De Thi Va Bai Giai Olympic Toan Giai Tich 20062012)

1

TNG HP THI V

LI GII CHI TIT

THI OLYMPIC TON SINH VIN

MN GII TCH T NM 2006 N NM 2012

(L Phc L tng hp v gii thiu)

Thnh ph H Ch Minh, ngy 26 thng 3 nm 2013

2

Phn A.

CC THI

CHNH THC

3


MN GII TCH NM 2006

Bi 1.

Cho dy s ( )nx xc nh theo h thc sau 2

1 1 2 32, ... , 2n nx x x x x n x n .

Tnh 2006 .x

Bi 2.

Cho hm s ( )f x kh vi trn . Gi s tn ti cc s 0p v (0;1)q sao cho

( ) , ( ) ,f x p f x q x . Chng minh rng dy s ( )nx c xc nh bi h thc

0 10, ( )n nx x f x hi t.

Bi 3.

Tm tt c cc a thc ( )P x tha mn iu kin (0) 0,0 ( ) ( ), (0;1)P P x P x x .

Bi 4.

Cho hm s lin tc : [0;1] [0; )f . t 0

( ) 1 2 ( )x

g x f t dt v ta gi s rng lun c 2

( ) ( ) , [0;1]g x f x x . Chng minh rng 2( ) (1 )g x x .

Bi 5.

Tn ti hay khng hm s lin tc : [ ; ] [ ; ]f a b a b vi a b v tha mn bt ng thc

( ) ( ) , , [ ; ]f x f y x y x y a b v .x y

Bi 6.

Xc nh cc dy s ( )nx bit rng

2 1 3 2,n nx x vi 0,1,2,...n

4


MN GII TCH NM 2007

Bi 1.

Tnh tch phn 2 20 ln sin 1 sinI x x dx

.

Bi 2.

Cho dy s ( )nx c xc nh bi: 0 1 2 1

0

...2007, 2007 , 1nn

x x x xx x n

n

.

Tm lin h gia 1,n nx x vi 1.n T , tnh tng 2007

0 1 2 20072 4 ... 2S x x x x .

Bi 3.

Tm tt c cc hm s ( )f x tha mn iu kin sau 1 32 ( ) , 1.1 1

xf f x xx x

Bi 4.

Cho , , ,a b c l cc s thc .c b Dy s ( ),( )n nu v c xc nh bi cng thc 2

1 11 1

, , , 1n

n n kn n

k k

u bu uu a u v n

c u b c

.

Bit rng lim .nu Tnh gii hn ca lim .nv

Bi 5.

Cho hm s ( )f x xc nh v kh vi trn [0; ). Bit rng tn ti lim ( ) ( ) 1x

f x f x

.

Tnh lim ( )x

f x

.

Bi 6.

Chng minh rng nu tam thc bc hai 2( )f x ax bx c vi , ,a b c v 0a c hai nghim thc phn bit th c t nht mt nguyn hm ca n l a thc bc ba c cc nghim u l s thc.

5


MN GII TCH NM 2008

Bi 1.

Dy s ( )na c xc nh nh sau 1 2 21

11, , 1,2,3...n nn

a a a a na

Tnh 2008a .

Bi 2.

Tnh gii hn 2008 2008 2008 2008

2009

1 2 3 ...limn

nn

.

Bi 3.

Gi s hm s ( )f x lin tc trn [0; ] v (0) ( ) 0f f tha mn ( ) 1f x vi

(0; )x . Chng minh rng:

i. Tn ti (0; )c sao cho ( ) tan ( )f c f c .

ii. ( )2

f x vi mi (0; ).x

Bi 4.

Cho hm s ( )f x lin tc trn [0;1] tha mn iu kin ( ) ( ) 1xf y yf x vi , [0;1]x y .

Chng minh rng 1

0( )

4f x dx .

Bi 5.

Gi s hm s ( )f x lin tc trn [0;1] v (0) 0, (1) 1f f , kh vi trong (0;1) . Chng

minh rng vi mi (0;1) , lun tn ti 1 2, (0;1)x x sao cho 1 2

1 1( ) ( )f x f x

.

Bi 6.

Cho hm s ( )g x c ( ) 0g x vi mi .x Gi s hm s ( )f x xc nh v lin tc

trn tha mn cc iu kin (0) (0)f g v 20

(0)( ) (0)2

gf x dx g

. Chng minh

rng tn ti [0; ]c sao cho ( ) ( ).f c g c

6


MN GII TCH NM 2009

Bi 1.

Gi s dy s ( )nx c xc nh bi cng thc 1 2

1 2

1, 1,( 1) , 3,4,5,...n n n

x xx n x x n

Tnh 2009x .

Bi 2. Cho hm s : [0;1]f c o hm cp hai, lin tc v c ( ) 0f x trn [0;1] . Chng

minh rng 1 1 2

0 02 ( ) 3 ( ) (0)f t dt f t dt f .

Bi 3.

Tm tt c cc hm s :f tha mn iu kin ( ) 4 2009 ,( ) ( ) ( ) 4, ,

f x x xf x y f x f y x y

.

Bi 4. Gi s ( ), ( )f x g x l cc hm s lin tc trn tha mn ( ( )) ( ( )), .f g x g f x x Chng minh rng nu phng trnh ( ) ( )f x g x khng c nghim thc, phng trnh

( ( )) ( ( ))f f x g g x cng khng c nghim thc.

Bi 5. Cho hai dy s ( ),( )n nx y xc nh bi cng thc

21 1 1 1 2

3 , 1 , , 1,2,3,...1 1

nn n n n

n

yx y x x x y n

y

Chng minh rng (2; 3)n nx y vi 2,3,4,...n v lim 0nn y .

Bi 6. (Th sinh chn mt trong hai cu)

a) Cho ( )P x l a thc bc n c h s thc. Chng minh rng phng trnh 2 ( )x P x c khng qu 1n nghim thc.

b) Cho 3( ) , ( )f x x f x x l nhng hm s n iu tng trn . Chng minh rng hm

s 23( )2

f x x cng l hm s n iu tng trn .

7


MN GII TCH NM 2010

Bi 1. Cho hm s ( ) ln( 1).f x x

a. Chng minh rng vi mi 0,x tn ti duy nht s thc c tha mn ( ) ( )f x xf c m

ta k hiu l ( ).c x

b. Tnh gii hn 0

( )lim .x

c xx

Bi 2.

Cho dy ( )nx xc nh bi 20101 11, 1n n nx x x x vi 1,2,3,...n

Tnh gii hn sau 20102010 2010

1 2

2 3 1

lim ... nx

n

xx xx x x

.

Bi 3. Cho s thc a v hm s ( )f x kh vi trn [0; ) tha mn cc iu kin (0) 0f v

( ) ( ) 0f x af x vi mi [0; )x . Chng minh rng ( ) 0f x vi mi 0.x

Bi 4.

Cho hm s ( )f x kh vi lin tc trn [0;1] . Gi s rng 1 1

0 0( ) ( ) 1f x dx xf x dx . Chng

minh rng tn ti im (0;1)c sao cho ( ) 6.f c

Bi 5.

Cho a thc ( )P x bc n c h s thc sao cho ( 1) 0P v ( 1)( 1) 2

P nP

. Chng minh

rng ( )P x c t nht mt nghim 0x vi 0 1.x


a. Xc nh hm s ( )f x kh vi lin tc trn [0;1] m (1) (0)f ef v 2

1

0

( ) 1( )

f x dxf x

.

b. Tm tt c cc hm s :f lin tc tha mn (1) 2010f v

( ) 2010 ( ) 2010 ( )yxf x y f y f x vi mi , .x y

8

THI OLYMPIC TON SINH VIN MN GII TCH NM 2011

Bi 1.

