1989年IMO中国国家队选拔考试试题

1989

Day 11A triangle of sides 32,√52,√2is folded along a variable line perpendicular to the side of 32.Find the maximum value of the coincident area.2Let v 0=0,v 1=1and v n +1=8·v n −v n −1,n =1,2,....Prove that in the sequence {v n }there aren’t terms of the form 3α·5βwith α,β∈N .3Find the greatest n such that (z +1)n =z n +1has all its non-zero roots in the unitary circumference,e.g.(α+1)n =αn +1,α=0implies |α|=1.4Given triangle ABC ,squares ABEF,BCGH,CAIJ are constructed externally on side AB,BC,CA ,respectively.Let AH ∩BJ =P 1,BJ ∩CF =Q 1,CF ∩AH =R 1,AG ∩CE =P 2,BI ∩AG =Q 2,CE ∩BI =R 2.Prove that triangle P 1Q 1R 1is congruent to triangle P 2Q 2R 2./This file was downloaded from the AoPS −MathLinks Math Olympiad Resources Page Page 1http://www.mathlinks.ro/

1989

Day 21Let N ={1,2,...}.Does there exists a function f :N →N such that ∀n ∈N ,f 1989(n )=2·n ?2AD is the altitude on side BC of triangle ABC .If BC +AD −AB −AC =0,find the range of ∠BAC .31989equal circles are arbitrarily placed on the table without overlap.What is the least number of colors are needed such that all the circles can be painted with any two tangential circles colored differently.4∀n ∈N ,P (n )denotes the number of the partition of n as the sum of positive integers (disregarding the order of the parts),e.g.since 4=1+1+1+1=1+1+2=1+3=2+2=4,so P (4)=5.”Dispersion”of a partition denotes the number of different parts in that partitation.And denote q (n )is the sum of all the dispersions,e.g.q (4)=1+2+2+1+1=7.n ≥1.Prove that

(1)q (n )=1+n −1 i =1

P (i ).

(2)1+n −1

i =1P (i )≤√·n ·P (n )./This file was downloaded from the AoPS −MathLinks Math Olympiad Resources Page Page 2http://www.mathlinks.ro/