1

LI GII V BNH LUN

CHN I TUYN QUC GIA

D THI IMO NM 2013

Nhng ngi thc hin: Trn Nam Dng

Trn Quang Hng V Quc B Cn

L Phc L

Xin chn thnh cm n thy Nguyn Tng V, thy Trn Quc Lut, cc bn Nguyn Vn Qu, V Anh c, Hong Kin, Phm Tun Huy, Nguyn Huy Tng cng nhiu thnh vin ca cc din n ng gp kin chng ti hon tt ti liu ny!

Thnh ph H Ch Minh, ngy 07 thng 05 nm 2013

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Phn 1.

THI CHNH THC

Ngy thi th nht.

Bi 1.

Cho t gic ABCD c cc cnh i khng song song ni tip ng trn ( , )O R . Gi E l giao im hai ng cho v ng phn gic gc AEB ct cc ng thng

, , ,AB BC CD DA ln lt ti cc im , , ,M N P Q . 1. Chng minh rng cc ng trn ( ),( ),( ),( )AQM BMN CNP DPQ cng i qua mt im duy nht. Gi im l K .

2. t min ,AC BD m . Chng minh rng 2

2 2

2 .4

ROKR m

Bi 2.

1. Chng minh rng tn ti v s s nguyn dng t sao cho 2012 1t v 2013 1t u l cc s chnh phng.

2. Xt ,m n l cc s nguyn dng sao cho 1mn v ( 1) 1m n u l cc s chnh phng. Chng minh rng n chia ht cho 8(2 1)m .

Bi 3.

Vi mi s n nguyn dng, t 0,1,2,...,2 1nS n . Xt hm s : ( ) [0;1]nf S tha mn ng thi cc iu kin sau:

i/ ( ,0) ( ,2 1) 0f x f x n vi mi s nguyn .x

ii/ ( 1, ) ( 1, ) ( , 1) ( , 1) 1f x y f x y f x y f x y vi ,x y v 1 2y n .

Gi F l tp hp tt c cc hm s f tha mn.

1. Chng minh rng F l v hn.

2. Vi mi hm s f F , t fv l tp hp nh ca f . Chng minh rng fv hu hn.

3. Tm gi tr ln nht ca fv vi f F .

3

Ngy thi th hai.

Bi 4.

Tm hng s k nguyn dng ln nht tha mn: Vi mi , ,a b c dng m 1abc th ta c bt ng thc sau

1 1 1 31 4

k ka b c a b c

.

Bi 5.

Cho tam gic ABC nhn khng cn c A bng 45 . Cc ng cao , ,AD BE CF ng quy ti trc tm H . ng thng EF ct ng thng BC ti P . Gi I l trung im ca BC ; ng thng IF ct PH ti .Q

1. Chng minh rng IQH AIE .

2. Gi K l trc tm ca tam gic AEF v ( )J l ng trn ngoi tip tam gic KPD . ng thng CK ct ( )J ti G , ng thng IG ct ( )J ti M , ng thng JC ct ng trn ng knh BC ti N . Chng minh rng cc im , , ,G M N C cng thuc mt ng trn.

Bi 6.

Cho mt khi lp phng 10 10 10 gm 1000 vung n v mu trng. An v Bnh chi mt tr chi. An chn mt s di 1 1 10 sao cho hai di bt k khng c chung nh hoc cnh ri i tt c cc sang mu en. Bnh th c chn mt s bt k ca hnh lp phng ri hi An cc ny c mu g. Hi Bnh phi chn t nht bao nhiu vi mi cu tr li ca An th Bnh lun xc nh c nhng no l mu en?

4

Phn 2.

LI GII CHI TIT V BNH LUN

Bi 1.

Cho t gic ABCD li c cc cnh i khng song song ni tip ng trn ( , )O R .

Gi E l giao im hai ng cho v ng phn gic gc AEB ct cc ng thng , , ,AB BC CD DA ln lt ti cc im , , ,M N P Q .

1. Chng minh rng cc ng trn ( ),( ),( ),( )AQM BMN CNP DPQ cng i qua mt im duy nht. Gi im l K .

2. t min ,AC BD m . Chng minh rng 2

2 2

2 .4

ROKR m

Li gii.

R

S

K

P

M

Q

N

E

O

AB

CD

1. Gi R l giao im ca ,AD BC v S l giao im ca ,AB CD (do cc cnh i ca t gic ABCD khng song song nn cc im ny hon ton xc nh). Gi s B nm

5

gia ,A S v nm gia ,C R nh hnh v. Gi K l giao im ca ng trn ngoi tip cc tam gic ,RAB SBC th

180BKR BKS BAD BCD hay , ,R K S thng hng.

Suy ra RK RS RB RC RA RD v SK SR SB SA SC SD nn cc t gic ADSK v CDRK cng ni tip hay K cng thuc v cc ng trn ( )RCD v ( ).SDA

Do , ta c

AKD ASD BSC BKC v ADK ASK BSK BCK

nn cc tam gic KAD v KBC ng dng. Suy ra KA AD AE AMKB BC BE BM

, theo tnh

cht ng phn gic th KM l phn gic ca gc .AKB

Mt khc, ta c RNQ BNE CBD BEN CAD AEQ RQN nn ta c

2ARB BNM .

T suy ra 1 12 2

BKM AKB ARB BNM hay t gic BMNK ni tip, tc l

K thuc ng trn ( )BMN .

Chng minh tng t, ta cng c K thuc cc ng trn ( ),( ),( )AQM CNP DPQ .

Tip theo, ta s chng minh rng K l im chung duy nht ca cc ng trn ny. Tht vy, cc ng trn ( ),( )AMP BMQ c hai im chung l ,K M cn cc ng trn ( ),( )DNP CNP c hai im chung l ,K N . Do , nu bn ng trn ny c hai im chung th ,M N trng nhau, v l.

Ta c pcm.

2. Theo tnh cht phng tch th

2 2 2 2,RK RS RB RC RO R SK SR SB SA SO R nn

2 2 2 2RO SO RK RS SK SR RK SK .

T suy ra .OK RS

Hn na, theo nh l Brocard trong t gic ni tip ABCD th E chnh l trc tm ca tam gic ORS , suy ra .OE RS

Do , , ,O E K thng hng.

6

Ta li c 2RKA SKC RBA SBC ADC AOC nn 180AKC AOC hay t gic AOCK ni tip. Suy ra

2 2 2( )EO EK EA EC R OE EO EO EK R hay 2ROK

EO .

Mt khc, theo bt ng thc c bn v ng xin v ng vung gc th

2 2 2 2 2 21 1 1max ( , ), ( , ) max 4 , 4 42 2 2

EO d O AC d O BC R AC R BD R m

.

Vy ta c 2

2 2

24

ROKR m

. y chnh l pcm.

Nhn xt.

Li gii ca bi ton da trn hai nh l rt c bn ca hnh hc l im Miquel ca t gic ton phn v nh l Brocard.

Ni n nh l Miquel th nhiu ngi quen thuc n dng pht biu sau

nh l Miquel. Cho tam gic ABC v cc im , ,D E F ln lt thuc cc ng thng , ,BC CA AB . Khi cc ng trn ngoi tip cc tam gic ( ),( ),( )AEF BFD CDE c mt

im chung M gi l im Miquel.

H qu 1. Cc im , ,D E F thng hng khi v ch khi im Miquel M thuc ng trn ( )ABC .

H qu trn rt c ngha nu ta coi mt tam gic v mt ng thng l mt t gic ton phn khi ta c th pht biu li h qu nh sau

H qu 2. Cho t gic ABCD c AB giao CD ti E , AD giao BC ti F . Khi cc ng trn ( ),( ),( ),( )EAD EBC FAB FCD c mt im chung M. im M gi l im Miquel ca t gic ABCD .

im Miquel M c bit c rt nhiu tnh cht th v khi t gic ABCD ni tip. Mt trong nhng tnh cht quan trng l nh sau

H qu 3. Cho t gic ABCD ni tip ng trn ( )O . Gi s AB giao CD ti E , AD giao BC ti F , AC giao BD ti G . Khi cc ng trn ( ),( ),( ),( )EAD EBC FAB FCD c mt im chung M v , ,O G M thng hng.

nh l Brocard cng l mt nh l kh quen thuc.

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nh l Brocard. Cho t gic ABCD ni tip ng trn ( )O . Gi s AB giao CD ti E , AD giao BC ti F , AC giao BD ti G . Khi O l trc tm tam gic EFG .

Mt trong nhng bi ton ng dng hay ca nh l ny chnh l bi thi quc gia Vit Nam nm 2012 va qua.