Cho hm s 2( ) ( 1)

xef xx

.

a. Chng minh rng ( )f x x c nghim duy nht trong 1 ;12

v ( )f x ng bin.

b. Chng minh rng dy ( )nu xc nh bi 1 11, ( )n nu u f u tha mn 1 ;1 ,2n

u n

.

Bi 2.

Tnh tch phn 1

0 2 4 21 3 1dx

x x x x .

Bi 3.

Cho hai dy s ( )nx v ( )ny tha mn 1 2n n

n

x yx

v

2 2

1 2n n

n

x yy

vi n .

a. Chng minh rng cc dy ( )n nx y , ( )n nx y l nhng dy n iu tng.

b. Gi s rng ( ),( )n nx y b chn. Chng minh rng chng cng hi t v mt im.

Bi 4.

Cho , tha mn iu kin *1 11 1 ,n n

e nn n

. Tm min .

Bi 5. Ta gi on thng [ , ] l on thng tt nu vi mi b s , ,a b c tha mn iu kin

2 3 6 0a b c th phng trnh 2 0ax bx c c nghim thc thuc on [ , ] . Trong tt c cc on thng tt, tm on c di nh nht.


a. Tm hm s :f tha mn 2 2( ) ( ) ( ) ( ) 4 ( ), ,x y f x y x y f x y xy x y x y .

b. Cho hm s f lin tc trn 1 ; 22

v tha mn iu kin 1 1( ) 2xf x fx x

vi mi

1 ; 22

x

. Chng minh rng 212

( ) 2 ln 2f x dx .

9


MN GII TCH NM 2012

Bi 1.

Cho dy s ( )na tha mn iu kin 1a v 11 2

n nna a

n n

vi 1,2,3,...n Tm

dy ( )na hi t.

Bi 2. Cho a thc ( )P x c bc khng nh hn 1 c h s thc v a thc ( )Q x xc nh bi

2 2 2( ) (2012 1) ( ) ( ) 2012 ( ( )) ( ( ))Q x x P x P x x P x P x . Gi s ( ) 0P x c ng n

nghim thc phn bit trong khong 1 ;2

, chng minh ( ) 0Q x c t nht 2 1n

nghim thc phn bit. Bi 3.

Tnh tch phn 1

21 (2012 1)(1 )xdx

x .

Bi 4.

Tm tt c cc hm s :f tha mn 1 , ,2012 2 2013 2014x y yxf f f x y

.

Bi 5. Gi s hm s ( )f x lin tc trn on [0; 2012] v tha mn ( ) (2012 ) 0f x f x vi

mi [0; 2012]x . Chng minh 2012

0( ) 0f x dx v

2012

0( 2012) ( ) 2012 ( )

xx f x f u du

c

nghim trong khong (0; 2012). Bi 6. (Th sinh chn mt trong hai cu)

a. Cho hm s ( )f x kh vi lin tc cp 2 trn . Gi s (1) 0f v 10

( ) 0f x dx . Chng

minh rng vi mi (0;1) , ta c 0 0 1

2( ) max ( )81 x

f x dx f x

.

b. Cho hm s : [0;1]f l hm lm (cn gi l li ln pha trn), kh vi lin tc tha

mn (0) (1) 0.f f Chng minh 1 22

00 1 0 11 4 max ( ) 1 ( ) 1 2 max ( )

x xf x f x dx f x

.

10

Phn B.

LI GII CHI TIT V BNH LUN

11

LI GII THI OLYMPIC TON SINH VIN

MN GII TCH NM 2006

Nhn xt chung.

Cc dng Ton c trong nhiu ti liu n thi v l kin thc ti thiu cn phi nm tip cn vi k thi ny xut hin cc cu 1, 5, 6. Cc cu cn li ni chung ch i hi dng cc k thut quen thuc nhng tinh t hn. Cu 2 l mt nh l hu ch x l cc bi Ton v gii hn dy s truy hi dng 1 ( )n nu f u nhng c pht biu dng tng qut nn gii y l

khng d dng. Cu 3, 4 l cc cu phn loi kh tt v cc hng tip cn c gii thiu bn di c l l con ng duy nht x l cc bi ny.

Bi 1.

Cho dy s ( )nx xc nh theo h thc sau:

21 1 2 32, ... , 2n nx x x x x n x n .

Tnh 2006 .x

Li gii.

Trong cng thc truy hi cho, thay n bi 1n ta c

21 2 3 1 1... ( 1)n n nx x x x x n x .

Suy ra 2 21 1 1 1( 1) ( 2) 2n n n n n n nnn x x n x nx n x x x

n

.

Ly tch hai v, ta c 1 1 11 1

! 42 ( 2)!/ (1 2) ( 1)( 2)

n n

i i ni i

i nx x x xi n n n

.

Do ta c cng thc tng qut ca dy cho l 4( 1)n

xn n

v 20064

2006 2007x

.

Nhn xt.

12

cc bi dy s c dng truy hi lin quan n tng hoc tch ca cc s hng lin trc nh trn, ta ch cn i n thnh 1n , li dng tnh cht ng vi mi n nh gi v trit tiu c mt lng kh ln cc s hng khc, hu ht cc trng hp l s a c v cng thc truy hi gia hai s hng lin tip. Di y l mt bi ton c cng dng:

Cho dy s thc ( )nx xc nh bi

1

1 21

21,( 1)

n

n ii

nx x xn

vi 2,3,4,...n

Vi mi s nguyn dng ,n t 1n n ny x x .

Chng minh rng dy s ( )ny c gii hn hu hn khi n .

Bi 2.

Cho hm s ( )f x kh vi trn . Gi s tn ti cc s 0p v (0;1)q sao cho

( ) , ( ) ,f x p f x q x .

Chng minh rng dy s ( )nx c xc nh bi h thc 0 10, ( )n nx x f x hi t.

Li gii.

Hm s ( )f x cho kh vi nn lin tc trn .

Vi mi ,x y m x y , theo nh l Lagrange th tn ti ( , )z x y sao cho

( ) ( ) ( )( ) ( ) ( ) ( ) ( )f x f y f z x y f x f y f z x y .

Do ( ) ,f x q x nn ta c ( ) ( ) ( )f x f y q x y .

Xt hm s ( ) ( )g x f x x th

( ) ( ) 0g p f p p v ( ) ( ) 0g p f p p (do ( ) ,f x p x )

Hm s ( )g x lin tc trn [ ; ]p p nn phng trnh ( ) 0g x c nghim trn [ ; ]p p .

13

Gi s phng trnh ( )f x x c hai nghim l u v th theo nh l Lagrange, tn ti s

( , )t u v sao cho ( ) ( ) ( ) ( ) ( ) 1f u f v f t u v u v f t u v f t , mu thun

do theo gi thit th ( ) 1,f x q x .

Do , phng trnh ( )f x x c nghim duy nht, t l .L R rng ( ) .f L L

Tip theo, ta thy rng 1 1( ) ( )n n nu L f u f L p u L . Lp lun tng t, ta c

0n

nu L p u L vi mi .n

Do (0;1)p nn 0np khi n , theo nguyn l kp th dy ( )nu hi t v .L

Nhn xt.

