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Mt S Kin Thc V Hm S Tun Hon Cao Minh Quang, THPT chuyn Nguyn Bnh Khim, Vnh Long

Trn Minh Hin, THPT chuyn Quang Trung, Bnh Phc

Trong chng trnh THPT, kin thc v hm s tun hon (HSTH) c cp rt t, ch yu khi hc sinh c hc v cc tnh cht ca cc hm s lng gic lp 11. Tuy nhin, trong cc k thi hc sinh gii, vn thng hay xut hin nhng bi ton lin quan n ni dung ny. Bi vit sau s trnh by mt s kin thc v l thuyt cng nh cc bi ton v HSTH. 1. nh ngha

Hm s ( )y f x= c tp xc nh D c gi l HSTH nu tn ti t nht mt s 0T sao cho vi mi x D ta c:

i) x T D ii) ( ) ( )f x T f x = . S thc dng T tha mn cc iu kin trn c gi l chu k (CK) ca HSTH ( )f x . Nu

HSTH ( )f x c CK nh nht 0T th 0T c gi l chu k c s (CKCS) ca HSTH ( )f x . Ta s tm hiu mt s tnh cht c bn ca HSTH.

2. Mt s tnh cht

2.1. Gi s ( )f x l HSTH vi CK T . Nu 0x D th 0x nT D+ , 0x D th 0x nT D+ , vi mi n .

2.2. Gi s ( )f x l HSTH vi CK T v ( )0f x a= , 0x D , khi ( )0f x nT a+ = , vi mi n .

2.3. Nu 1 2, 0T T > l cc CK ca HSTH ( )f x trn tp D th cc thc dng 1 2 1, ,mT nT mT nT+ , vi ,m n + , u l CK ca ( )f x trn tp D . 2.4. Nu ( )f x l HSTH vi CKCS 0T th 0 ,T nT n += l mt CK ca HSTH ( )f x .

2.5. Nu 1 2,T T l cc CK ca cc HSTH ( ) ( ),f x g x v 12

TT

l s hu t th cc hm s ( ) ( )f x g x+ ,

( ) ( ) ( ) ( ), .f x g x f x g x cng l cc HSTH vi chu k 1 2 , ,T mT nT m n += = . Vic chng minh cc tnh cht 2.1 2.4 tng i n gin. Ta s chng minh tnh cht 2.5.

Chng minh. V 12

TT

l s hu t nn tn ti ,m n + sao cho 12

T nT m

= . t 1 2T mT nT= = ,

vi mi x D , ta c

( ) ( ) ( ) ( ) ( )1 1 12 ...f x f x T f x T f x mT f x T= + = + = = + = + , ( ) ( ) ( ) ( ) ( )2 2 22 ...g x g x T g x T g x nT g x T= + = + = = + = + .

Do ,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), . .f x T g x T f x g x f x T g x T f x g x+ + = + + = . Vy ( ) ( ) ( ) ( ), .f x g x f x g x l cc HSTH vi chu k 1 2 , ,T mT nT m n += = .

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Vic kt lun mt hm s c phi l HSTH hay khng ph thuc rt nhiu vo vic xc nh CK hoc CKCS (nu c) ca hm s. Ta cp n CK (CKCS) ca mt s hm s thng gp. 3. Chu k v chu k c s ca mt s hm s 3.1. Hm s ( )f x c= ( c l hng s) l HSTH vi CK l s dng bt k nhng khng c CKCS.

3.2. Hm Dirichlet ( )1, 0, \

xf xx

=

l HSTH vi CK l s hu t dng bt k nhng khng

c CKCS.

3.3. Hm s ( ) { } [ ]f x x x x= = l HSTH c CKCS 0 1T = . 3.4. Cc hm s ( ) ( )sin , cosf x x f x x= = l cc HSTH c CKCS 0 2T = . Cc hm s ( ) ( ) ( ) ( )tan , cot , sin , cosf x x f x x f x x f x x= = = = l cc HSTH c CKCS 0T = .

