MathDB

3

Part of 2014 Turkey Team Selection Test

Problems(3)

Ttriangles with the same area

Source: Turkey TST 2014 Day 1 Problem 3

3/12/2014
Let r,Rr,R and rar_a be the radii of the incircle, circumcircle and A-excircle of the triangle ABCABC with AC>ABAC>AB, respectively. I,OI,O and JAJ_A are the centers of these circles, respectively. Let incircle touches the BCBC at DD, for a point E∈(BD)E \in (BD) the condition A(IEJA)=2A(IEO)A(IEJ_A)=2A(IEO) holds. Prove that ED=AC−AB  ⟺  R=2r+ra.ED=AC-AB \iff R=2r+r_a.
geometrycircumcircleparallelogramgeometry proposed
Hard ineq

Source: Turkey TST 2014 Day 2 Problem 6

3/12/2014
Prove that for all all non-negative real numbers a,b,ca,b,c with a2+b2+c2=1a^2+b^2+c^2=1 a+b+a+c+b+c≥5abc+2.\sqrt{a+b}+\sqrt{a+c}+\sqrt{b+c} \geq 5abc+2.
inequalitiesinequalities proposed
Worms that are allowed to move one way

Source: Turkey TST 2014 Day 3 Problem 9

3/12/2014
At the bottom-left corner of a 2014×20142014\times 2014 chessboard, there are some green worms and at the top-left corner of the same chessboard, there are some brown worms. Green worms can move only to right and up, and brown worms can move only to right and down. After a while, the worms make some moves and all of the unit squares of the chessboard become occupied at least once throughout this process. Find the minimum total number of the worms.
geometrycombinatoricsTSTcombinatorial geometryTurkey