MathDB

4

Part of 2002 IMO Shortlist

Problems(3)

IMO ShortList 2002, geometry problem 4

Source: IMO ShortList 2002, geometry problem 4

9/28/2004
Circles S1S_1 and S2S_2 intersect at points PP and QQ. Distinct points A1A_1 and B1B_1 (not at PP or QQ) are selected on S1S_1. The lines A1PA_1P and B1PB_1P meet S2S_2 again at A2A_2 and B2B_2 respectively, and the lines A1B1A_1B_1 and A2B2A_2B_2 meet at CC. Prove that, as A1A_1 and B1B_1 vary, the circumcentres of triangles A1A2CA_1A_2C all lie on one fixed circle.
geometryIMO ShortlistcirclesCircumcenter
IMO ShortList 2002, number theory problem 4

Source: IMO ShortList 2002, number theory problem 4

9/28/2004
Is there a positive integer mm such that the equation 1a+1b+1c+1abc=ma+b+c {1\over a}+{1\over b}+{1\over c}+{1\over abc}={m\over a+b+c} has infinitely many solutions in positive integers a,b,ca,b,c?
algebranumber theoryequationIMO Shortlistinfinitely many solutionsVieta JumpingPell equations
IMO ShortList 2002, combinatorics problem 4

Source: IMO ShortList 2002, combinatorics problem 4

9/28/2004
Let TT be the set of ordered triples (x,y,z)(x,y,z), where x,y,zx,y,z are integers with 0≤x,y,z≤90\leq x,y,z\leq9. Players AA and BB play the following guessing game. Player AA chooses a triple (x,y,z)(x,y,z) in TT, and Player BB has to discover AA's triple in as few moves as possible. A move consists of the following: BB gives AA a triple (a,b,c)(a,b,c) in TT, and AA replies by giving BB the number ∣x+y−a−b∣+∣y+z−b−c∣+∣z+x−c−a∣\left|x+y-a-b\right |+\left|y+z-b-c\right|+\left|z+x-c-a\right|. Find the minimum number of moves that BB needs to be sure of determining AA's triple.
combinatoricsgamegame strategyalgorithmIMO Shortlist