3.5. Cc hm s ( ) ( ) ( ) ( )sin , cosf x ax b f x ax b= + = + , 0a l cc HSTH c CKCS 0 2Ta

= .

Cc hm s ( ) ( ) ( ) ( )tan , cotf x ax b f x ax b= + = + , 0a l cc HSTH c CKCS 0Ta

= .

Chng minh. Ta s chng minh cho hm s ( ) { } [ ]f x x x x= = v ( ) ( )sinf x ax b= + , cc hm s cn li xin dnh cho bn c nh bi tp t luyn.

Vi mi n , ta c ( ) { } { } ( )f x n n x x f x+ = + = = . Do ( ) ( )1f x f x+ = . Mt khc, nu 00 1T t< = < l CKCS ca ( )f x th vi 1x t= , ta c 0 1x< < , do .

( ) ( ) ( ) { }1 0 1f x t f f x x t+ = = = = . Vy hm s ( ) { } [ ]f x x x x= = l HSTH c CKCS 0 1T = .

Trc ht, ta chng minh 0 2T a= , 0a l CK ca ( ) ( )sinf x ax b= + . Tht vy, ta c

( ) ( ) ( ) ( ) ( )2 sin 2 sin 2 sinf x a a x a b ax b ax b f x + = + + = + = + = .

Gi s tn ti s dng 2t a< sao cho ( ) ( )f x t f x+ = , vi mi x . Khi , vi 2 b

xa

= , ta c

( ) ( ) ( )2sin sin 2 cos cos 1bf x t a t b at at t aa

+ = + + = + = =

3

c) ( ) 3cos cos2 2x xf x =

d) ( ) cos cos 2f x x x= + e) ( ) 2sinf x x= Li gii.

a) Theo tnh cht 3.5, d thy rng ( ) cosf x x= l HSTH vi CKCS 2T = .

b) Tp xc nh ca hm s l [ )0,D = + . Gi s ( ) cosf x x= l HSTH vi CK 0T > . Nu 0x D th 0x nT D+ , vi mi n . Tuy nhin, iu ny khng th xy nu cho 0n< b th 0 0x nT+ < . Do ( ) cosf x x= khng l HSTH.

c) Ta c ( ) ( ) ( )3 1cos cos cos cos2 22 2 2x xf x x x f x = = + = + . Ta s chng minh 0 2T =

l CKCS ca hm s ny. Tht vy, vi 0 2a < < th cos 1,cos2 1a a< , suy ra

( ) ( ) ( )1 cos cos2 1 02

f a a a f= + < = .

Do , ( ) ( )f x a f x+ = khng th xy ra vi mi x , tc l 0 2T = l s dng nh nht sao cho ( ) ( )0f x T f x+ = vi mi x hay 0 2T = l CKCS.

d) Gi s ( ) cos cos 2f x x x= + l HSTH, tc l tn ti 0T > sao cho ( ) ( )f x T f x+ = , vi mi x , hay ( ) ( )cos cos 2 cos cos 2x T x T x x+ + + = + .

Vi 0x = , ta c cos cos 2 2T T+ = , suy ra cos cos 2 1T T= = hay 2 , 2 2T k T m = = ,

trong ,k m + . Do 2 mk

= (v l). Vy ( ) cos cos 2f x x x= + khng l HSTH.

e) Gi s ( ) 2sinf x x= l HSTH, tc l tn ti 0T > sao cho ( ) ( )f x T f x+ = , vi mi x , hay ( )2 2sin sinx T x+ = .

Vi 0x = , ta c 2sin 0T = hay 2T k= , k + hay T k= . Suy ra ( ) ( )f x k f x+ = .

Vi 2x k= , ta c ( ) ( ) ( )2 2sin 2 sin 2 sin 2 0k k k k + = = = , v l v

( ) ( ) ( )2sin 2 sin 2 2 2 sin 2 2 0k k k k k k + = + + = . Vy ( ) 2sinf x x= khng l HSTH.