Chng minh ca nh l Miquel v cc h qu c th coi l mt trong nhng th hin quan trng nht ca vic phi s dng gc nh hng trong hnh hc. bi ny, nu khng dng di i s cng nh gc nh hng, ta phi quy c v tr cc im

nh trnh by li gii c cht ch.

Bi thi l mt trong nhng tng hay cho vic kt hp nh l Miquel v nh l Brocard c bit l cu 2). tng ca cu ny c l bt ngun t ng thc

2OK OE OR OS R

; tuy nhin, giu i s hin din im E th bi thi a v mt bt ng thc lin h gia ng xin v hnh chiu, l

2 2

2min ,

max ( , ), ( , )4

AC BDOE d O AC d O BD R .

Thc s tng mun giu i im E kh hay xong vic phi dng n mt bt ng thc hnh hc khin cho bi ton mt i kh nhiu v p ca n.

Bi ton ny cng c th gii bng cch s dng b sau lin quan n php bin hnh nh sau:

Cho hai on thng ,AB CD sao cho ABCD khng phi l hnh thang. Khi , tn ti mt php v t quay tm O bin AB thnh .CD Nu P l giao im ca AB v ,CD Q l giao im ca AD v BC th cc t gic , , ,ADPK BCPK ABQK CDQK ni tip.

Trong bi ton cho, im K cng chnh l tm ca php v t quay .

tng s dng bt ng thc y kh ging vi bi hnh s 5 trong thi TST 2006, cng l s so snh gia ng xin v ng vung gc. Tuy nhin, y ta khng cn ch ra vi trng hp no th ng thc xy ra.

Nu tng qut ln, thay ng phn gic gc E thnh ng thng bt k qua E , ta c bi ton sau:

Cho t gic ABCD li c cc cnh i khng song song ni tip ng trn ( , )O R .

Gi E l giao im hai ng cho v mt ng thng bt k i qua E ct cc ng

8

thng , , ,AB BC CD DA ln lt ti cc im , , ,M N P Q . Chng minh rng giao im khc , , ,M N P Q ca cc cp ng trn ( ),( )AQM BMN ; ( ),( )BMN CNP ; ( ),( )CNP DPQ v ( ),( )DPQ AQM cng thuc mt ng trn qua im Miquel K ca t gic .ABCD

K

N

P

M

Q

R

S

E

O

AB

CD

Trong trng hp ng thng bt k trn tr thnh phn gic th bn giao im trn trng nhau v trng vi .K

Bi 2.

1. Chng minh rng tn ti v s s nguyn dng t sao cho 2012 1t v 2013 1t u l cc s chnh phng.

2. Xt ,m n l cc s nguyn dng sao cho 1mn v ( 1) 1m n u l cc s chnh phng. Chng minh rng n chia ht cho 8(2 1)m .

Li gii.

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1. t (2012 1,2013 1)d t t th d thy 1d . Do , 2012 1t v 2013 1t u l cc

s chnh phng khi v ch khi 2(2012 1)(2013 1)t t y vi y l s nguyn dng no

. Ta bin i ng thc trn

2

2 2 2 2

2 2

(2012 1)(2013 1)4 2012 2013 4 2012 2013 4025 4 2012 2013 4 2012 2013(2 2012 2013 4025) 1 4 2012 2013

t t yt t y

t y

t 2 2012 2013 4025x t th ta c phng trnh 2 24 2012 2013 1x y .

D thy 4 2012 2013 khng phi l s chnh phng nn phng trnh Pell loi 1 ny c v s nghim. Nghim nh nht ca phng trnh ny l ( , ) (4025,1)x y nn cc

nghim ca n c cho bi cng thc

0 1 2 1

0 1 2 1

1, 4025, 8050, 0

1, 1, 8050n n n

n n n

x x x x xn

y y y y y

.

Bng quy np, ta chng minh c 2 1ix chia 2 2012 2013 d 4025 vi mi i v mi

gi tr nguyn dng 2 14025

2 2012 2013ix

s cho ta mt gi tr t tha mn bi.

Vy tn ti v s gi tr nguyn dng t sao cho 2012 1t v 2013 1t u l cc s chnh phng. Ta c pcm.

2. t ( 1, 1)d mn mn n th

1 1d mn n mn hay d n , suy ra ( 1 )d mn mn hay 1d .

Do 1d hay cc s 1,( 1) 1mn m n nguyn t cng nhau.

Khi , 1mn v ( 1) 1m n u l cc s chnh phng khi v ch khi

( 1) ( 1) 1mn m n l s chnh phng.

Gi s 2( 1) ( 1) 1mn m n y vi y . Bin i biu thc ny, ta thu c

2 2

2 2 2 2

2 2

( 1) (2 1) 14 ( 1) 4 ( 1)(2 1) 4 ( 1) 4 ( 1)

2 ( 1) (2 1) 1 4 ( 1)

m m n m n ym m n m m m n m m m m y

m m n m m m y

10

t 2 ( 1) (2 1)x m m n m th ta c phng trnh sau

2 24 ( 1) 1x m m y (*)

y chnh l phng trnh Pell loi 1 v do 4 ( 1)m m khng l s chnh phng vi mi m nguyn dng nn (*) c v s nghim.

Phng trnh (*) c nghim nh nht l ( , ) (2 1,1)x y m nn cng thc nghim ( , )i ix y

ca n c th c vit di dng

0 1 2 1

0 1 2 1

1, 2 1, 2(2 1), 0

0, 1, 2(2 1)i i i

i i i

x x m x m x xi

y y y m y y

.

Bng quy np, ta s chng minh rng 2 ix chia 2 ( 1)m m d 1 v 2 1ix chia 2 ( 1)m m d

2 1m vi mi 0,1,2,...i (**)

Tht vy,

- Vi 0,i theo cng thc truy hi ca dy ( )ix th ta thy khng nh (**) ng.

- Gi s (**) ng n i , tc l 2 ix chia 2 ( 1)m m d 1 v 2 1ix chia 2 ( 1)m m d 2 1m .

Ta c

2 2 2 1 22(2 1) 2(2 1)(2 1) 1 8 ( 1) 1 1 (mod 2 ( 1))i i ix m x x m m m m m m v

2 3 2 2 2 12(2 1) 2(2 1) (2 1) 2 1 (mod 2 ( 1))i i ix m x x m m m m m .

Do , (**) cng ng vi 1i .

Theo nguyn l quy np, (**) ng vi mi s t nhin .i

Tip theo, ta s xy dng cng thc truy hi cho dy 2 1i ir x vi 0,1,2,...i

Ta c

2 2 5 2 4 2 3 2 3 2 2 2 3

2 22 3 2 2 2 3 2 3 2 1

2 22 3 2 1 1

2(2 1) 2(2 1) 2(2 1)

4(2 1) 1 2(2 1) 4(2 1) 1 ( )

4(2 1) 2 4(2 1) 2

i i i i i i i

i i i i i

i i i i

r x m x x m m x x x

m x m x m x x x

m x x m r r

t 2 ( 1) (2 1)i ir m m s m th dy ( ), 0is i nguyn dng v xc nh duy nht.

Thay vo cng thc truy hi ca ( ),ir ta c

11

22 1

2 22 1

22 1

2 ( 1) (2 1) 4(2 1) 2 2 ( 1) (2 1) 2 ( 1) (2 1)

2 ( 1) 2 ( 1) 4(2 1) 2 2 ( 1) 4(2 1) (2 1) 1

4(2 1) 2 8(2 1)

i i i

i i i

i i i

m m s m m m m s m m m s m

m m s m m m s m m s m m

s m s s m

Ta tnh c 0 1 2 1r x m nn 0 0s v

21 3 2(2 1) 2(2 1) 1 (2 1) 16 ( 1)(2 1) 2 1r x m m m m m m m nn 1 8(2 1)s m .

Ta c cng thc truy hi ca dy s ( )is l 0 1

22 1

0, 8(2 1),

4(2 1) 2 8(2 1), 0i i i

s s m

s m s s m i

T suy ra dy s ( )is c cc s hng chia ht cho 8(2 1)m vi mi 0,1,2,...i

Hn na, vi cch t 2 ( 1) (2 1)x m m n m th d dng thy rng n tha mn bi

khi v ch khi , 1,2,3,...in s i (do 0 0s khng phi l s nguyn dng).

Vy tt c cc gi tr n u chia ht cho 8(2 1).m y chnh l pcm.