Nh nu trn, y l mt nh l tng qut x l cc bi ton gii hn c dng

1 ( )n nu f u , trong ( ) 1f x q . Trong tnh hung c th, s q rt quan trng v nu

ta khng ch c s tn ti ca n m mi ch c ( ) 1f x th li gii vn cha th

thnh cng. Hy th p dng lp lun trn, gii cc bi ton sau :

(1) Cho dy s nx c xc nh bi:

1

2 2 21

,2011 ln 2011 2011

3n n

x a

x x

Chng minh rng dy s nx c gii hn.

(2) Cho s thc a v dy s thc { }nx xc nh bi

1 1, ln(3 cos sin ) 2008n n nx a x x x vi mi 0,1,2,...n

Chng minh rng dy s nx c gii hn hu hn khi n tin n dng v cng.

Bi 3.

Tm tt c cc a thc ( )P x tha mn iu kin

(0) 0,0 ( ) ( ), (0;1)P P x P x x .

14

Li gii.

Trong bt ng thc cho, tnh lim cc v, ta c

10 lim ( ) (1) 0

xP x P

.

Xt hm s ( ) ( )xf x e P x th

( ) ( ) ( ) ( ( ) ( )) 0x x xf x e P x e P x e P x P x vi mi (0;1)x .

Do , hm s ( )f x nghch bin trn (0;1) .

Ta suy ra 0 10 1

lim ( ) ( ) lim ( ) (0) ( ) (1) (0) ( ) (1)xx x

f x f x f x f f x f e P e P x e P

.

Do (0) 0P v (1) 0P nn 0 ( ) 0P x dn n ( ) 0P x vi mi (0;1).x

iu ny c ngha l ( )P x nhn tt c cc gi tr (0;1)x lm nghim, nhng a thc ( )P x bc dng ch c hu hn nghim nn suy ra ( ) 0.P x

Vy tt c cc a thc cn tm l ( ) 0P x vi mi .x

Nhn xt.

y ta cn ch rng hm a thc lin tc trn c min s thc nn c th thoi mi p dng cc tnh cht ca hm s lin tc. Thm vo , k thut chn hm s ( )f x c

dng nh trn rt thng gp trong cc thi Olympic Sinh vin v trong nhiu trng hp, cc hm chn ra c kh rc ri. a thm xe c hai im li: th nht l gi tr ca xe lun dng nn du ca ( ), ( )f x P x lun nh nhau; th hai l khi o hm th

chng ta nhn c biu thc dng ( ) ( )P x P x v tn dng thnh cng gi thit.

Bi 4.

Cho hm s lin tc : [0;1] [0; )f . t 0

( ) 1 2 ( )x

g x f t dt v ta gi s rng lun c 2

( ) ( ) , [0;1]g x f x x . Chng minh rng 2( ) (1 )g x x .

Li gii.

15

t ( )F x l hm s tha mn 0

( ) ( )x

F x f t dt . Suy ra ( ) 1 2 ( )g x F x v ( ) ( )F x f x .

Theo gi thit th

21 2 ( ) ( ) ( )F x g x f x nn ( ) 2 ( )1 11 2 ( ) 2 1 2 ( )

f x F xF x F x

.

Ta cn chng minh 21 2 ( ) (1 ) 1 2 ( ) (1 ) 0F x x F x x .

Xt hm s ( ) 1 2 ( ) (1 )h x F x x th ta c 2 ( )( ) 1 02 1 2 ( )

F xh xF x

nn ( )h x nghch

bin trn [0;1] . Suy ra ( ) (0) 1 2 (0) 1h x h F .

Ch rng 0

0

(0) ( ) 0F f t dt nn (0) 0h . Do ( ) 0h x vi mi [0;1]x hay

2( ) (1 )g x x vi mi [0;1]x .

Nhn xt.

Mt s bn n on 2 ( ) 12 1 2 ( )

F xF x

s tnh nguyn hm hai v v suy ra

1 2 ( )F x x vi [0;1]x , dn n 2 2( ) 1 2 ( ) ( 1)g x F x x x .

y l mt sai lm rt nghim trng!

Bi ton c sng to ra kh th v khi kt hp gia cc iu kin lin h gia hm s v tch phn ca n t a v kho st hm s v o hm. trn ta xt o hm ca cn bc 2, ta hon ton c th thay bng cn bc n v to ra cc bi ton tng t.

Bi 5.

Tn ti hay khng hm s lin tc : [ ; ] [ ; ]f a b a b vi a b v tha mn bt ng thc

( ) ( ) , , [ ; ]f x f y x y x y a b v x y ?

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16

Li gii.

Ta c ( )a f a b v ( )a f b b nn ( )b f b a . Do , ta c

( ) ( ) ( )a b f a f b a b

Suy ra ( ) ( )f a f b a b .

Tuy nhin, trong iu kin ( ) ( ) , , [ ; ]f x f y x y x y a b , nu ta thay ,x a y b th

c bt ng thc ( ) ( )f a f b a b , mu thun.

Vy khng tn ti hm s tha mn bi.

Nhn xt.

Cu hi dnh cho bi ton ny n gin n bt ng. Nu i iu kin trong bi trn

thnh ( ) ( ) , [ ; ]f x f y x y x y a b th ta c th chng minh c rng phng

trnh ( )f x x c nghim duy nht trn [ ; ]a b . ng thi, hm s tng ng trong

trng hp l tn ti, chng hn ( )2

a bf x vi mi [ ; ]x a b .

Bi 6.

Xc nh cc dy s ( )nx bit rng

2 1 3 2,n nx x vi 0,1,2,...n

Li gii.

T cng thc xc nh dy, ta c

2 1 1 3( 1)n nx x vi 0,1,2,...n

t 1n ny x th ta c

2 1 3n ny y vi 0,1,2,...n

Thay 1n m th ta c 2 2 13m my y hay 2 3m my y vi mi 1,2,3,...m

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17

Tip tc t 2log 3m my m u th ta c 2 2 22 2

2 log 3 log 3 log 3(2 ) 3m m m

m m

y y yu u

m m m .

Khi , ( )mu l hm nhn tnh chu k 2 v ta c c

2 1n ku u nu n c dng 2 (2 1)mn k vi m v k ;

nu l hm ty vi cc trng hp cn li.

Do , ta c c 2log 31 1m m mx y m u v mu xc nh nh trn.

Nhn xt.

x l cc bi ton xc nh dy s dng ny, ta ch cn thc hin ln lt cc thao tc:

(1) Kh s hng t do.

(2) a ch s v dng kn nx x , tc l dy s y c dng mt hm nhn tnh.

(3) Vit cng thc tng qut cho hm nhn tnh v kt lun.

Trong mt s trng hp, vic kh h s t do cng khng n gin, ta cn s dng thm mt s kin thc phi hp. Chng hn, nu dy c dng 3 1n nx x , ta c th t

3( )n ny x v n vi 3( )v n l s m ln nht ca 3 trong khai trin n thnh tha s

nguyn t. Khi , d thy 3 3(3 ) ( ) 1v n v n v 3 .n ny y

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18


MN GII TCH NM 2007

Nhn xt chung.

ny cha nhiu yu t ca Ton s cp, cc phn dy s, phng trnh hm v thm ch l bi tch phn cng ch i hi cc k thut x l quen thuc ca THPT, c th l trong chng trnh thi HSG. thi nhn chung c tnh phn loi kh cao nhng cha mang nhiu du n ca mt thi Olympic Ton cao cp dnh cho Sinh vin H.

Bi 1.