Bi ton 2. Chng minh rng hm s ( ) ( )[ ]{ }1 xf x x= l HSTH. Li gii. Ta s chng minh 0 2T = l CKCS ca hm s. Tht vy, ta c

( ) ( )[ ]{ } ( ) [ ]{ } ( )[ ]{ } ( )2 22 1 2 1 1x x xf x x x x f x+ ++ = + = = = . Gi s tn ti 0 2a< < sao cho ( ) ( )f x a f a+ = , vi mi x . Ta s xt ba trng hp. (i). 0 1a< < . Chn 2x a= th 1 2x< < . Do ( ) { } 0f x x= ; ( ) ( )2 0f x a f+ = = ,

suy ra ( ) ( )f x a f x+ .

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(ii). 1a = . Chn 0 1x< < , ta c ( ) { } ( ) { };f x x x f x a x x= = + = = , ( ) ( )f x a f x+ . (iii). 1 2a< < . Chn 2x a= th 0 1x< < , ta c ( ) { } ( ) ( ); 2 0f x x x f x a f= = + = = ,

suy ra ( ) ( )f x a f x+ . Vy khng tn ti 0 2a< < sao cho ( ) ( )f x a f a+ = , vi mi x hay 0 2T = l CKCS. Bi ton 3. [Vit Nam 1997, bng B] Cho , , ,a b c d l cc s thc khc 0 . Chng minh rng

( ) sin cosf x a cx b dx= + l HSTH cd

l s hu t.

Li gii.

( ) Gi s ( )f x l HSTH, tc l tn ti 0T > sao cho ( ) ( )f x T f x+ = , vi mi x . Vi 0x= ta c ( ) ( )0f T f= hay sin cosa cT b dT b+ = . Vi x T= , ta c ( ) ( )0f T f= hay sin cosa cT b dT b + = . Cng theo tng v cc ng thc trn, ta nhn c cos 1dT = , suy ra { }2 , \ 0dT k k= .

Tr theo tng v cc ng thc trn, ta nhn c sin 0cT = , suy ra { }, \ 0cT m m= .

T suy ra 2

c m

d k= .

( ) Ngc li, gi s cd

l s hu t, tc l tn ti { }, \ 0m n sao cho c md n= . Ta chn s

dng 2 2m nTc d

= = , khi vi mi x , ta c

( ) ( )2 2sin cos sin cosm nf x T a c x b d x a cx b dx f xc d + = + + + = + =

.

Do , ( )f x l HSTH vi CK 2 2m nTc d

= = .

Bi ton 4. Chng minh rng nu th hm s ( )f x c hai trc i xng ( ),x a x b a b= = , th ( )f x l HSTH.

Li gii. Trc ht, ta gi ( )C l th ca hm s. Khng mt tnh tng qut, ta gi s rng

a b< . Tnh tin ( )C theo vector ( ),0v a=

. Bi ton tr thnh: Chng minh rng nu th ca hm s ( )f x c hai trc i xng 0,x x c b a= = = th ( )f x l HSTH.

V th ca hm s ( )f x i xng qua 0x = nn ( ) ( )f x f x= . Mt khc, th ca hm s ( )f x cng i xng qua x c= nn ( ) ( )2f x f c x= . Suy ra ( ) ( )2f x f c x = , vi mi x , tc l ( )f x l HSTH vi CK ( )2 2T c b a= = .

Bi ton 5. Cho hm s ( )f x xc nh trn D v ( ) ( )( )

1, 0

1f xf x a af x

+ =

+. Chng minh

rng ( )f x l HSTH. Li gii. Vi mi , 0x D a , ta c

5

( ) ( ) ( )( )

( )( ) ( )( )( )( ) ( )( ) ( )

1 1 11 121 1 1 1

f x f xf x af x a f x a a f x a f xf x f x + + + = + + = = = + + + +

.

Suy ra, ( )( )

( )142

f x a f xf x a

+ = =+

. Do ( )f x l HSTH.

Bi ton 6. Cho hm s ( )f x xc nh trn v tha mn iu kin

( ) ( ) ( )4 4f x f x f x+ + = , vi mi x . Chng minh rng ( )f x l HSTH. Li gii. Vi mi x , t iu kin bi ton, ta c ( ) ( ) ( )8 4f x f x f x+ + = + . Suy ra

( ) ( )8 4f x f x+ = . Do ( ) ( )( ) ( )( ) ( )12 4 8 4 4f x f x f x f x+ = + + = + = , v

( ) ( )( ) ( ) ( )24 12 12 12f f x f x f x= + + = + = .