Nhn xt.

th 1 ca bi ton thc ra l mt trng hp c bit v cng l mt s dn dt cho th 2. Ta c th gii 1 theo cch t tnh ton hn bng nhn xt:

Nu , 1x y l nghim ca phng trnh 2 22013 2012 1x y th 2 2 *t y x v cc s 2012 1,2013 1t t u chnh phng.

Hn na, mi nghim ( , )x y ca phng trnh 2 22012 2013 1x y cho ta mt nghim

( 2012 , 2013 )x y x y ca phng trnh 2 22013 2012 1x y nn d thy pcm.

Cch tip cn bng phng trnh Pell hon ton t nhin v khi xy dng thnh cng th ta ch cn cn thao tc trn cc dy s nguyn. Trong nhng nm gn y, cc bi ton v dy s nguyn kh c a chung, c bit dy s nguyn lin quan n cng thc nghim ca phng trnh nghim nguyn. Chng hn nh

(Vit Nam TST 2012) Cho dy s nguyn dng ( )nx c xc nh bi

1 2

2 1

1, 2011,4022 , 1,2,3,...n n n

x xx x x n

12

Chng minh rng 20121

2012x

l s chnh phng.

Trong vic gii quyt th 2, nu ta lp lun nh sau:

Do 1mn v ( 1) 1m n u l cc s chnh phng nn ta c th t

2 21 ,( 1) 1mn a m n b .

T ta c phng trnh 2 2( 1) 1m a mb . i bin , ( 1)a x my b x m y a

v phng trnh Pell lin kt l 2 2( 1) 1x m m y .

Cch gii ny cng hon ton ph hp nhng ch rng phng trnh Pell tng qut dng 2 2Ax By n ni chung c nhiu hn mt nghim c s. Ta cn chng minh

trong trng hp c bit ny ( 1A B v 1n ) th n ch c duy nht mt nghim c s. Nu khng th li gii cng thc s vn cn thiu st.

Ngoi cch s dng phng trnh Pell nh trn, ta c th lp lun nh sau:

Do ( 1, 1) 1mm mn n nn ta cn c ( 1)( 1)mn mn n l s chnh phng hay tn

ti a sao cho 2 24 ( 1) 4(2 1) 4m m n m n a , a v

2 2 2(2 1) 2m n n a

Ta chng minh c rng n phi l s chn v y l phng trnh Pythagore nn phi tn ti cc s , ,p q k sao cho

2 2

2 ,(2 1) 2 ( )n kpq

m n k p q

trong ( ,(2 1) 2) 2k n m n nn 1k hoc 2.k

Nu 1k th d thy 2 2p q chn, suy ra ,p q cng tnh chn l. Nu ,p q cng l th n

chia 4 d 2 v (2 1) 2m n chia ht cho 4 trong khi 2 2p q chia 4 d 2, mu thun. Nu

,p q cng chn th n chia ht cho 4 dn n (2 1) 2m n chia 4 d 2 trong khi 2 2p q

chia ht cho 4, cng mu thun.

Nu 2k th 4n pq v 2 2(2 1) 2 2( )m n p q , suy ra

2 2 2(2 1) 1 0p q m pq .

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n y ta chng minh c (s dng phng php Vite Jumping) ,p q l hai s hng

lin tip ca dy s cho bi cng thc

1 2

2 1

1, 2(2 1),2(2 1) , 1i i i

x x mx m x x i

.

T suy ra pq chia ht cho 2(2 1)m hay n chia ht cho 8(2 1)m .

Phng php xy dng nghim y c nt ging vi bi 6 VMO 2012.

Xt cc s t nhin l ,a b m a l c s ca 2 2b v b l c s ca 2 2a . Chng minh rng ,a b l cc s hng ca dy s t nhin ( )nv xc nh bi

1 2 1v v v 1 24n n nv v v vi mi 2n .

Cch gii ny c phn ngn gn hn v x l nh nhng hn cch nu li gii ban u nhng trnh by cht ch khng phi l iu n gin.

Bi 3.

Vi mi s n nguyn dng, t 0,1,2,...,2 1nS n . Xt hm s : ( ) [0;1]nf S tha mn ng thi cc iu kin sau:

i/ ( ,0) ( ,2 1) 0f x f x n vi mi s nguyn .x

ii/ ( 1, ) ( 1, ) ( , 1) ( , 1) 1f x y f x y f x y f x y vi ,x y v 1 2y n .

Gi F l tp hp tt c cc hm s f tha mn.

1. Chng minh rng F l v hn.

2. Vi mi hm s f F , t fv l tp hp nh ca f . Chng minh rng fv hu hn.

3. Tm gi tr ln nht ca fv vi f F .

Li gii.

1. Trong ng thc ii/ cho, ta thy rng

( 1) ( 1) ( 1) ( 1) (mod 2)x y x y x y x y

14

iu ny c ngha l cc gi tr ca hm ( , )f x y vi ,x y cng tnh chn l v ,x y khc

tnh chn l l khng c lin h vi nhau.

Ta s tm cc xc nh hm f trong c hai trng hp.

Trong mt phng ta ,Oxy ta xt mt li nguyn nm ngang c chiu cao l 2 1n ,

chiu rng v hn v im c ta ( , )i j s c gn gi tr ( , )f i j .

iu kin i/ c hiu l tt c cc s thuc hai bin (trn v di) ca li u gn s 0; cn cc im nguyn bn trong li u c gn gi tr thuc [0;1] .

iu kin ii/ chnh l vi mi hnh vung con nm nghing (cc ng cho song song

vi cc trc ta ) c cc nh nguyn v cnh l 2 u c tng cc s gn cho cc nh bng 1.

Vi mi im c ta A nguyn thuc li nguyn ang xt, ta t 1 2( ), ( )f A f A l gi

tr gn cho cc im c cng tung vi A , ln lt c honh ln hn v nh hn honh ca A ng 2 n v.

Tip tc t ( , )ka f k k vi 1,2,3,...,2k n l gi tr gn cho im ( , )kA k k . Theo iu

kin xc nh hm s th

1 2 1 1 1 1 1 2

2 3 1 2 1 1 1 2 1 3

2 1 2 1 2 2 1 2 1 1 2 1 1 2

2 1 2 1 1 2 1 2 1

( ) 0 1 ( ) 1 ,( ) ( ) 1 ( ) ,

...( ) ( ) 1 ( ) ,

0 ( ) ( ) 1 ( )n n n n n n

n n n n

a a f A f A a aa a f A f A f A a a

a a f A f A f A a aa f A f A f A a

Tng t, ta cng c

2 2 2 1 2

2 2 1 2 2 2

2 2 1 1 2

2 2 2

( ) 1( )

...( ) 1 ,( )

n n n

n n n

n n

n n

f A a af A a a

f A a af A a

T y ta thy rng nu dy ka xc nh th cc gi tr 1 2( ), ( )k kf A f A cng hon ton

xc nh. Ta s chn cc gi tr ka sao cho 1 2( ), ( )k kf A f A u thuc [0;1] .

Ta chn 1 3 5 2 1 2 4 6 2 1 2... , ... , 1n n na a a a a a a a a a th 1 2( ), ( ) [0;1]k kf A f A .

15

Hn na, ta cng thy rng

1 1 1 3 1 5 1 2 1 1 2 1 4 1 6 1 2 1 1 1 2

2 1 2 3 2 5 2 2 1 2 2 2 4 2 6 2 2 2 1 2 2

( ) ( ) ( ) ... ( ), ( ) ( ) ( ) ... ( ), ( ) ( ) 1,( ) ( ) ( ) ... ( ), ( ) ( ) ( ) ... ( ), ( ) ( ) 1

n n n

n n n

f A f A f A f A f A f A f A f A f A f Af A f A f A f A f A f A f A f A f A f A

Do , tnh n iu ca hai dy con c ch s chn v l vn c bo ton.

Tng t, t dy 1( )kf A , ta xc nh c 1 1( ( ))kf f A tha mn, t dy 2( ),kf A ta cng

xc nh c 2 2( ( ))kf f A v c th, tc l xy dng c tt c gi tr cho hm s

( , )f x y trong trng hp x y chn.

Tip tc t ( 1, )kb f k k vi 1,2,3,...,2k n l gi tr gn cho im ( 1, )kB k k th

hon ton tng t nh trn, ta xy dng c tt c cc gi tr ca ( , )f x y vi x y l.

D thy c v s cch chn dy ( ),( )k ka b tha mn cc iu kin trn nn c v s hm

f tha mn bi, tc l F v hn. Ta c pcm.