Tnh tch phn 2 20 ln sin 1 sinI x x dx

.

Li gii.

Ta c

2 2

0

22 2

0

02 2

0

12 2

0 0 0

ln sin 1 sin

ln sin 1 sin ln sin 1 sin

ln sin 1 sin ln sin(2 ) 1 sin (2 ) (2 )

ln sin 1 sin ln sin 1 sin ln1 0

I x x dx

x x dx x x dx

x x dx x x d x

x x dx x x dx dx

Vy tch phn cn tnh l 0.I

Nhn xt.

Ta cng c th gii bi ton bng hm sinh2

x xe ex

v 1 2sinh ln 1x x x . Ni chung cc nguyn hm ca hm s dng hm lng gic nm trong hm logarit thng rt phc tp v ta ch tnh c tch phn vi cc cn thch hp. Trong nhiu trng hp, ta cn cn phi s dng n cc k thut kh hn, chng hn nh a thm tham s vo ri i vai tr gia tham s v bin.

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19

Bi 2.

Cho dy s ( )nx c xc nh nh sau:

0 1 2 10

...2007, 2007 , 1nn

x x x xx x n

n

.

Tm h thc lin h gia 1,n nx x vi 1.n T , tnh tng

20070 1 2 20072 4 ... 2S x x x x .

Li gii.

T cng thc xc nh dy, ta c

0 1 2 12007 ...n nnx x x x x .

Thay n bi 1,n ta c

1 0 1 2 1( 1) 2007 ...n n nn x x x x x x

Tr tng v cc ng thc, ta c

1 12007( 1) 2007

1n n n n nnn x nx x x x

n

vi mi n .

Do , ta c lin h gia nx v 1nx l 12008

n nnx x

n

vi mi 1.n

Ta tnh c 2 1 2 21 2007 2 20072006 2006 20072007 2007 , ( 2007 ) 2007 20072 1 2

x C x C

.

Ta s chng minh bng quy np rng 20072007( 1)k k

kx C vi 0 2007.k (*)

Tht vy,

- Vi 0k th (*) ng.

- Gi s (*) ng vi 0k , tc l 20072007( 1)k k

kx C , ta c

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20

1 11 2007

2007 2007 (2008 ) ... 20072007( 1) 2007( 1)1 1 1 2 3...

k k kk k

k k kx x Ck k k

.

Do , (*) cng ng vi 1k .

Theo nguyn l quy np th (*) ng vi mi 0 2007.k

T , ta tnh c 2007 2007

2007 20072007

0 02 2007 1 ( 2) 2007(1 2) 2007i k k ki

i iS x C

.

Vy biu thc cn tnh c gi tr l 2007.S

Nhn xt.

Nhiu bn s x l c cng thc truy hi trn v y l dng quen thuc v t nht l nm 2006 trc c mt bi tng t. Tuy nhin, vic rt gn c tng S i hi ta phi tm c cng thc tng qut ca nx v y chnh l im th v ca bi

ton ny. R rng nu chng ta chu kh tnh th vi s hng u 0 1 2 3, , ,x x x x th c th

d on c v cng vic cn li l quy np d dng (ta cng phi cn c trn dng

ca biu thc S l c cha cc ly tha ca 2 tng dn mi c c s ngh n 2007kC ).

Bi 3.

Tm tt c cc hm s ( )f x tha mn iu kin sau

1 32 ( ) , 1.1 1

xf f x xx x

Li gii.

t 111 1

yxy xx y

, iu kin 1.y

Thay vo phng trnh hm cho, ta c

1 3( ) 211 11

yf y fyyy

hay 1 3( 1)( ) 21 2

y yf y fy

.

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21

i v bin ,x ta c 1 3( 1)( ) 21 2

x xf x fx

.

Nhn ng thc cho vi 2 ri cng vi ng thc ny, ta c

1 6 1 3( 1)2 ( ) 4 ( ) 21 1 1 2

x x xf f x f x fx x x

Suy ra 3( 1) 6 1 23 ( ) 0 ( )2 1 2 1

x xf x f xx x

vi mi 1x .

Th li ta thy tha.

Vy tt c cc hm s cn tm l 1 2( ) , 12 1

xf x xx

v (1)f l mt s ty .

Nhn xt.

Trong trng hp ny, ch bng mt php t n ph l ta a c phng trnh hm cho v dng mt h phng trnh hm hai bin v gii d dng. iu ny c

ngha l hm s 1( )1

xxx

tha mn ( ( ))x x . Trong nhiu trng hp, vic t n

ph ny cn phi thc hin nhiu ln v ta phi gii mt h gm nhiu phng trnh

hn, hy thay 1( )1

xxx

trong phng trnh hm cho bi 1( )1

xxx

thy r

vn (nu kin nhn tnh ton, ta s thy rng ( ( ( ( ))))x x ).

Mt c im cn ch ca bi ton l vic kt lun gi tr (1).f Do khng c d

kin xc nh nn ta s cho n nhn gi tr ty . Ta th xt mt bi ton tng t:

Xc nh tt c cc hm s ( )f x tha mn 3 31 1f x xx x

vi mi 0.x

Bi ny gii c d dng bng cch t thm 1t xx

v c c 3 331 3x t tx

. Tuy

nhin, nu kt lun 3( ) 3f x x x th r rng l rt thiu st v gi thit ch c lin quan

n 2t , tng ng vi 2x nn ta khng c d kin xc nh ( )f x vi

2x . Do , ta ni ( )f x nhn gi tr ty vi 2x .

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22

Bi 4.

Cho , , ,a b c l cc s thc tha .c b Dy s ( ),( )n nu v c xc nh bi cng thc

2

1 11 1

, , , 1n

n n kn n

k k

u bu uu a u v n

c u b c

.

Bit rng lim .nu Tnh gii hn ca lim .nv

Li gii.

Ta biu din 1

k

k

uu b c

di dng 1

1 1

k k

ru s u s

vi ,r s c nh s xc nh sau.

Ta c

2

1

1 1 1 1

( )1 1( )( ) ( )( ) ( )( )

k kk

k kk k

k k k k k k k k

u bur u

c ru u b cr u ur

u s u s u s u s u s u s u s u s

.

So snh vi biu thc ban u, ta chn s b c v 1r ; khi , ta c

1 1

1 1kk k k

uu b c u b c u b c

.

Do 1 11 1 1 1

1 1 1 1n nkn

k kk k k n

uv

u b c u b c u b c u b c u b c

.

Do lim nu c b v 1u a nn ta c

1 1lim( )( )n

ava b c b c a b c b c

.

Nhn xt.

Bi ton tuy cha nhiu tham s nhng ta ch cn nu ra c biu thc dng sai phn trn l c th gii quyt n nhanh chng. Thng thng th cc bi ny s c nu di dng cc s c th v chng ta on biu thc dng sai phn (hu ht l phn s nh trn) ri s dng tham s bin thin vic lp lun c r rng.

Di y l mt s bi ton tng t:

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23

(1) Cho dy s nu xc nh bi 1

21

31 4 , 1,2,3,...5n n n

u

u u u n

t 1

1 , 1,2,3,...3

n

nk k

v nu

. Tnh lim nn v .

(2) Cho dy s ( )nu xc nh bi 1

21

2 3

3 2 2 6 5 3 3 3 2 , 1.n n n

u

u u u n

t 1

1 , 1,2,3,...2

n

nk k

v nu

Tm lim nv .

Bi 5.

Cho hm s ( )f x xc nh v c o hm trn [0; ). Bit rng tn ti gii hn

lim ( ) ( ) 1x

f x f x

.