Vy ( )f x l HSTH vi CK 24T = . Bi ton 7. Cho hm s ( )f x xc nh trn v tha mn iu kin

( ) ( ) ( )3 3f x f x f x= + , vi mi x . Chng minh rng ( )f x l HSTH. Li gii. Vi mi x , t iu kin bi ton, ta c ( ) ( ) ( )3 6f x f x f x+ = + . Suy ra

( ) ( ) ( ) ( )3 3 3 6f x f x f x f x+ = + + , tc l ( )3 0f x+ = hoc ( ) ( )3 6 1f x f x + = . Nu ( )3 0f x+ = , vi mi x , th ( ) 0f x = , v vy ( )f x l HSTH. Nu ( ) ( )3 6 1f x f x + = th ( ) ( )9 1f x f x+ = , do ( ) ( )9 18 1f x f x+ + = . T suy ra

( ) ( )18f x f x= + hay ( )f x l HSTH CK 18T = . Bi ton 8. Cho hm s ( )f x xc nh trn v tha mn iu kin

( ) ( ) ( )1 1 2f x f x f x+ + = , vi mi x . Chng minh rng ( )f x l HSTH. Li gii. Vi mi x , t iu kin bi ton, ta c

( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 1 2 2 1 2 2 1f x f x f x f x f x f x f x + + = + = = .

Do ( ) ( ) ( )2 2 1f x f x f x+ = . T ng thc ny, suy ra

( ) ( ) ( ) ( ) ( )3 1 2 1 1f x f x f x f x f x+ + = = + hay ( ) ( )3 1f x f x+ = . Suy ra ( ) ( ) ( )4 8f x f x f x= + = + . Vy ( )f x l HSTH vi CK 8T = Bi ton 9. Cho hm s ( )f x xc nh trn , tha mn cc iu kin ( ) ( )3 3f x f x+ + ,

( ) ( )2 2f x f x+ + , vi mi x . Chng minh rng ( ) ( )g x f x x= l HSTH. Li gii. Ta s chng minh ( ) ( )6g x g x+ = , vi mi x . Tht vy, ta c

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( ) ( ) ( )6 6 6 3 3 6g x f x x f x x+ = + = + + ( ) ( ) ( ) ( )3 3 6 3 3 6f x x f x x f x x g x + + + + = = . Mt khc, ( ) ( ) ( ) ( )6 6 6 4 2 6 4 2 6g x f x x f x x f x x+ = + = + + + + ( ) ( ) ( ) ( )2 4 6 6 6f x x f x x x f x x g x + + + = = . Suy ra ( ) ( )6g x g x+ = , vi mi x hay ( ) ( )g x f x x= l HSTH vi CK 6T = . Bi ton 10. Chng minh rng nu HSTH ( )f x tha mn iu kin ( ) ( ),kf x f kx= vi mi

x , , 0, 1k k k th ( )f x khng c CKCS. Li gii. Gi s 0T l CKCS ca HSTH ( )f x . Khi , vi mi , 1, 0x k k , ta c

( ) ( ) ( )0f kx T f kx kf x+ = = v ( ) 0 00 T Tf kx T f k x kf xk k + = + = +

.

Do , ( ) 0Tf x f xk

= + . Ta s xt hai trng hp sau:

(i) 1k > . Nu 1k > th 0 0T Tk< (v l v 0T l CKCS). Nu 1k , bng cch

t 0Ty xk

= + , ta c ( ) 0Tf y f yk

= (v l v 0 0T Tk < ).

(ii) 1k < . Vi mi x , ta c ( ) ( )0 0 0x x xf x kT f k T kf T kf f xk k k

+ = + = + = = .

t 1

'kk

= , ta nhn c ( ) 0'

Tf x f xk

= + , vi ' 1k > . Theo (i), ta cng nhn c iu v l.