2. Trong ng thc ( 1, ) ( 1, ) ( , 1) ( , 1) 1f x y f x y f x y f x y , ta thay ,x y ln lt

bi 1, 1x y , ta c ( , 1) ( 2, 1) ( 1, ) ( 1, 2) 1f x y f x y f x y f x y .

T suy ra ( 1, ) ( , 1) ( 1, 2) ( 2, 1)f x y f x y f x y f x y (*) hay

(1,1) (2,0) (3,3) (4,2) ... (2 1,2 1) (2 2,2 )f f f f f n n f n n ,

(3,1) (4,0) (5,3) (6,2) ... (2 3,2 1) (2 4,2 )f f f f f n n f n n .

Do (1,1) (2 2,2 ), (3,1) (2 4,2 )f f n n f f n n .

16

Tng t, ta c c (2,2) (2 2,2 1), (4,2) (2 5,2 1)f f n n f f n n .

Tip tc p dng nhiu ln ng thc (*), suy ra

( , ) (2 1 ,2 1 ), ( 2, ) (2 3 ,2 1 )f k k f n k n k f k k f n k n k .

Chng minh tng t, ta c

(2 ,2 1 ) (4 2 , ), (2 3 ,2 1 ) (4 4 , )f n k n k f n k k f n k n k f n k k .

Do , ( , ) (2 1 ,2 1 ) (4 2 , )f k k f n k n k f n k k vi mi 1,2,3,...,2k n v nh th,

bng quy np, ta c c

( , ) ((2 1) ,2 1 ) ((4 2) , )f k k f n i k n k f n i k k vi i v 1,2,3,...,2k n .

Hn na, s xc nh cc gi tr trong ng cho tip theo (nm v pha phi) hon ton ging nhau di cc im ( , )k k v ((2 1) ,2 1 )n i k n k nn cc gi tr trn cng tng ng bng nhau. Suy ra gi tr ca f c s tun hon v c minh ha nh

hnh bn di, tc l cc gi tr ca ( , )f x y vi x y chn l s lp li cc gi tr c

gn cho cc im nguyn trong tam gic c ta cc nh (1,1),(2 1,2 ),(4 1,1)n n n .

1

2n+12n

O 4n+24n+1 4n+31 2n+1 6n+32n+2 8n+3

Tuy nhin, cc gi tr ny l hu hn v ta tnh c tng cng c khng qu

1 2 3 ... 2 (2 1)n n n gi tr.

Tng t vi gi tr ca ( , )f x y m x y l, ta cng c thm khng qu (2 1)n n gi tr

na. Kt hp vi 0, ta c 2 (2 1) 1fv n n .

Do , fv hu hn vi mi f F , ta c pcm.

17

3. Ta s xy dng mt hm s f c 2 (2 1) 1fv n n v ch ra y chnh l gi tr ln

nht ca fv .

Bng quy np, ta chng minh c rng

1 ( 1) 1 ( 1)( 2 , ) ( 1) ( 1)

4

i kk k i

k i kf i k i a a

vi mi 1,2,...,2i n v k m 0 2 1k i n . (**)

Tht vy, vi 0k th (**) hin nhin ng.

Gi s (**) ng vi mi ( , )i k m k m v 1,k m i j . T iu kin ii/, ta c

( 1 2( 1), 1) ( 1 2 , 1) ( 2( 1), ) ( 2 2 , 2) 1f j m j f j m j f j m j f j m j .

S dng gi thit quy np, ta tnh c

1 1 212 1

1 ( 1) 1 ( 1)( 1 2( 1), 1) ( 1) ( 1)

4

j mm jm

m j mf j m j a a

.

Suy ra (**) cng ng vi 1, 1i j k m .

Nh th, cc gi tr c gn cho cc im trong tam gic u c dng ij i ja a vi

0,1ij , trong gi tr ij cng nh du ca ,i ja a xc nh duy nht theo ,i j .

Tip theo, ta chn 2 1 22 1 2( 1 )1 1,

3 3k kk n ka a vi 1,2,...,k n th do trong h c s 3, mt

s nguyn bt k c duy nht mt cch biu din di dng tng 0

3r

ii

i

vi 0r v

1,0,1ie ) nn suy ra tt c cc gi tr ca ( , )f x y dng nh s cho cc im nm trong tam gic l i mt phn bit v khc 0, tc l c ng (2 1)n n gi tr nh th, tc l ta xy dng c gi tr cho hm s ( , )f x y vi x y chn.

Tng t, xy dng cho ( , )f x y vi x y l, ta chn dy s

2 1 22 1 2( 1 )

1 1,3 3 3 3k kk n k

b b vi 1,2,...,2k n .

th cc gi tr dng ij i jb b cng i mt phn bit v khc vi cc gi tr ij i ja a .

18

T , ta xy dng c ton b gi tr cho hm s ( , )f x y v c tt c 2 (2 1) 1n n gi

tr i mt khc nhau.

Vy gi tr ln nht cn tm ca fv trn min F l 2 (2 1) 1n n .

Nhn xt.

y l mt bi ton i hi nhng kin thc tng hp v mt phong cch lm vic t tn, bi bn. M hnh li hoc bng l mt m hnh t nhin m ta ngh n, v hm s c th c xc nh theo hng, ct hoc ng cho.

Trong cc bi ton c cha tham s n , ta nn bt u t nhng gi tr n nh hnh dung bi ton mt cch tt nht, c th nht, t tm cch tip cn tng qut. Nu chn cch ny th c l cu 1) v 2), thm ch c phn chn trn ng cu 3) khng phi l qu kh. im kh cu 1) l cch xy dng hm phi tha mn iu kin

( , ) [0;1]f x y . c c iu ny, ta phi tm c mt iu kin c tnh bt bin i

vi dy gi tr trn ng cho (khng ch dy k n thuc [0;1] , m cn cc dy tip theo cng nh th).

lm cu 3), ta phi c nhng nhn xt tinh t hn, i hi nhiu thi gian hn. C l cu ny ch dnh cho nhng bn gii quyt tt hai bi 1, 2 trong vng 1,5 - 2 gi, cn kh nhiu thi gian tp trung ton lc cho bi ny. tng c bn y l d on v chng minh c cng thc tng qut ca hm da theo cc phn t thuc mt ng cho. T cng thc tng qut ny ta mi tm cch chn cc gi tr khi to tt c cc gi tr nm trong tam gic tun hon i mt khc nhau.

chn cc gi tr khc nhau, ta c mt s nh hng c bn sau:

+ Dng bt ng thc

+ Dng tnh c lp tuyn tnh ca cc s v t, c th l: Nu 1 2, ,..., np p p l cc s

nguyn t phn bit th khng tn ti cc s hu t 1 2, ,..., nc c c khng ng thi bng 0

sao cho 1 1 2 2 ... 0n nc p c p c p .

Tuy nhin, vic chng minh b ny l khng n gin. Ta cng c th thay cc s

ip ny bng cc s v t khc da vo tnh khng m c ca , nhng y cng l kin thc nm ngoi khun kh chng trnh ph thng.

19

Mt cch tip cn th v v s cp cho vn ny c trnh by trn l s dng tnh cht ca h m c s 3: Mi s nguyn 0N bt k u c th biu din duy nht di dng

11 03 3 ...

i ii iN

trong 1, 1; 0;1 , 0,1,.., 1.i j j i

Mnh ny c th chng minh kh d dng bng quy np ton hc.

Vic ( , ) [0;1]f x y khng thnh vn v ta c th chia cc gi tr khi to cho mt

hng s ln t c iu ny.

Ngoi cch xy dng theo ng cho vi 4n gi tr khi to nh trn ( 2n cho trng hp ( , )f x y m x y chn v 2n cho trng hp ( , )f x y m x y l), ta cng c th xy

dng theo hai cnh dc hoc theo hai cnh ngang c di 2n .

Chng hn, trong trng hp xy dng theo cnh dc, song song vi trc Oy , ta c th

lm nh sau:

Bng quy np, ta chng minh c rng

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

( ,2 1) ( 2 ,1) ( 2 2,1) ... ( ,1) ... ( 2 2,1) ( 2 ,1)

f x i i f x i f x i f x i

f x i f x i f x i f x f x i f x i i

Ta cn chn ( ,1), [ 2 1; 2 ]f x x n n sao cho:

i/ 1 ( 2 1,1) ( 2 3,1) ... ( 2 1,1)i f x i f x i f x i i .

ii/ ( 2 ,1) ( 2 2,1) ... ( ,1) ... ( 2 2,1) ( 2 ,1) 1i f x i f x i f x f x i f x i i .

iii/ Cc gi tr ( ,1)f x l phn bit vi [ 2 1; 2 ]x n n .

iv/ ( ,2 1) ( 2 ,1) ( 2 2,1) ... ( ,1) ... ( 2 2,1) ( 2 ,1)f x i f x i f x i f x f x i f x i i

vi i n th lun bng .n

T a ln ng trn v kt hp vi bt ng thc chn, tc l xt mt s gi tr nh thch hp cho chng i mt phn bit. Cch xy dng cho trng hp cnh ngang cng thc hin tng t.

