MathDB

1

Part of 2011 Bosnia Herzegovina Team Selection Test

Problems(2)

triangleABC BC=1/2(AB+AC)

Source: Bosnia and Herzegovina 2011

5/16/2011
In triangle ABCABC it holds ∣BC∣=12(∣AB∣+∣AC∣)|BC|= \frac{1}{2}(|AB|+|AC|). Let MM and NN be midpoints of ABAB and ACAC, and let II be the incenter of ABCABC. Prove that A,M,I,NA, M, I, N are concyclic.
geometryincentercircumcirclegeometric transformationhomothetyratioanalytic geometry
Numbers on circle

Source: Bosnia and Herzegovina 2011

5/16/2011
Find maximum value of number aa such that for any arrangement of numbers 1,2,…,101,2,\ldots ,10 on a circle, we can find three consecutive numbers such their sum bigger or equal than aa.
combinatorics proposedcombinatorics