de thi china tst 2013

7/28/2019 De Thi China TST 2013

1/6

THI CHN I TUYN TRUNG QUC THI IMO

NM 2012 2013

Ngy 13 thng 3 nm 2013

Bi 1.

Cho t gic ABCD ni tip trong ng trn c hai ng cho ,AC BD l ct nhau ti F,

cc ng thng ,AB CD ct nhau ti E. Gi hnh chiu vung gc ca im F ln ,AB CD

tng ng l ,G H v cc im ,M Nln lt l trung im ca cc on ,BC EF. ng trn

ngoi tip tam gic MNG ct on BFti P, ng trn ngoi tip tam gic MNHcton CF ti Q . Chng minh rng PQ song song vi BC.

Bi 2.

Vi n l s nguyn dng, ta nh ngha ( ) min 2m

mf n

n

. Gi s ( )in l mt dy cc s

nguyn dng tng ngt v C l mt hng s sao cho2

( ) , 1, 2,3,...iC

f n in

Chng minh rng

tn ti mt s thc 1q 1, 1,2,3,...iin q i

Bi 3.

Cho snguyn dng 3.n nh s 1, 2,3,..., n ln n qubng t mu cho cc qu bng,

mi qubng c t mt trong bn mu , vng, lam v lc theo cch sau: u tin sp xp nqu bng mt cch ty thnh vng trn, vi ba qu bng lin tip bt k theo chiu kim ngh, ta c 3 s , ,i j ktng ng ghi trn chng. Ta xt cc iu kin sau:

o Nu i j k th qu bng j c t mu .o Nu i j k th qu bng j c t mu vng.o Nu ,i j k j th qu bng j c t mu lam.o Nu ,i j k j th qu bng j c t mu lc.

Hai cch t mu n qubng c gi l khc nhau nu c t nht mt qu bng trong hai ccht mu c mu khc nhau. Tm s tt c cc cch t mu khc nhau.


2/6


Bi 4.

Gi ,n k l cc s nguyn ln hn 1, cc s thc khng m 1 2, , ..., na a a ; 1 2, ,..., nc c c tha mn

ng thi cc iu kin:

1) 1 2 ... na a a v 1 2 ... 1na a a .2) Vi 1,2,3,...,m n th 1 2 ... kmc c c m .

Tm gi tr ln nht ca biu thc 1 1 2 2 ...k k k

n nc a c a c a t gi tr ln nht.

Bi 5.

Gi P l mt im nm bn trong ABC. Gi , ,L M N ln lt l trung im ca cc cnh

, ,BC CA AB . Gi s rng : : : :PL PM PN BC CA AB . Ko di cc tia , ,AP BP CPct ng

trn ngoi tip tam gic ABC ln th hai , ,D E F. Chng minh tm ng trn ngoi tip ca

cc tam gic , , , , ,APF APE BPF BPD CPD CPEcng thuc mt ng trn.

Bi 6.

Tm tt c cc s thc dng 1r , sao cho tn ti mt tp hp S cc s thc tha mn tnh chtsau: Vi bt k s thc t no , c ng mt trong cc s , , 1t t r t thuc S v c ng mt

trong cc s 1, ,t t r t thuc S.


3/6


Bi 1.

Vi s nguyn 2k , gi ( , ) | , 0,1,2,..., 1kT x y x y k l mt tp hp gm2k im trong

mt phng ta vung gc. Cc khong cch gia cc im trong kT c sp xp t ln n

nhnh sau 1 2 3( ) ( ) ( ) ...d k d k d k Gi ( )iS k l s cc cp im thuc kT c khong cch

bng ( )id k . Chng minh nu ,m n l cc snguyn dng m m n i th ( ) ( ).i iS m S n

Bi 2.

Chng minh tn ti snguyn dng k v dy snguyn dng tng ngt ( )na sao cho vi bt

k snguyn dng n no, ta u c (1,01)nna K v tng ca hu hn cc s hng bt k ca

( )na u khng phi l schnh phng.

Bi 3.

Gi A l tp hp 6 im no trong mt phng. t ( )n A l scc ng trn n vi qua t

nht 3 im trong A . Tm gi tr ln nht ca ( )n A .


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Bi 4.

Cho s nguyn 1N phn tch c thnh 1 21 2k

kp p p , t 1 2 ... kN .

Vi 1 2, ,..., na a a l cc snguyn dng , xt 1 2( ) nP x x a x a x a . Gi s rng vi

bt k snguyn dng kno th ( )P k l s chn. Chng minh n l s chn.

Bi 5.

Tm snguyn dng m ln nht tha mn cc tnh cht sau: Vi mi hon v 1 2 3, , ,...a a a ca

cc tt c cc s nguyn dng, u tn ti cc s nguyn dng 1 2 ... mi i i sao cho

1 2 3, , ,...,

mi i i ia a a a lp thnh cp s cng vi cng sai l.

Bi 6.

Vi s nguyn dng 1n , xt cc s thc khng m 0 1 2, , ,..., na a a a , t0

ki

k k i

i

S C a

vi

0,1,...,k n v quy c 00 1C . Chng minh rng

2

122

020 0

1 1 4

45

n n

k k n

k k

S S S S n n

.


5/6


Bi 1.

Cho s nguyn 2n v cc s nguyn dng 1 2, ,..., na a a c 1 2gcd( , ,..., ) 1na a a . Ta t

1 2 ... nA a a a v gi gcd( , ), 1,i id A a i n . Vi mi 1,i n , gi iD l c chung ln

nht ca 1n s trong n s cho m khng tnh ia . Tm gi tr nh nht ca1

ni

i i i

A a

d D

.

Bi 2.

Cho tam gic ABC ni tip ng trn tm O c Pl trung im cung BAC, Q l im i

xng vi Pqua O . Gi Il tm ng trn ni tip ABC. Gi sPI ct BC ti D . ngtrn ngoi tip tam gic AID ct ng thng PA ti F. Gi El im trn on PD sao cho

DE DQ . Gi bn knh ng trn ngoi tip, ni tip ABCl ,R r. Chng minh nu AEF APE th 2

2sin

rBAC

R .

Bi 3.

C 101 ngi c nh s l 1,2,3,...,101 c gi s tm thng bng s th t ca h. Nhng

ngi ny ngi xung quanh bn trn mt cch ngu nhin. Mi ln chuyn qua ch khc thngi phi trao mt tm th trong tay mnh cho mt trong hai ngi ngi cnh. Tm snguyn dng k nh nht sao cho c th thng qua khng qu kln i ch sao cho mi ngi

u c mt con stng ng.


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Bi 4.

Cho s nguyn t p , ,a kl snguyn dng tha mn 2a ap k p . Chng minh tn ti s

nguyn dng n tha mn 2an p v modk anC n k p .

Bi 5.

Cho s nguyn 2n v 1 2 1 2, ,..., , , ,...,n na a a b b b l cc s nguyn khng m. Chng minh rng

21 1

2 2 2

11 1

1 1

1

n n n n

ni i i i

ii i

na b a b

n n n

.

Bi 6.

Trn mt phng ta vung gc, xt cc tp hp im nguyn P,Q sao cho mi tp hp im

u to thnh mt a gic li. t T P Q . Chng minh nu T khng rng m cng khng

cha im nguyn no th tp hp T khng suy bin thnh t gic li.

de thi china tst 2013

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