Bi ny nhc ta nh n mt phng trnh hm tng t xut hin trong k thi chn i tuyn cch y 10 nm:

20

(Vit Nam TST 2003) Cho hm s :f tha mn ng thi cc iu kin sau: i) 2003(0,0) 5 , (0, ) 0f f n vi mi n l s nguyn khc 0.

ii) ( 1, ) ( 1, 1) ( 1, 1)( , ) ( 1, ) 22 2 2

f m n f m n f m nf m n f m n

vi mi s nguyn dng m v mi s nguyn n. Chng minh rng tn ti s nguyn dng M sao cho ( , ) 1f M n vi mi s nguyn n

tha 20035 1

2n v ( , ) 0f M n vi mi s nguyn n tha

20035 12

n .

Tuy nhin bi ton ny li l mt cu chuyn khc !

Bi 4.

Tm hng s k nguyn dng ln nht tha mn: Vi mi , ,a b c dng m 1abc th ta c bt ng thc sau

1 1 1 31 4

k ka b c a b c

(*)

Li gii.

Gi s k l s nguyn dng sao cho bt ng thc cho ng vi mi , ,a b c m

1.abc Thay 2 9,3 4

b c a vo (*), ta c

3 4 8802 3 142 2 92 9 4 6313 3 4

k k k

.

Hn na, v k l s nguyn dng nn t nh gi trn, ta c 13.k Ta s chng minh rng vi 13k th bt ng thc (*) ng.

Tht vy, vi 13k ta c bt ng thc 1 1 1 13 25 .1 4a b c a b c

(**)

t 1 1 1 1( , , )1

f a b ca b c a b c

. Khng mt tnh tng qut, ta c th gi s

max , ,a a b c , khi

21

2

1 1 2 1 1( , , ) , , 131 2 1

1 13 .( 1) 2 1

f a b c f a bc bcb c a b cbc a bc

b cbc a b c a bc

Do max , ,a a b c v gi thit 1abc nn ta c 1,bc suy ra 1 1bc

.

Mt khc, s dng bt ng thc AM-GM v bin i cho biu thc trong ngoc ng thc trn, ta c

3 313 13 13 1,

16( 1) 2 1 3 1 3 1a b c a bc abc abc

nn hin nhin ( , , ) , ,f a b c f a bc bc . Ta a c bi ton v chng minh

2

1 25, ,4

f x xx

vi , 0 1.x bc x

Nu 1x th bt ng thc trn tr thnh ng thc. Trong trng hp 0 1,x bng cch s dng li bin i thc hin trong qu trnh tm iu kin cn cho ,k ta thy bt ng thc tng ng vi

3 23 2

4 3 2 2 4 3 2

( 2)(2 1) 13 4( 2)(2 1) 13 (2 1)(2 1) 4

4(2 5 2 2) 26 13 8 20 18 9 8 0.

x x x x x x x xx x

x x x x x x x x x x

Ta c 4 3 2 4 2 3 2 2

2 2 2 2

8 20 18 9 8 (8 8 2) (20 20 5 ) (10 14 6)2(2 1) 5 (2 1) 2(5 7 3) 0

x x x x x x x x x x xx x x x x

do 2 2 22(2 1) 0, 5 (2 1) 0x x x v 25 7 3 0x x (tam thc bc hai c h s cao nht dng v bit thc 11 0 ). Nh vy, bt ng thc cui hin nhin ng. Ta i n kt lun cui cng 13k l gi tr cn tm.

Nhn xt.

Bi ton ny c dng pht biu kh ging vi cc bi ton tm hng s tt thng thng nn rt t nhin, ta ngh n li tip cn tng t nh cc dng ton ny: Tm mt b s thch hp thay vo d on iu kin cn cho k ri sau i chng minh cng l iu kin .

22

Do bt ng thc c dng i xng nn ta ngh n vic chn mt b s m c hai

bin nhn gi tr bng nhau. iu ny l gii cho vic chn a b x v 21cx

nh

trong li gii trn. Mt cch khc chng minh bt ng thc (**) dng phng php dn bin nh sau:

Bin i (**) v dng a thc 25( )( 1) 13 ( )4

ab bc ca a b c a b c .

Gi s max , ,a a b c th 1a v t x b c th bt ng thc trn c th vit li

thnh 25( )( 1) 13 ( )4

ax bc a x a x . Bt ng thc ny tng ng vi

2 2

2222

25 25( ) 14 04 4

25254 254 14 0

2 2 4

aax bc a a x

bc a abc a aa xa a

Do 22x b c bca

nn

2 222 25 1 2525 2 44 44 0

2 2 2

a bc a a a abc aa ax

a a a

Ta a v chng minh

2222

1 251 254 4 254 14 02 2 4

a aa a a a aaaa a

.

t 0a t th ta a v

224 22 4

2222

2 2

1 251 254 4 254 14 042 2

t tt t t ttttt t

hay 2 4 3 2( 1) 8 9 18 20 8 0t t t t t . t 1t u th ta a c v bt ng thc cp trong cch ban u.

Ta cng c th chng minh bt ng thc 1 1 1 13 251 4a b c a b c

khng dng n

dn bin nh sau:

23

Do 1abc nn tn ti cc s thc dng , ,x y z sao cho 22 2

, ,yx za b cyz zx xy

, ta cn

chng minh rng

2 2 2 3 3 3

13 1334

yz xy xyzzxx y z x y z xyz

.

Do , ,x y z bnh ng nn ta c th gi s x y z . Vit bt ng thc trn thnh dng

tng ng 3 3 3 3 3 3 2 2 2 3 3 3

2 2 2 3 3 3

3 13( 3 )4( )

x y y z z x x y z x y z xyzx y z x y z xyz

.

Ta c cc nh gi 3 3 3 23 ( ) ( ) ( )( )x y z xyz x y z x y z x z y v

3 3 3 3 3 3 2 2 2 2 2

2

3 ( ) ( ) ( )( )

( ) ( ) ( )( )

x y y z z x x y z xy yz zx z x y xy z x z y

xy xy yz zx x y z x z y

Ta cn chng minh 2 2

2 3 3 3

( ) ( ) ( )( ) 13( ) ( ) ( )( )

4( )

xy xy yz zx x y z x z y x y z x y z x z y

xyz x y z xyz

hay

3 3 3 24( )( ) 13 ( )xy yz zx x y z xyz xyz x y z .

Do tnh thun nht nn ta c th gi s 2x y th dn n 3

3 3 2 22

x yx y

.

Suy ra 3 3 3 3 3 3( )( ) ( 2 )(2 ) (2 ) 2 (2 )xy yz zx x y z xyz xy z z xyz xy z z z xyz .

Ta a v 3 3 2 3 3 2

4 3 2 3 2

4 (2 ) 8 (2 ) 13 ( 2) 8 (2 ) (9 18 8)8 9 18 16 8 (1 )(9 18 8) 0

xy z z z xyz xyz z z z xy z zz z z z xy z z

Ch rng 1xy v 1z nn 3 2(1 )(9 18 8) 0xy z z ; tip tc t 1z t vi 0t

v thay vo 4 3 2 4 3 28 9 18 16 8 8 23 3 15 5z z z z t t t t vi 0t . D dng chng minh c biu thc ny khng m nn bt ng thc cn chng minh trn l ng. T ta c pcm. Ta thy rng vic x l bt ng thc (**) khng qu kh nhng vn l ti sao li

ngh ra cch chn 2 9,3 4

b c a c c 14k . im tinh t v cng l kh nht

ca bi ton chnh l y.

24

Trong bt ng thc cho, ng thc xy ra khi 1a b c nn ta d on l gim s

bin bng cch t 11, ,a b x cx

th thu c ngay 14 2k xx

, tip tc cho

1x th c 16k . Tuy nhiu iu kin ny cn cha cht!

Ta tip tc cho 21 ,b c a xx

th thay vo bt ng thc ban u, ta c

4 3 2 3 2

2 2

4(2 5 2 2) 4( 2)(2 1) ( )2 2

x x x x x x xk f xx x x x

vi 0x . (***)

Khi , ta phi c 0

min ( )x

k f x

.