Tnh lim ( )x

f x

.

Li gii.

Xt hm s ( ) ( ) xg x f x e th ( )g x lin tc v c o hm trn [0; ). Ta c

( )lim ( ) lim xx xg xf xe

.

Theo quy tc LHospital th

( ) ( )( ) ( )lim lim lim lim ( ) ( ) 1

( )

x

x x xx x x x

f x f x eg x g x f x f xe e e

.

T suy ra lim ( ) 1x

f x

.

Nhn xt.

Bi ton khng thay i nu thay 1 bi mt s dng ty . Hy th gii bi ton tng t gii hn sau:

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Cho hm s ( )f x xc nh v c o hm trn [0; ). Bit rng tn ti gii hn

lim ( ) 2 ( ) 1x

f x x f x

.

Tnh gii hn lim ( )x

f x

.

Bi 6.

Chng minh rng nu tam thc bc hai 2( )f x ax bx c vi , ,a b c v 0a c hai nghim thc phn bit th c t nht mt nguyn hm ca n l a thc bc ba c cc nghim u l s thc.

Li gii.

Xt hm s 3 2

( )3 2

ax bxg x cx th r rng ( ) ( )g x f x c hai nghim thc nn hm s

bc ba ( )g x c hai im cc tr.

Gi 1 2x x l hai nghim ca phng trnh ( ) 0f x th y cng chnh l hai im cc

tr ca hm s ( )g x .

Khi , vi cc gi tr m nm gia 1 2( ), ( )g x g x th ng thng y m s ct ng

cong ( )y g x ti 3 im v phng trnh ( )g x m c ng 3 nghim.

Do hm s 3 2

3 2ax bx cx m c 3 nghim u thc v y cng chnh l nguyn

hm cn tm ca ( ).f x Ta c pcm.

Nhn xt.

y l mt bi ton nh kim tra kin thc v kho st hm s. Ngoi cch gii trn, ta cn nhiu cch chng minh s tn ti ca nguyn hm c 3 nghim thc, chng hn ta c th ch trc tip nguyn hm , c th l

( ) ( )2bh x g x ga

.

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25


MN GII TCH NM 2008

Nhn xt chung.

thi kh hay v bao qut cc dng, i hi phi nm vng y cc kin thc lin quan nh: x l dy s, tnh gii hn dng k thut tch phn xc nh, kho st tnh n iu ca hm s, nh l Lagrange v khai trin Taylor. Cc cu cng c sp theo kh tng dn v ch dng li mc trung bnh ch khng c bi kh. Tuy mt s cu cha tht mi m, sng to nhng i hi phi tp trung khai thc su gi thit cng nh c im ca cc kt lun th mi c th gii quyt trn vn c.

Bi 1.

Dy s ( )na c xc nh nh sau

1 2 21

11, , 1,2,3...n nn

a a a a na

Tnh 2008a .

Li gii.

Theo gi thit, ta c 2 1 1 1n n n na a a a vi mi 1,2,3,...n Do dy s 1n n nu a a l

mt cp s cng vi s hng u l 1 1u v cng sai l 1. Khi ,

21

1 1n n

n

n na aa n

vi 1,2,3,...n

T suy ra 2008 22007 2005 3 3 5 7 ... 2007 2007!!... .2006 2004 2 2 4 6 ... 2006 2006!!

a a

Nhn xt.

Trong cc bi ton dy s phi tuyn tnh dng ny, ta khai thc c im ca cng thc a v mt biu thc d x l hn, y ta a c v sai phn dng tch.

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26

Ch cc k hiu (2 )!! 2 4 6 ... 2n n v (2 1)!! 1 3 5 ... (2 1)n n .

T cng thc ca dy trn, hy kim tra th kt qu lim 0nn

an ?

Bi 2.

Tnh 2008 2008 2008 2008

2009

1 2 3 ...limn

nn

.

Li gii.

Ta c 2008 2008 2008 20082008 2008 2008 2008

20091

1 2 3 ... 1 1 2 1...n

i

n n iSn n n n n nn

.

Xt hm s 2008( )f x x th r rng ( )f x kh tch trn [0;1] . Chia on [0;1] thnh cc

on con bi cc im iixn

v chn 1[ , ]i i iic x xn

vi 1,2,3,...,i n . Ta c

20081 1 2008

0 01 1

1 1 1lim lim ( )2009

n n

n ni i

i if f x dx x dxn n n n

.

Vy gii hn cn tnh l 1 .2009

Nhn xt.

Phng php s dng trong bi ny l dng nh ngha tch phn xc nh tnh gii hn. c im ca cc bi ton dng ny rt d nhn bit, chnh l ch cn mt s

bin i, ta c th a c v 1( )

n

if i

n

. y c l l phng php tt nht v duy nht

gii bi ny, trong nhiu trng hp, hm s ( )f x cng khng d dng nhn ra.

Ta th xem xt cc bi ton sau:

(1) Tnh gii hn sau 1 1 1lim ...1 2n n n n n

.

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27

(2) Tnh gii hn sau 1 2 ( 1)lim sin sin ... sinn

nn n n n

.

(3) Tnh gii hn sau 2 2 2 2

2 2 3 3 3 3 2 3

1 2 3lim ...2 4 8 (2 )n

nn n n n n

.

Bi 3.

Gi s hm s ( )f x lin tc trn [0; ] v (0) ( ) 0f f tha mn iu kin

( ) 1f x vi mi (0; )x .

Chng minh rng

i. Tn ti (0; )c sao cho ( ) tan ( )f c f c .

ii. ( )2

f x vi mi (0; ).x

Li gii.

i. Xt hm s ( ) sin ( )xg x e f x th r rng ( )g x lin tc trn [0; ] , kh vi trong (0; )

v (0) ( ) 0g g . Theo nh l Rolle th tn ti (0; )c sao cho ( ) 0g c .

Ta cng c ( ) sin ( ) cos ( ) ( )xg x e f x f x f x nn

sin ( ) cos ( ) ( ) 0 ( ) tan ( )f c f c f c f c f c .

Do , gi tr c ny tha mn bi.

ii. Vi mi (0; )x c nh, ch rng ( ) 1, (0; )f x x nn theo nh l Lagrange:

- Tn ti 1 (0; )c x sao cho 1( ) ( ) (0) ( ) 0f x f x f f c x x .

- Tn ti 2 ( , )c x sao cho 2( ) ( ) ( ) ( )f x f f x f c x x .

Do (0; )x nn min , 2x x

. Suy ra ( ) min , 2f x x x

. Ta c pcm.

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28

Nhn xt.

cu hi u tin, ta phn tch ngc li t biu thc cn chng minh

sin ( )( ) tan ( ) ( ) cos ( ) ( ) sin ( )cos ( )

f cf c f c f c f c f c f cf c

.

Ch rng sin ( ) cos ( ) ( )f c f c f c nn ng trn trn c dng ti mt im no , gi tr ca hm s bng gi tr ca o hm. Nhng x l iu ny th ni chung khng cn xa l g vi vic s dng hm s c dng ( )xe f x . th hai ca bi ton i

hi phi nh gi min gi tr ca hm s thng qua min gi tr cho trc ca o hm, vic ny thng c gii quyt nh tnh n iu ca hm s hoc nh l Lagrange. Do , ty vo tnh hung m chng ta c th la chn cc cng c ph hp.

Bi 4.