Tm li, ( )f x khng c CKCS.

Bi ton 11. Cho 0>a v hm s :f + tha iu kin ( ) ( ) ( )212

+ = + f x a f x f x , vi mi 0>x . Chng minh rng ( )f x l HSTH.

Li gii. V ( ) ( ) ( )21 ,2

f x a f x f x+ = + vi mi 0>x nn ( ) 12

f x a+ .

Do , ( ) 12

f x . Suy ra

( ) ( ) ( ) ( ) ( )21 12 12 2

f x a f x a f x a f x a f x a + = + + + = + + +

( ) ( ) ( ) ( ) ( ) ( )2 2 21 1 1 1 12 2 2 2 4

f x f x f x f x f x f x = + + = + +

( ) ( ) ( )1 1 1 12 2 2 2

f x f x f x= + = + = .

Vy tn ti 2 0T a= > sao cho ( ) ( )f x T f x+ = , vi mi x + nn ( )f x l HSTH.

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Bi ton 12. Tn ti hay khng cc hm s , :f g , vi g l HSTH tha mn iu kin [ ]( ) ( )3x f x g x= + , vi mi x , k hiu [ ]i ch phn nguyn.

Li gii. Gi s tn ti cc hm ,f g tha mn yu cu bi ton. Gi 0T l CK ca g . Vi mi x , ta c

( ) [ ]( ) ( ) [ ]( ) ( )30 0 0 0x T f x T g x T f x T g x+ = + + + = + + .

Suy ra [ ]( ) [ ]( ) ( ) ( )3 3 2 2 30 0 0 0 03 3 *f x T f x x T x T x T x T+ = + = + + .

Vi mi [ ] )0 00, 1x T T + th v tri ca (*) bng 0, do (*) l a thc bc 2 c v s nghim, suy ra 2 30 0 03 3 0T T T= = = hay 0 0T = (v l).

Vy khng th tn ti cc hm ,f g tha mn yu cu bi ton. Bi ton 13. Gi s ( )f x l mt HSTH c CKCS 0T . Tn ti hay khng ( )10lim xx f ?

Li gii. Trc ht, ta nhn thy rng ( )f x c ( c l hng s), v hm hng khng c CKCS. Do , s tn ti hai s thc ,a b sao cho ( ) ( )f a f b . t 0 0,n na a nT b b nT= + = + . Khi , ta c 1 10, 0n n

n na b . Do

( ) ( ) ( ) ( )01lim lim lim lim

1 nn n n nnf f a f a nT f a f a

a

= = + = = ,

( ) ( ) ( ) ( )01lim lim lim lim

1 nn n n nnf f b f b nT f b f b

b = = + = =

.

Suy ra, 1 1lim lim1 1n nn n

f fa b

hay khng tn ti ( )1

0lim

xx

f

.

Bi ton 14. Cho ( )f x l HSTH v lin tc trn , c CK 2T . Chng minh rng tn ti 0x sao cho ( ) ( )0 0f x T f x+ = .

Li gii. t ( ) ( ) ( )g x f x T f x= + . Ta c ( ) ( ) ( ) ( ) ( )2g x T f x T f x T f x f x T+ = + + = + .

Do , ( ) ( ) ( ) ( )2

. 0g x g x T f x T f x + = + .

V ( )f x l hm s lin tc nn ( )g x cng l hm s lin tc, do , theo nh l Cauchy Bolzano, tn ti [ ]0 ,x x x T + sao cho ( )0 0g x = hay ( ) ( )0 0f x T f x+ = .

Bi ton 15. Cho hm s :f tha mn cc iu kin sau: (i) ( ) ( ) ( ) ( )2f x y f x y f x f y+ + = , vi mi ,x y ; (ii) Tn ti s thc 0x sao cho ( )0 1f x = . Chng minh rng ( )f x l HSTH.

Li gii. Cho 0x y= = , ta nhn c ( ) ( )( )22 0 2 0f f= . Do ( )0 0f = hoc ( )0 2f = .