Di y ta s phn tch mt s cch t (***) c th suy ra 14.k

(1) Ta c ( ) 0f x tng ng vi mt phng trnh bc 5:

5 4 3( ) 4 8 5 4 1 0g x x x x x .

Ta khng gii c phng trnh ny, v th khng th tm c chn trn chnh xc cho k . Tuy nhin, do bi ton yu cu tm k nguyn dng ln nht, nn ta cng khng cn i tm gi tr min, m ch nh gi n nhm tm ra mt chn trn cho k .

Ch phng trnh ( ) 0g x c th vit di dng 2 2 34 14 8 5x xx x

.

Trn min (0; ) th v tri l hm tng, v phi l hm gim nn ( ) 0g x c nhiu

nht 1 nghim dng.

Ch rng 1 7 0, (1) 12 02 4

g g

nn nghim 0x ca phng trnh ny nm gia

12

v 1. Ta khng c nh i tm 0x m thng tin ny ch dng c nh hng

chn gi tr x thay vo bt ng thc 3 2

2

4( 2)(2 1)2

x x x kx x

.

Ta thay cc gi tr c bit nm trong 1 ;12

.

Thay 1x , ta c 16k .

Thay 12

x , ta c 15k .

25

Thay 23

x , ta c 13,98k .

Thay 34

x , ta c 14,2.k

n y, ta ch ra c 14.k

(2) cch tip cn ny, ta s chng minh rng 14k khng ng (do cc nh gi n gin d a v trng hp 14k nhiu hn).

Vi 14k , ta c bt ng thc 4 3 24 10 10 5 4 0x x x x . Ta s chng minh rng bt ng thc ny khng ng vi mi 0.x t ( )h x l v tri ca n, ta c

3 2 2( ) 16 30 20 5, ( ) 48 60 20h x x x x h x x x .

Ta thy ( ) 0h x c nghim duy nht v (0) 0 (1)f f nn kho st hm s ( ),h x ta

c phng trnh ( ) 0h x c nghim duy nht 0x tha 0 (0;1)x v 00min ( ) ( )x h x h x .

Ch rng 0 03 3 30 ;14 4 4

f a a

v nn 3 20 0 016 30 20 5 0x x x v ta c

4 3 2 3 20 0 0 0 0 0 0 0 0

3 20 0 0 0

7 ( ) 7(4 10 10 5 4) (16 30 20 5)( 1)(12 52 2 28)f x x x x x x x x xx x x x

Ta c 3 2

3 20 0 0

3 312 52 2 28 12 52 30 04 4

x x x

v 0 1 0x nn 0( ) 0f x , mu

thun (ch rng 3 212 52 2 28x x x ng bin trn 3 ;14

).

Do 14k khng tha mn.

(3) Do ( )f x c dng kh cng knh nn nu nguyn nh vy m kho st th s kh

phc tp. Do , ta ngh n vic bin i ( )f x v dng n gin hn d tnh o

hm v v ( )f x c dng phn thc nn ta ngh n vic tch v chia a thc:

2 2 3( ) 2 .2 1

f x x xx x

n y th vic ly o hm tr nn kh d dng, ta tnh c

26

2 2

1 3( ) 2 1 .(2 1)

f x xx x

Ta cn gii phng trnh ( ) 0.f x Quan st mt cht, ta pht hin c : 3 2 2

2 2 2 2 2

1 1 ( 1)( 1) 3 4( 1), 1 .(2 1) (2 1)

x x x x x xxx x x x x

Nh vy, ta c th phn tch c nhn t:

22 2

1 4( ) ( 1) ,(2 1)

xf x x xx x

t ta a c vic xt ( ) 0f x v xt mt phng trnh n gin hn:

3 22 2

1 4 0 4 4 3 1 0.(2 1)

x x x xx x

n y, ta gp phi mt kh khn kh ln, l phng trnh 3 24 4 3 1 0x x x khng c nghim p. Khi kho st mt phng trnh, ta quan tm n hai vic: phng trnh c bao nhiu nghim v l nhng gi tr no. Vic th nht c th d dng thc hin bng cch vit li phng trnh di dng:

2 14 4 3 0.x xx

V tri ca phng trnh trn l mt hm lin tc v ng bin vi mi 0,x ng thi bng kim tra trc tip ta d thy phng trnh c t nht mt nghim thuc (0, 1) nn bng cch kt hp 2 iu ny li, ta c th khng nh phng trnh ( ) 0f x c nghim

duy nht thuc (0, 1).

Vic th hai thc s rt kh khn trong trng hp ca bi ton ny. R rng vi vic ch bng tnh tay, ta rt kh tnh c gi tr chnh xc nghim ca phng trnh

3 24 4 3 1 0.x x x V li, nu tnh c th chc chn trong cng thc nghim s c cn thc chng cht v khi thay vo ( ),f x ta s rt kh nh gi xp x sang dng

thp phn t a ra nhn nh v .k C v nh ta ang i vo ng ct Tuy nhin, cc bn hy ch rng ta ang cn tm gi tr nguyn dng ln nht ca ,k do ta hon ton khng cn phi tnh ra ng gi tr nh nht ca ( )f x lm g. Thay vo ,

ta c th ngh n vic tm mt gi tr st vi gi tr nh nht cng c. Mun vy, ta cn tm mt gi tr x st vi nghim ca phng trnh ( ) 0f x thay vo tnh

ton. T nhu cu mi ny, ta ngh n vic nh gi chn min cho nghim ca phng

27

trnh 3 24 4 3 1 0.x x x Gi 0 (0, 1)x l nghim ca phng trnh. tng ca ta l lm sao kh c dng bc ba c c mt phng trnh bc thp c th gii c

nhanh chng bng tnh tay. u tin ta c rng vi 0 (0,1)x th 3 20 04 4 ,x x do

2 3 20 0 0 0 08 3 1 4 4 3 1 0.x x x x x

T y, ta tm c chn di cho 0x l 03 41 1

16 2x Tip theo, ta s tm chn trn

cho 0 .x Bng ch nh rng vi 0 (0,1)x th cc ly tha ca n kh nh v xp x

vi nhau nn kh bc ba, ta mnh dn s dng bt ng thc AM-GM nh sau:

3 2 3 2 2 2 20 0 0 0 0 0 0 0 0 0 0 00 4 4 3 1 (4 ) 4 4 1 4 4 4 1 8 4 1.x x x x x x x x x x x x

T y, ta tm c chn trn cho 0x l 03 14

x . Nh vy 03 41 3 1

16 4x . V

cc nh gi kh st nn ta mnh dn chn 3 14

x thay vo ( ).f x rng x l

nghim ca phng trnh 28 4 1 0,x x do 2 4 1 1 3, 8 4, 4 4 .8 2 1

xx x xx x

Suy ra

4 1 45 95 45 3 1 95 45 3 50( ) 2 2(8 4) (4 4 ) .8 2 8 2 4 8 8

xf x x x x x

Vi kt qu ny, ta thu c 45 3 50 345 252 2

45 0,866 25 13,97.k

Tt nhin, cng c th la chn mt gi tr no p hn cho x thun tin hn na cho vic tnh ton nhng y ta nn thn trng v ch cn chn mt s x lch hi xa so vi 0x thi l c th a n vic k s b lch i my n v sang 14, 15 thm ch l 16. Tt nht, ta vn c nn s dng gi tr no m ta bit chc rng n st vi

0x d l mt cht cng khng sao, b li ta s c th yn tm hn v kt qu.

Sau khi tm c 13,k ta c th th i chng minh bt ng thc ng vi 13k bng nhiu cch nhng v cht ca bt ng thc ( k l hng s tt nht) nn tt nhin ta s chn gii php no an ton m hiu qu nht v phng php m ta chn y l lm gim s bin bng dn bin.

28

Cui cng, ta s ch ra mt cch tm hng s thc k rt gn vi hng s tt nht bt ng thc (*) ng.

t 4 3 2 2( ) 4(2 5 2 2) (2 2) 0,F x x x x x k x x . V vi hng s tt nht th

im cc tiu 0x ca a thc gn vi 23

nn ta gi s rng vi k cn tm th im cc

tiu 0x bng ng 23

, tc l o hm 3 2( ) 32 60 16 4 (4 1)F x x x x k x bng 0 ti

23

x , ta tnh c 137299

k . Vi gi tr k ny th ( )F x t c cc tiu ti 23

nn

2 152( ) 03 891

F x F

.

Do , bt ng thc (*) ng vi 1372 13,8585999

k .

