABCABC. Let OO be the circumcenter of this triangle ABCABC. Let HH, KK, LL be the feet of the altitudes of triangle ABCABC from the vertices AA, BB, CC, respectively. Denote by A0A_{0}, B0B_{0}, C0C_{0} the midpoints of these altitudes AHAH, BKBK, CLCL, respectively. The incircle of triangle ABCABC has center II and touches the sides BCBC, CACA, ABAB at the points DD, EE, FF, respectively. Prove that the four lines A0DA_{0}D, B0EB_{0}E, C0FC_{0}F and OIOI are concurrent. (When the point OO concides with II, we consider the line OIOI as an arbitrary line passing through OO.)
geometrycircumcircleratiogeometric transformationreflectionanalytic geometryhomothety
Vietnam TST 2003 set of all permutations

Source: Vietnam TST 2003 for the 44th IMO, problem 5

6/26/2005
Let AA be the set of all permutations a=(a1,a2,…,a2003)a = (a_1, a_2, \ldots, a_{2003}) of the 2003 first positive integers such that each permutation satisfies the condition: there is no proper subset SS of the set {1,2,…,2003}\{1, 2, \ldots, 2003\} such that {ak∣k∈S}=S.\{a_k | k \in S\} = S. For each a=(a1,a2,…,a2003)∈Aa = (a_1, a_2, \ldots, a_{2003}) \in A, let d(a)=∑k=12003(ak−k)2.d(a) = \sum^{2003}_{k=1} \left(a_k - k \right)^2. I. Find the least value of d(a)d(a). Denote this least value by d0d_0. II. Find all permutations a∈Aa \in A such that d(a)=d0d(a) = d_0.
inequalitiestriangle inequalitycombinatorics unsolvedcombinatorics