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Nu ( )0 0f = , vi 0 , 0x x y= = , ta c ( )02 0f x = , mu thun iu kin (ii). Vy ta phi c c ( )0 1f = , v nh vy 0 0x .

Cho x y= , ta c ( ) ( ) ( )( )22 0 2f x f f x+ = hay ( ) ( )22 2 1f x f x= , vi mi x .

Cho 0x y x= = , ta c ( ) ( ) ( )( )2

02 0 2f x f f x+ = hay ( )02 1f x = .

Thay x bi 02x x+ v y bi 02x x , ta c ( ) ( ) ( ) ( )0 0 02 4 2 2 2f x f x f x x f x x+ = + . Nhng ( ) ( )20 04 2 2 1f x f x= , do

( ) ( ) ( ) ( ) ( ) ( )20 0 02 2 2 4 2 1f x x f x x f x f x f x f x+ = + = + = . Mt khc, vi x v 02y x= , ta c ( ) ( ) ( )0 02 2 2f x x f x x f x+ + = . Suy ra

( ) ( )2

0 02 2f x x f x x + =

( ) ( ) ( ) ( ) ( ) ( )2 2 2

0 0 0 02 2 4 2 2 4 4 0f x x f x x f x x f x x f x f x + + = + + = = .

Do , vi mi x th ( ) ( ) ( )0 02 2f x x f x x f x+ = = hay ( )f x l HSTH.

Bi ton 16. Cho hm s :f tha mn iu kin ( ) 2008f x v

( )13 1 142 6 7

f x f x f x f x + + = + + + , vi mi x .

Chng minh rng ( )f x l HSTH.

Li gii. t 1 1,6 7

a b= = th 1342

a b+ = . Khi , mi quan h ca hm s c th c vit

li nh sau

( ) ( ) ( ) ( )f x a b f x f x a f x b+ + + = + + + , vi mi x . Trong ng thc trn, ta thay lin tip x bi , 2 , 3 , 4 , 5x a x a x a x a x a+ + + + + , ri cng li,

ta thu c

( ) ( ) ( ) ( )1 1f x b f x f x f x b+ + + = + + + , vi mi x . Trong ng thc va nhn c, ta thay lin tip x bi , 2 , 3 , 4 , 5x b x b x b x b x b+ + + + + ,

ri cng li, ta thu c

( ) ( ) ( )2 2 1f x f x f x+ + = + hay ( ) ( ) ( ) ( )2 1 1f x f x f x f x+ + = + , vi mi x . t ( ) ( )1f x f x a+ = . Bng phng php quy np, ta chng minh c, vi mi s n

nguyn dng th ( ) ( )1f x n f x n a+ + = . Do , ( ) ( )f x n f x na+ = . Ta s chng minh 0a= , v khi ( )f x l HSTH vi CK 1T = . Tht vy, gi s 0a , th

( ) ( ) 2.2008n a na f x n f x= = + , vi mi x . Nhng iu ny khng xy ra khi 0n> ln. Do 0a = .

Bi ton 17. Cho hm s :f + + v s nguyn dng a tha mn cc iu kin sau: (i) ( ) ( ) ( ) ( ) ( ) ( )1995 , 1 1996 , 2 1997f a f f a f f a f= + = + = ;

9

(ii) ( ) ( )( )

11

f nf n a f n

+ =+

, vi mi n + .

(a) Chng minh rng ( ) ( )4f n a f n+ = vi mi n + . (b) Xc nh gi tr nh nht ca a tha mn bi ton. Li gii. (a) Trc ht ta nhn thy rng, c c iu kin (ii), ta phi c ( ) 1f n , vi

mi n . Hn na, nu tn ti n sao cho ( ) 0f n = th ( ) 1f n a+ = , suy ra ( )2f n a+ l khng xc nh. Do ( ) { }1,0f n , vi mi n . Ta c

( )( )( )

( )( ) ( )( )( )( ) ( )( ) ( )

1 1 11 121 1 1 1

f n f nf n af n a f n a f nf n f n + +

+ = = =+ + + +

, vi mi n .

Do ( )( )

( )142

f n a f nf n a+ = =+ , vi mi n .