Ch rng vi 1880 13,9682563

k k v vi 2 13,85859k th (*) cng ng nn hng

s thc tt nht k tha mn 2 1k k k (ta tnh c 13,96764k ). Mt iu th v

y l d 2k d ln hn 13 kh nhiu nhng bt ng thc trong trng hp tng

ng li d chng minh hn.

Nhn xt chung, bi ton ny c hnh thc kh n gin nhng li i hi nhiu x l trung gian tinh t, nht l phn tnh ton trong iu kin thi gian c gii hn v khng c my tnh h tr. Nhiu bn ch quan khi gp bi ny, nh gi k vi vng ra

8k hoc 16k ri t kt lun lun v mt im ng tic. Mt suy ngh thng thy l vic chng minh bt ng thc mt bin l chuyn n gin (cc bi dn bin thng a v cc bt ng thc 1 bin hin nhin ng) nhng thc ra khng phi vy; cng nh cc phng trnh i s, bt ng thc mt bin cng c th kh v thm ch l rt kh nu nh trong qu trnh x l, chng ta khng thu c nghim c bit no.

C 2 bi ton c cch gii kh ging vi bi 4 (phn chng minh), tuy nhin n gin hn. C l bi ton 4 c pht trin t cc bi ton di y

(Vasile Cirtoaje, Algebraic Inequalities, Chapter 1, bi ton 50 v 55)

29

1. Cho , ,a b c l cc s thc dng sao cho 1abc . Chng minh rng

( )( )( ) 7 5( )a b b c c a a b c .

2. Cho , ,a b c l cc s thc dng sao cho 1abc . Chng minh rng

2 1 33a b c ab bc ca

.

Bi 5.

Cho tam gic ABC nhn khng cn c gc A bng 45 . Cc ng cao , ,AD BE CF ng quy ti trc tm H . ng thng EF ct ng thng BC ti P . Gi I l trung im ca BC ; ng thng IF ct PH ti .Q

1. Chng minh rng IQH AIE .

2. Gi K l trc tm ca tam gic AEF v ( )J l ng trn ngoi tip tam gic KPD . ng thng CK ct ( )J ti G , ng thng IG ct ( )J ti M , ng thng JC ct ng trn ng knh BC ti N . Chng minh rng cc im , , ,G M N C cng thuc mt ng trn.

Li gii.

1. Gi s AB AC , khi B s nm gia ,P C . Trng hp AB AC c chng minh hon ton tng t.

Trc ht, ta s chng minh rng PH vung gc vi AI .

Tht vy, gi ,U V ln lt l trung im ca ,AH IH th ta c .UV AI

D thy ( , , , ) 1P D B C nn theo tnh cht ca hng im iu ha th .PB PC PD PI

Ta cng c PE PF PB PC nn PE PF PD PI hay P nm trn trc ng phng ca ng trn ng knh AH (tm U ) v IH (tm V ).

Hn na H cng nm trn trc ng phng ca hai ng trn ny nn PH UV .

Do .PH AI

V 45BAC nn 90EIF , suy ra 90 .IQH AIF EIF AIF AIE

Vy ta c IQH AIE .

30

T

N

M

G

J K

Q

IP

E

D

HF

A

B C

2. Ta thy rng 180EKF ECF EKF EAF nn K thuc ng trn ng

knh .BC Do hng im , , ,D P B C iu ha nn ta c 2ID IP IC , m IM IG ID IP

(cng bng phng tch ca I n ( )J ) nn 2IM IG IC hay

( . . )IMC ICG c g c nn 45 .IMC ICG ICK (1)

Gi T l trung im PD th theo h thc Maclaurin, ta c CB CT CD CP CK CG hay t gic GTBK ni tip v do 90BKG nn cng c 90GTD hay GT PD .

Tam gic JPD cn ti J v c T l trung im PD nn JT vung gc vi .PD

Do , , ,G J T thng hng v 45KGJ .

Mt khc CN CJ CB CT CK CG nn t gic KNJG ni tip v dn n

45JNG JKG JGK . (2)

T (1) v (2) suy ra 135GMC GNC hay cc im , , ,G M N C cng thuc mt ng trn. y chnh l pcm.

31

Nhn xt.

th 1, vic chng minh PH vung gc vi AI c th c thc hin bng nhiu cch. Chng hn, xt t gic BCEF ni tip ng trn tm .I Theo nh l Brocard th H chnh l trc tm ca tam gic AIP nn PH vung gc vi .AI

Cng c th dng cch tnh ton chng minh AD PDID HD

chng minh kt qu ny.

th 2 l mt ng dng rt p ca hang im iu ha trong c kt hp c hai h thc c bn l h thc Newton v h thc Maclaurin. S bi ton v hng im iu ha vn dng n c hai h thc ny trong mt bi ton l khng nhiu, nn cu 2) l mt chng minh rt hay v th v. Ngoi cch gii trn, ta c th s dng hng tip cn khc nh sau:

Gi L l giao im th hai khc K ca hai ng trn ( ),( )I J . Khi , chng minh , , ,G M N C cng thuc mt ng trn, ta a v chng minh , ,GI CJ LK ng quy.

Trc ht, ta c th i theo hng hon ton thun ty nh sau:

L

N

M

G

J K

IP

E

D

HF

A

B C

32

Ch ( , , , ) 1P D B C v 90BKC nn ,KB KC ln lt l phn gic trong v ngoi ca gc PKD . T dn n DKI DPK hay IK tip xc vi ng trn ( )J , iu ny c ngha l cc ng trn ( ),( )I J trc giao v dn n 90ILJ GLC .

Hn na, GI v CJ ln lt l cc ng i trung trong cc tam gic ,GKL CKL . Gi

U l trung im KL th ,GI CJ i xng vi ,GU CU qua phn gic cc gc ,LGC LCG ca tam gic .LGC

Do cc ng thng , ,LK GU CU ng quy ti U v LK l phn gic gc GLC nn cc ng thng , ,LK GI CJ ng quy.

T li gii bi ton hon tt.

Tuy nhin, ngay t u, ta c th pht biu li bi ton thnh:

Cho tam gic vung KIJ K v dng pha ngoi cc tam gic KJG vung cn nh J v KIC vung cn nh I . Chng minh ,GI CJ ng quy ti mt im nm trn ng cao nh K ca tam gic KIJ .

(Ch rng cc tam gic ,GJK CIK vung cn l c th chng minh hon ton d dng).

gii bi ton ny, ta ch cn bin i i s thng qua nh l Ceva, Thales l c th x l nhanh chng.

Tuy rng mt iu cha hon ho lm trong bi ton ny l cu 1) v cu 2) hu nh khng lin quan g ti nhau.

Thc cht vai tr ca gi thit gc A bng 45 cng khng tht s cn thit lm trong bi ton ny. Cc bn c th tm hiu mt m rng n gin nht nh sau

Cho tam gic ABC ng cao , ,AD BE CF ng quy ti H , EF ct BC ti G . Gi ( )K l ng trn ng knh BC . Trung trc ca BC ct ( )K ti im L sao cho ,A L nm cng pha i vi BC . Gi ( )N l ng trn ngoi tip tam gic GDL . ng thng

CL ct ( )N ti M khc L . ng thng MK ct ( )N ti P khc M . ng thng CN ct ( )K ti Q khc C . Chng minh rng , , ,M P Q C cng thuc mt ng trn.

Cch chng minh hon ton tng t nh li gii bi ton gc.

Thc s bi ton ny c nh gi rt cao khng nhng v pht biu p v chng minh hay ca n m cn bi v tnh pht trin ca n. Trong bi ton ny, cn c rt nhiu nhng khm ph mi ang i cc bn cng suy ngm.

33

Bi 6.

Cho mt khi lp phng 10 10 10 gm 1000 vung n v mu trng. An v Bnh chi mt tr chi. An chn mt s di 1 1 10 sao cho hai di bt k khng c chung nh hoc cnh ri i tt c cc sang mu en. Bnh th c chn mt s bt k ca hnh lp phng ri hi An cc ny mu g. Hi Bnh phi chn t nht bao nhiu vi mi cu tr li ca An th Bnh lun xc nh c nhng no l mu en?

Li gii.

Trc ht, ta s chng minh nhn xt tng qut:

Cho mt khi lp phng 2 2 2n n n gm 38n vung n v mu trng. An v Bnh chi mt tr chi. An chn mt s di 1 1 n sao cho vi hai di bt k th chng khng c chung nh hoc cnh ri i tt c cc sang mu en. Bnh th c chn mt s bt k ca hnh lp phng ri hi An cc ny c mu g. Khi , Bnh cn chn t nht 26n mi c th xc nh c no c mu en.