Cho hm s ( )f x lin tc trn [0;1] tha mn iu kin

( ) ( ) 1xf y yf x vi mi , [0;1]x y .

Chng minh rng 1

0( )

4f x dx .

Li gii.

t sinx vi [0; ]2

th ta c

12

0 0( ) (sin )cosI f x dx f d

.

Mt khc, nu t cos , [0; ]2

x th ta c

12

0 0( ) (cos )sinI f x dx f d

.

Do 2 2 20 0 0

2 (sin )cos (cos )sin (sin )cos (sin )cosI f d f d f t t f t t dt

.

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29

Theo gi thit th ( ) ( ) 1xf y yf x vi mi , [0;1]x y nn suy ra

20

22

I dt

hay 4I

.

Nhn xt.

Nhiu bn cho rng nh gi theo cch i bin thnh hm lng gic nh trn hi thiu t nhin v c v gi thit c s dng cha trit (gi thit cho bt ng thc ng vi mi ,x y v ta ch s dng mt ln khi t sin , cosx t y t ); tuy nhin, gi

thit c a ra hng ti ng thc c sn

1

20 0

2 ( ) (sin )cos (sin )cosf x dx f t t f t t dt

Bng chng l s 1 trong bt ng thc ( ) ( ) 1xf y yf x hon ton c th thay bng s

khc. V v th, rt kh khn khi tip cn bi ton theo tng dng bt ng thc i

s vi kinh nghim l 1

20 41dx

x

, a v chng minh 2

1( )1

f xx

vi mi [0;1]x .

T gi thit ( ) ( ) 1xf y yf x vi mi , [0;1]x y , trc mt, ta c 1( )2

f xx

vi [0;1]x .

Nhng nh gi ang yu cu trn li cht hn, khai thc tip l iu khng d dng!

Bi 5.

Gi s hm s ( )f x lin tc trn [0;1] v (0) 0, (1) 1f f , kh vi trong (0;1) . Chng

minh rng vi mi (0;1) th lun tn ti 1 2, (0;1)x x sao cho

1 2

1 1( ) ( )f x f x

.

Li gii.

Do ( )f x lin tc nn vi mi [0;1] th tn ti 0 (0;1)x sao cho 0( )f x .

Theo nh l Lagrange th tn ti 1 0(0; )x x v 2 0( ;1)x x sao cho

01

0

( ) (0)( )

0f x f

f xx

v 0 2

0

(1) ( )( )

1f f x

f xx

.

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30

T suy ra 1 20 0

1( ) , ( )1

f x f xx x

v

0 01 2 0 0

1 1 1 1( ) ( ) / (1 ) / (1 )

x xf x f x x x

.

Ta c pcm.

Nhn xt.

Bi ny th p dng nh l Lagrange l hon ton t nhin, nu ch ng gii th th c l im quan trng nht ta cn vt qua l nhn xt c tn ti 0 (0;1)x 0( )f x .

Nu bin i biu thc trn thnh 1 2 12 1

( ) ( ) ( )( ) ( )

f x f x f xa

f x f x

th li hon ton khng d x

l. Rt d hiu, l v chng minh tn ti mt s tha mn mt ng thc th d nhng chng minh tn ti hai s cng tha mn mt ng thc th khng n gin cht no.

Bi 6.

Cho hm s ( )g x c ( ) 0g x vi mi .x Gi s hm s ( )f x xc nh v lin tc trn tha mn cc iu kin

(0) (0)f g v 20

(0)( ) (0)2

gf x dx g

.

Chng minh rng tn ti [0; ]c sao cho ( ) ( ).f c g c

Li gii.

Xt hm s ( ) ( ) ( )h x g x f x th ( )h x cng lin tc. Theo gi thit th (0) 0h .

Ta cng c ( ) 0g x nn khai trin Taylor cho hm s ( )h x ti 0x v tnh tch phn

cho hm s ny, ta thu c

2

0 0 0 0 0

2

0 0 0 0

( )( ) ( ) ( ) (0) (0) ( )2

(0)(0) (0) ( ) (0) ( ) 02

gh x dx g x dx f x dx g g x dx f x dx

gg dx g dx f x dx g f x dx

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31

Suy ra tn ti [0; ]m sao cho ( ) 0h m .

Do tnh lin tc ca hm s ( )h x trn on [0; ]m th tn ti (0; ) [0; ]c m sao cho ( ) 0h c . T suy ra ( ) ( ).f c g c Ta c pcm.

Nhn xt.

Ta thy rng gi thit cho trc rt l liu 20

(0)( ) (0)2

gf x dx g

vi tng s

dng khai trin Taylor cho hm s. Do , trong bi ny, ch cn nm vng cng thc khai trin Taylor ca hm s l c th gii quyt tt bi ton:

2( ) ( )( ) ( ) ( )! ( ) ... ( )1! 2! n

f a f af x f a x a x a R x

vi ( )

1( )( ) ( )( 1)!

nn

nfR x x an

Chng ta hon ton c th da trn tng ny m pht biu mt bi ton tng t dng tng qut nh sau:

Cho cc s dng 0 a b v cc s nguyn dng r s . Xt hm s :g tha

mn ( )( ) 0sg x vi mi x v s l s nguyn dng no . Gi s hm s ( )f x xc nh v lin tc trn tha mn cc iu kin

( ) ( )f a g a v 1

( )( )!

rb i

ai

g af x dx bi

.

Chng minh rng tn ti [ ; ]c a b sao cho ( ) ( ).f c g c

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32


MN GII TCH NM 2009

Nhn xt chung.

Ngoi cu 2 kh ra th cc cu cn li thuc dng c bn, khng i hi nhiu k thut x l v thm ch l hon ton gii bng cc kin thc v gii tch s cp mt cch nh nhng. Cc cu 1, 3 v 5 cn nng tnh i s; cha th hin c vai tr ca hm s, i tng c bn trong Gii tch. Ni chung cc bn no nm vng thm k thut tch phn tng phn bin i tch phn th c th gii quyt trn vn thi ny kh nhanh chng.

Bi 1.

Gi s dy s ( )nx c xc nh bi cng thc

1 2

1 2

1, 1,( 1) , 3,4,5,...n n n

x xx n x x n

Tnh 2009x .

Li gii.

T iu kin cho, ta c

1 1 2( 1)n n n nx nx x n x vi mi 3,4,5,...n

t 1n n ny x nx th ta c 1n ny y v 2 2 12 1y x x nn 1( 1)nny

.

Suy ra 11 ( 1)n

n nx nx

hay 1

1 ( 1)! ( 1)! !

nn nx x

n n n

, t cng thc sai phn ny, tnh tng

hai v, ta c c

111

2 2

( 1)! 1! ! ( 1)! !

in nn i i

i i

x x xxn i i i

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33

Do 11 1 1 ( 1)1 ...

! 2! 3! 4! !

nnx

n n

hay

11 1 1 ( 1)! 1 ...2! 3! 4! !

n

nx n n

.

T ta tnh c 20091 1 1 1 12009! 1 ...2! 3! 4! 2008! 2009!

x

.

Nhn xt.

Cng nh nhiu bi ton truy hi phi tuyn khc, y ta cn tm cch t thm dy s mi n gin ha quan h truy hi (ch cn truy hi gia hai s hng lin tip).