(b) Xt 1a = , th ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )

11 1995 , 2 1996 , 3 1997 , 1

1f nf f f f f f f n f n

= = = + =

+.

Khi , ( ) ( )4f n f n+ = v ( )( )12f n f n

+ = , vi mi n + . Bng phng php quy np,

ta chng minh c ( ) ( )4f n k f n+ = , vi mi ,n k + .

V 1995 3 4.498= + nn ( ) ( ) ( )1 1995 3f f f= = (v l v ( )( )131

f f

= ).

Xt 2a = , th ( ) ( ) ( ) ( ) ( ) ( )2 1995 , 3 1996 , 4 1997f f f f f f= = = ; v vi mi n + th ( ) ( )8f n f n+ = , ( )

( )14f n f n

+ = , vi mi n + . Bng phng php quy np, ta chng minh

c ( ) ( )8f n k f n+ = , vi mi ,n k + . Ta c 1995 3 8.249= + nn ( ) ( ) ( )2 1995 3f f f= = . 1996 4 8.249= + nn ( ) ( ) ( )3 1996 4f f f= = . 1997 5 8.249= + nn ( ) ( ) ( )4 1997 5f f f= = .

Suy ra ( ) ( )3 5f f= .

Mt khc ta c ( )( )151

f f= hay ( ) ( )5 1 1f f = . Suy ra

( ) ( )( )( )1 1

5 31 1

ff f f

= =+

hay ( ) ( ) ( ) ( )5 1 5 1 1f f f f+ = hay ( ) ( )5 1f f= (v l).

Ta s chng minh 3a = l gi tr nh nht cn tm.

Ta xy dng hm f nh sau: ( ) ( ) ( )1 , 2 , 3f f f nhn cc gi tr ty khc 0 v 1. Vi mi

3n> , ta xc nh ( )( )( )

3 13 1

f nf n f n

=+ +

. Ta chng minh hm f va c xy dng nh trn s

tha mn iu kin ca bi ton.

10

Ta c ( )( )( )

13

1f nf n f n

+ =

+. Suy ra ( )

( )( )

11

f nf n a f n

+ =+

, vi mi n .

Mt khc ( ) ( )12f n f n+ = , vi mi n . Bng phng php quy np, ta chng minh c ( ) ( )12f n k f n+ = , vi mi ,n k + . Ta c

( ) ( ) ( )1995 3 12.166 3f f f= + = . ( ) ( ) ( )1996 4 12.166 4f f f= + = . ( ) ( ) ( )1997 5 12.166 5f f f= + = .

Vy 3a = l gi tr nh nht cn tm.

Bi ton 18. Dy s { }nx c cho bi cng thc 1 122, ,1 2

nn

n

xx x n Z

x

++

+= =

. Chng minh

rng

(a) 0nx , vi mi *n . (b) Dy { }nx khng tun hon. Li gii. (a) Trc ht, d nhn thy rng cc s hng ca dy ny u l s hu t. Ta t 1 tan 2x = = , th

12 2

1

2 tan tantan 2

1 2 1 tanx

xx

+ += = =

, , ( )1

tan tantan 1

1 tan tannn

x nn

+

+= = +

.

Nu 0nx = vi 2 1,n k k= + th 22 12

2 01 2

kk

k

xx

x+

+= =

, hay 2 2kx = . Khi

2 22 tan 1 5

tan 2 2 2 11 tan 1 2

kk

k

xkk xk x

= = = =

(v l).

Nu 0nx = vi 2 ,n k k= th 22 tan

tan 2 01 tan

kkk

= =

. Do tan 0kx k= = , suy ra

k l mt s chn (v 0nx , vi mi n l s l). V vy ( )2 2 1tk m= + . Tip tc l lun nh trn, ta c

( ) ( )1 2 12 2 12 2 10,..., 0, 0t mmmx x x +++ = = = (v l).

Vy 0nx , vi mi *n .

(b) Gi s dy { }nx tun hon, tc l tn ti m + sao cho ,m n nx x n ++ = . Suy ra

( )( )

sintan tan 0 sin 0

cos cos

mm n n m

m n n

+ = = =+

.