Tht vy, gi nS l tp hp cc khi m Bnh cn phi chn hi An v vi mi u

c Bnh chn, t uR l hp ca cc thuc ba di ngang, dc, cho i qua u . Ta

thy rng:

34

Do iu kin hai dy c chn bt k khng chung cnh v nh nn vi mi cu tr li v mu cho mi u m Bnh chn. Ta thy rng nu gi uR l hp ca cc c

chn th khi mu en, s c ng mt trong ba di ngang, dc, cho i qua c t mu en. Trong trng hp ny, ta cn chn thm mt s thuc uR bit

chnh xc di . Nu ch chn thm mt thi th khi An tr li mu trng, Bnh s khng xc nh c di no trong hai di cn li c t en. Do , Bnh phi chn thm t nht hai na trong uR mi c kh nng tr li c. Thm vo , hai

phi thuc hai di khc nhau v nu chng cng thuc mt di th cng tng t nh trng hp chn mt nu trn.

Vi nhn xt ny, ta gn cho mi u ca hnh lp phng mt b s ( , , )a b c vi nh ngha nh sau:

2a nu di hnh hp theo chiu ngang i qua u khng c thm im no thuc nS v 1a nu ngc li.

2b nu di hnh hp theo chiu dc i qua u khng c thm im no thuc

nS v 1b nu ngc li.

35

2c nu di hnh hp theo chiu cho i qua u khng c thm im no thuc

nS v 1c nu ngc li.

Khi , c hai trong ba s , ,a b c c gi tr bng 1 v s cn li khng vt qu 2 nn 4a b c .

t T l tng cc s dng gn cho cc ca hnh lp phng. Khi , ta c

( ) 4n

nu S

T a b c S

.

Mt khc, vi mi dy 1 1 n ca khi lp phng (theo c ba chiu) u c t nht mt khi thuc tp hp nS v nu khng, Bnh s khng c thng tin g v di v

trong trng hp An tr li rng tt c cc c Bnh chn u c t mu trng th Bnh s khng bit c di cn li c c t mu en hay khng.

D thy rng c tt c 2(2 )n di 1 1 2n nm ngang v tt c cc di ny s ng gp t

nht 22(2 )n n v vo T (ng gp vo cc s a theo nh ngha nh trn).

Tng t vi 2(2 )n di 1 1 2n dc v 2(2 )n di 1 1 2n cho nn ta suy ra

2 23 2(2 ) 24T n n .

T , ta c 24 24nS n hay 26 ,nS n n

n y, ta suy ra hai iu sau:

- Trong hnh lp phng 2 2 2 , Bnh cn chn t nht 6 .

- Trong hnh lp phng 10 10 10 , Bnh cn chn t nht 150 .

Ta s ch ra cch t mu 6 tha mn bi v t ch ra cch xy dng cho hnh lp phng 10 10 10 cho. (Trn thc t, ta hon ton c th xy dng cho trng hp tng qut cho hnh lp phng 2 2 2n n n ).

Tht vy, trong hnh lp phng 2 2 2 , tr hai no i xng nhau qua tm, Bnh chn 6 cn li nh hnh bn di.

Ta s chng minh cch chn ny tha mn yu cu.

36

D thy rng trong hnh lp phng 2 2 2 , ch c khng qu 1 di c t mu nn hoc khng c no hoc c 2 ca hnh lp phng c t mu v 2 phi thuc cng 1 di. Do , trong 6 c chn, ta c 3 trng hp:

- Nu khng c no c t en th c hnh lp phng khng c.

- Nu c ng 1 c t en th en cn li s thuc trong 2 khng c chn c cng 1 di vi en bit.

- Nu c 2 c t en th chnh l tt c cc en ca hnh lp phng.

Do , cch chn ny vi hnh lp phng 2 2 2 tha mn iu kin bi.

Tip theo, ta xy dng cho hnh lp phng 10 10 10 nh sau:

Ta chia hnh lp phng thnh 5 lp 10 10 2 v ta chia n thnh 25 phn, mi phn l mt hnh lp phng 2 2 2 ri nh s nh hnh bn di:

5

5

5

5

4

4

4

4

3

3

3

3

2

2

2

2

1

1

1

1

5

4321

37

lp th i vi 1,2,3,4,5i , ta chn cc khi c nh s i v vi mi khi , ta b i 2 bt k i xng nhau qua tm nh cch nu trn ri chn 6 cn li.

D thy rng vi cch t nh vy, ta chiu cc hnh lp phng c chn xung mt mt no th cc hnh chiu s ph kn mt . iu ny c ngha l vi mt di bt k m An chn th n u i qua mt trong cc hnh lp phng 2 2 2 m Bnh chn nh trn. Khi , nh chng minh trn, ta s xc nh c rng di c c t mu hay khng, tc l xc nh c mu ca tt c cc c t mu en ca hnh lp phng ban u. Do , cch chn cc ny tha mn bi.

Vy s t nht m Bnh cn chn l 150.

Nhn xt.

C th ni y l bi ton kh nht ca k thi, tuy nhin, ci kh ca n khng nng tnh k thut nh bi 3 m l ci kh v mt phng php. Thc t nhiu th sinh ni rng Em thc s khng bit phi xoay s th no!.

lm c bi ton ny, trc ht phi tht tp trung hiu r yu cu ca bi ton v phi dng trng hp 2n hoc xt bi ton 2 chiu hnh dung bi ton mt cch c th nht. Trong cc nhn xt trn th nht xt rng mi di u c mt c chn l kh hin nhin (v nhiu th sinh pht biu c nhn xt ny), cn nhn xt rng ngoi khi u th trong hp uR cn c t nht 2 khi na c chn th tinh t hn.

Khi c hai nhn xt ny th vic p dng k thut m bng hai cch nh gi l kh t nhin.

Vic xy dng cu hnh cch chn cng l mt thch thc.

y, trng hp 2n ng mt vai tr quan trng, nh nhng vin gch ta xy dng. R rng cc cu trc c bn nh hnh vung Latin trong bi gii gip chng ta gii quyt bi ton. Cc m hnh, cu trc, cch sp xp c bn v vy lun ng mt vai tr quan trng trong cc bi ton xy dng v d, phn v d. Nu c lm quen nhiu vi cc bi ton v cc tr ri rc th c th x l bi ny, t nht l vic xy dng mt cch ch ng hn.

Trong cc k TST trc y, c mt s bi ton s dng phng php m bng hai cch nh gi, chng hn nh:

38

(Vit Nam TST 2000) Trong mt phng cho 2000 ng trn bn knh 1 sao cho khng c hai ng trn no tip xc nhau v hp ca cc ng trn ny to thnh mt tp hp lin thng. Chng minh rng s cc im thuc t nht hai ng trn trong cc ng trn cho khng nh hn 2000.

(Vit Nam TST 2010) Gi mt hnh ch nht c kch thc 1 2 l hnh ch nht n v mt hnh ch nht c kch thc 2 3 , b i 2 gc cho nhau (tc c c 4 vung con) l hnh ch nht kp. Ngi ta ghp kht cc hnh ch nht n v hnh ch nht kp ny li vi nhau c mt bng hnh ch nht c kch thc l 2008 2010 . Tm s nh nht cc hnh ch nht n c th dng ghp.

Mt ln na th trong bi ton cc tr k thi chn i tuyn, du bng xy ra c th xy dng khng qu kh nhng ch ra l gi tr tt nht th l iu hon ton khng d dng, thm ch rt kh!

Mt vn th v t ra l nu kch thc ca hnh lp phng l l th ta s c kt qu th no? p dng cch nh gi chn di nh trng hp n chn, ta c nu

2 1n k th s hnh vung cn c nh du phi tha mn bt ng thc 23(2 1)

2kT . T y suy ra 26 6 2T k k . Tuy nhin, vic xy dng cch chn

26 6 2k k hnh lp phng n v tha mn yu cu bi ton vn l mt cu hi m. Vi 3n , tc l 1k , hin nay cha tm c cch chn 14 hnh lp phng tha mn yu cu bi ton v cng cha chng minh c l cn nhiu hn. (Ta c th ch c 15 hnh lp phng tha mn yu cu).

Ta th thay gi thit nu trong bi mt cht c bi ton mi: Bnh chn mt bt k ca hnh lp phng ri hi An ny c mu g ri sau tip tc nh th v cc d kin khc vn gi tng t.

Hoc bi ton trong trng hp hai chiu: Thay khi lp phng bi bng vung v thay cc di vung c chn bi cc hng, ct ca bng.

Khai thc cc trng hp ny s gip ta thu c nhiu kt qu th v khc!