Nu i s hng u tin thnh 1 0x th cng thc ca dy s c thay i mt t l:

0

1 1 1 1 ( 1) ( 1)! 1 ... !1! 2! 3! 4! ! !

n in

ni

x n nn i

vi mi .n

Xt dy s 0

1 1 1 1 1...0! 1! 2! ! !

n

ni

yn i

th ta c kt qu quen thuc lim nn y e (khai

trin Taylor ca xe ti 1x ). Lin h gia ,n nx y c cho bi cng thc

lim 1!

n nn

x yn

hay 1lim!n

n

xn e

.

Bi 2.

Cho hm s : [0;1]f c o hm cp 2 lin tc v ( ) 0f x trn [0;1] . Chng minh

rng 1 1 2

0 02 ( ) 3 ( ) (0)f t dt f t dt f .

Li gii.

Ta s dng tch phn tng phn vi 1 12

0 0( ) ( ) ( )f t dt f t d t .

t ( ), ( )u f t dv d t th ( )du f t v chn 1v t . Ta c

11 1 120 0 00( ) ( )( 1) ( ) 1 (0) ( ) 1f t dt f t t f t t dt f f t t dt .

Tip tc p dng tch phn tng phn vi 10 ( ) 1f t t dt , ta c

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34

31 12

0 0

(0) 2 1( ) 1 ( )3 3 3

ff t t dt f x x x dx

.

Do , 31 12 2

0 0

(0) 2 1( ) (0) ( )3 3 3

ff t dt f f x x x dx

.

Vi tch phn 1

0( )f t dt , ta t ( ),u f t dv dt th ( )u f t v chn 1v t , ta c

1 1 11

00 0 0( ) ( )( 1) ( 1) ( ) (0) (1 ) ( )f t dt f t t t f t dt f t f t dt

Tip tc p dng tch phn tng phn vi 1

0(1 ) ( )t f t dt , ta c

1 1 2

0 0

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

ft f t dt t f t dt

.

Do , 1 1 2

0 0

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

ff t dt f t f t dt

.

Bt ng thc cn chng minh chnh l

31 12 20 0

(0) (0)1 2 12 (0) (1 ) ( ) 3 (0) ( ) (0)2 2 3 3 3

f ff t f t dt f f t t t dt f

hay

31 12 20 0(1 ) ( ) ( ) 2 1 3t f t dt f t t t dt

.

Tuy nhin, d thy 3

2 2(1 ) 2 1 3 , [0;1]t t t t v bt ng thc ny tng ng vi

32 22 1 2t t t t t , ng theo BT Cauchy.

T ta c pcm.

Nhn xt.

Cch dng tch phn tng phn bin i tch phn v dng thch hp p dng gi thit cng kh ph bin v ng c ch . Nh n m ta chuyn hm s di du

tch phn dng 2( )f x thnh ( )f x v tn dng c ( ) 0, [0;1]f x x .

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35

Di y l mt bi tng t:

Cho : [0;1]f l mt hm kh vi cp 2 v tha mn ( ) 0, [0;1]f x x . Chng minh bt ng thc sau

1 1 2

0 02 (1 ) ( ) ( )x f x dx f x dx .

Bi 3.

Tm tt c cc hm s :f tha mn cc iu kin

( ) 4 2009 ,( ) ( ) ( ) 4, ,

f x x xf x y f x f y x y

.

Li gii.

t ( ) ( ) 4g x f x vi mi x th ta c ( ) 2009 , (1)( ) ( ) ( ), , (2)

g x x xg x y g x g y x y

.

Trong (1), thay 0x , ta c (0) 0g .

Trong (2), thay 0y , ta c ( ) ( ) (0) (0) 0g x g x g g .

Do , ta phi c (0) 0.g

Trong (1), thay x bi x , ta c ( ) 2009g x x , suy ra ( ) ( ) 0g x g x .

Trong (2), thay y x , ta c ( ) ( ) ( ) 0g x g x g x x .

Do , ta phi c ( ) ( ) 0g x g x hay ( ) ( )g x g x .

Kt hp vi ( ) 2009g x x , ta c ( ) 2009 ( ) 2009g x x g x x vi mi .x

T bt ng thc ny v (1), ta suy ra ( ) 2009g x x vi mi .x

Do ( ) 2009 4f x x vi mi x .

Th li, ta thy tha.

Vy tt c cc hm s cn tm l ( ) 2009 4f x x vi .x

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36

Nhn xt.

Bi ton ny c dng mt h bt phng trnh hm i s. T tng chnh gii quyt dng ny l tn dng cc bt ng thc :

- Hoc ch ra mt iu v l no .

- Hoc a v dng a b a th dn n ng thc a b phi xy ra.

Di y l mt s bi tng t:

(1) Tm tt c cc hm s :f tha mn

( ) ( ) ( ) 3 ( 2 3 )f x y f y z f z x f x y z vi mi , , .x y z

(2) Cho ,a b l cc s nguyn dng v nguyn t cng nhau. Xt hm s :f tha mn ng thi cc iu kin:

i) ( ) ( ) ,f x a f x a x .

ii) ( ) ( ) ,f x b f x b x .

Chng minh rng ( 1) ( ) 1f x f x vi mi x .

Bi 4.

Gi s ( ), ( )f x g x l cc hm s lin tc trn tha mn cc iu kin

( ( )) ( ( )), .f g x g f x x

Chng minh rng nu phng trnh ( ) ( )f x g x khng c nghim thc, phng trnh

( ( )) ( ( ))f f x g g x cng khng c nghim thc.

Li gii.

Xt hm s ( ) ( ) ( )h x f x g x , d thy ( )h x lin tc trn . T iu kin cho, ta thy ( )h x khng c nghim thc.

Gi s tn ti a b sao cho ( )h a v ( )h b tri du th ( ) 0h x c nghim nm gia ,a b , mu thun. Do ( )h x gi nguyn du trn c min .

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37

Khng mt tnh tng qut, ta c th gi s ( ) 0,h x x hay ( ) ( ),f x g x x .

Thay x bi ( )f x , ta c ( ) ( ) ( ) ( )f f x g f x f g x g g x vi mi .x

T suy ra phng trnh ( ( )) ( ( ))f f x g g x khng c nghim thc.

Nhn xt.

y ta p dng nh l trung gian cho hm lin tc, mt kt qu quen thuc l nu trn mt min ,D hm s v nghim th n khng i du trn .D nh l ny cn rt hu ch trong nhiu trng hp, chng hn gii mt bt phng trnh rc ri, ta chuyn v gii phng trnh tm cc nghim ri lp bng xt du l xong.

Di y l mt bi tng t vi bi ton trn:

Cho , : [0;1] [0;1]f g l cc hm s lin tc, tha mn ( ( )) ( ( ))f g x g f x vi mi .x

a. Chng minh rng tn ti 0 [0;1]x sao cho 0 0( ) ( ).f x g x

b. Gi s rng ( )f x n iu, chng minh tn ti 0 [0;1]x sao cho 0 0 0( ) ( ) .f x g x x

c. Hy cho phn v d trong trng hp thay min [0;1] bi .

Bi 5.

Cho hai dy s ( ),( )n nx y xc nh bi cng thc

21 1 1 1 2

3 , 1 , , 1,2,3,...1 1

nn n n n

n

yx y x x x y n

y

Chng minh rng (2; 3)n nx y vi 2,3,4,...n v lim 0nn y .

Li gii.

t 1 1cot 3x a vi 6a , ta c

2

22 2

2cos1 cos 1 2cot 1 cot cot cot cossin sin 2 2 32sin cos

2 2

aa ax a a a

a aa a

.

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38

Do , bng quy np, ta chng minh c cot2 3n n

x

vi 1n .