Do , tan 0mx m= = (v l). Vy dy { }nx khng tun hon.

11

Mt s bi tp t luyn Bi 1. Xt tnh tun hon v tm CKCS (nu c) ca cc hm s sau a) ( ) sin cosf x x x= + b) ( ) cos sin 2f x x x = + c) ( ) 2 cosf x x x=

d) ( ) xf x x nn

=

e) ( ) ( )1 cos3

xf x x = +

Bi 2. Chng minh rng nu th hm s ( )f x c tm trc i xng ( ),E a b v c trc i xng ( )x c c a= , th ( )f x l HSTH.

Bi 3. Cho ( )f x l HSTH v lin tc trn , c CKCS 0T . Chng minh rng, vi mi

a th ( ) ( )0 0

0

a T T

a

f x dx f x dx+

= .

Bi 4. Cho ( )f x l HSTH, lin tc trn v ( )lim ,x

f x a a+

= . Chng minh rng vi

mi x , ta c ( )f x a= .

Bi 5. Cho ( ) ( ),f x g x l cc HSTH, lin tc trn v ( ) ( )lim ,x

f x g x a a+

= . Chng

minh rng ( ) ( )f x g x a= + , vi mi x . Bi 6. Cho hm s { }: \ 3f , sao cho tn ti s thc 0a> tha mn iu kin

( )( )( )

53

f xf x a f x

+ =

. Chng minh rng ( )f x l HSTH.

Bi 7. Cho hm s :f l HSTH sao cho tp hp ( ){ }|f n n cha v s phn t. Chng minh rng chu k ca hm s ( )f x l mt s v t.

Bi 8. Cho hai hm s ( ) ( ),f x g x xc nh trn v tn ti s thc 0a sao cho (i) ( ) ( ) ( )f x a f x g x+ = + ;

(ii) ( ) ( )( )

, 2, 2

g x ng x na

g x n

+ =

;

(iii) ( ) 1f x = nu 0 x a .

Chng minh rng nu ( ) 1g x th ( )0 2f x . Bi 9. Cho hm s

( )2

1:

2 2,

1:

2 tan 2

x kf x k

x kx

= += + +

.

12

Chng minh rng hm s ( ) ( ) ( )g x f x f ax= + l HSTH khi v ch khi a l s v t. Bi 10. Tm tt c cc a thc ( )f x vi h s thc sao cho ( )cos ,f x x l HSTH. Bi 11. Cho hm s :f tha mn cc iu kin sau (i) ( )f x l hm khng gim; (ii) ( )f x l HSTH. Chng minh rng ( )f x l hm hng. Bi 12. Dy s nguyn { }, 1,2,...nu n= c xc nh nh sau:

1 2 31990, 1989, 2000u u u= = = , *

3 2 119 9 1991,n n nu u u n+ + += + + .

Vi mi n , gi nr l s d trong php chia nu cho 1992. Chng minh rng dy { }nr l dy s tun hon.

Ti liu tham kho [1] Don Minh Cng, Nguyn Huy oan, Ng Xun Sn. Nhng bi ton s cp chn

lc (tp 1). NXB Gio Dc, 1986. [2] Nguyn V Thanh. Phng php chn lc gii ton lng gic. NXB C Mau, 1993. [3] Nguyn V Thanh. Chuyn bi dng s hc. NXB Tin Giang, 1993. [4] Nguyn Qu Dy, Nguyn Vn Nho. Tuyn tp 200 bi ton gii tch. NXB Gio Dc,

2000. [5] Phan Huy Khi. Ton nng cao cho hc sinh THPT i S (tp 1). NXB Gio Dc,

2000. [6] Nguyn Vit Hi. Khai thc nh ngha hm s tun hon. Tp ch Ton Hc v Tui

Tr, s 1/2000. [7] L Sng. Dy s v cc vn lin quan. NXB Nng, 1994. [8] Nguyn Trng Tun. Bi ton hm s qua cc k thi Olympic. NXB Gio Dc, 2004.