Tng t, t 1 tan 3y b vi 3b th ta c

2 2 2

2sin costan tan sin 2 2 tan tan1 1 cos 2 3 21 1 tan 1 2cos

cos 2

b bb b b by

bbbb

.

Bng quy np, ta cng chng minh c rng 1tan 3 2n ny

vi 1n .

T suy ra 12 2

2 tan 23 2cot tan cot3 2 3 2 3 2 1 tan 1 tan

3 2 3 2

n

n n n n n

n n

x y

vi mi

1n . Ta thy 2tan 03 2n

v 2 2 1tan tan6 33 2n

nn

2 3n nx y vi mi 2,3,4,...n

Ta cng c 1lim lim tan tan 0 03 2n nn ny

. Ta c pcm.

Nhn xt.

Cc biu thc c dng 2( ) 1f x x x v 2

( )1 1

xg xx

rt d gi cho ta ngh

n cc hm lng gic v (cot ) cot2xf x v (tan ) tan

2xg x . Sau khi tm c cc

cng thc tng qut th cng vic li l ch cn x l i s l xong. Do , vic ch cc c trng hm ca cc hm s quen thuc cng l mt cng vic c ch khi n gip ta

nhn bit bn cht ca vn nhanh chng hn. Chng hn 3

4 2

4 4( )6 1

x xf xx x

tha

mn tnh cht sau (tan ) tan 4 .f x x

Mt bi ton tng t:

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39

Cho dy s ( )nu xc nh bi 1 2 1 11, 2, 3 , 2n n nu u u u u n v xt dy s sau

1arc cot , 1,2,3,...

n

n ii

v u n

Tnh gii hn lim nn v .

Bi 6.

a) Cho ( )P x l a thc bc n c h s thc. Chng minh rng phng trnh 2 ( )x P x c khng qu 1n nghim thc.

b) Cho 3( ) , ( )f x x f x x l nhng hm s n iu tng trn . Chng minh rng hm

s 23( )2

f x x cng l hm s n iu tng trn .

Li gii.

a) Xt hm s ( ) 2 ( )xg x P x th d thy do ( )P x l a thc bc n nn

( 1) ( 1) 1( ) 0 ( ) 2 (ln 2) 0n n x nP x g x .

o hm cp 1n ca hm s ( )g x khng i du nn theo nh l Rolle th phng

trnh ( ) 0g x c khng qu 1n nghim. Ta c pcm.

b) Gi s tn ti a b sao cho 2 23 3( ) ( )2 2

f a a f b b , suy ra

3 3 2 3 3 23 3( ) ( )2 2

f a a a a f b b b b

v

2 23 3( ) ( )2 2

f a a a a f b b b b

.

Theo gi thit th 3( ) , ( )f x x f x x l cc hm s n iu tng trn nn

3 3( ) ( )f a a f b b v ( ) ( )f a a f b b .

T suy ra ,a b nm trong min m cc hm s 3 232

x x v 232

x x nghch bin.

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Ta thy 3 232

x x nghch bin trn 30;3

, cn hm s 232

x x th nghch bin trn

3 ;3

nn ta phi c ng thi 3 3 3, 0; ;

3 3 3a b

.

Tuy nhin a b nn iu ny khng th xy ra nn iu gi s l sai.

Vy hm s 23( )2

f x x l n iu tng trn .

Nhn xt.

cu a, ta thy rng phng trnh cho c dng hm s m bng a thc. Ta bit

rng o hm cp 1n ca a thc bc n l bng 0 cn ca hm s m dng xa th li khng i du nn x l hon ton d dng. Mt cu hi t ra l ng vi mi s

nguyn dng ,n c lun tn ti a thc bc n tha mn phng trnh 2 ( )x P x c ng 1n nghim khng?

Cu b ca bi ny tuy khng mi nhng kh th v. D ch n thun p dng nh ngha ca hm n iu nhng cng i hi phi la chn hng tip cn ph hp l phn chng. Ta c th i theo mt con ng t nhin hn l kho st du o hm:

Gi thit cho c th vit li l

2( ) 1, ( ) 3f x f x x vi mi x .

Ta cn chng minh ( ) 3f x x vi mi .x

R rng ( ) 0f x nn ta ch cn xt vi 0x v nu 0x th hin nhin 3 ( )x f x , ta

nhn hai bt ng thc cho, v theo v, li th c

2 2( ) 3 ( ) 3f x x f x x vi mi .x

Vi cch tip cn ny, r rng kh ca hai bi a v b l nh nhau.

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41


MN GII TCH NM 2010

Nhn xt chung.

C th ni trong cc nm gn y th thi Gii tch 2010 l hay nht vi phn loi, tnh mi m ca dng Ton cng nh bao qut (bn cnh th i s 2010 cng l mt hay, nhiu cu kh th v). Nu nh cc cu 2, 3 l qu quen thuc v d th cu 1 v 6 s thuc dng trung bnh, cn phi u t thch hp mi gii quyt c trn vn. Cu 4 v 5 chnh l im nhn ca vi c trng gii tch m ta c th nhn nh l thc s dnh cho SV thi Olympic. S khng qu kh khn gii cc cu 1, 2, 3 v 6 nhng gii quyt trn vn hai cu cn li th ng l mt th thch khng nh.

Bi 1.

Cho hm s ( ) ln( 1).f x x

a. Chng minh rng vi mi 0,x tn ti duy nht s thc c tha mn ( ) ( )f x xf c m

ta k hiu l ( ).c x

b. Tnh 0

( )lim .x

c xx

Li gii.

a. Ta cn chng minh rng phng trnh ln( 1) 11

xx c

c nghim duy nht l c vi

mi 0.x D dng tnh c 1.ln( 1)

xcx

b. Ta cn tnh gii hn

0 0 0

1( ) ln( 1)ln( 1)lim lim lim

ln( 1)x x x

xc x x xx

x x x x

.

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Theo quy tc LHospital th

20 0 0 0 0

11ln( 1) ln(1 ) 1 11lim lim lim lim lim1ln( 1) ln(1 ) 2 2

1x x x x x

x x x x x xx x x xx

x

.

Vy gii hn cn tm l 1 .2

Nhn xt.

y l mt bi c bn kim tra cc kin thc v o hm, gii hn. Tuy khng kh nhng pht biu tng i l v nu y khng dng quy tc LHospital hoc dng nhng khng tch ra thnh 2 phn tnh ring th cng kh vt v mi ch ra c gii hn nh trn.

Bi 2.

Cho dy ( )nx xc nh bi 20101 11, 1n n nx x x x vi 1,2,3,...n

Tnh gii hn 20102010 2010

1 2

2 3 1

lim ... nx

n

xx xx x x

.

Li gii.

Trc ht, ta thy dy ny tng thc s v nu dy ny b chn th tn ti gii hn, t gii hn l 0L . Chuyn cng thc xc nh ca dy qua gii hn, ta c

2010(1 ) 0,L L L L mu thun.

Do lim .nx x Vi mi 1k , ta c

2010 20111

1 1 1 1

1 1k k k kk k k k k k k

x x x xx x x x x x x

.

Suy ra 20102010 2010

1 2

2 3 1 1 1

1 1lim ... lim 1nx x

n n

xx xx x x x x

.

Vy gii hn cn tnh l 1.

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43

Nhn xt.

Dng gii hn dy s ny kh ph bin v trong Gii tch trc cng c xut hin. Ta c th lit k ra cc bc chnh x l dng Ton ny l:

- Chng minh dy ( )nu khng b chn v tin ti v cc (thc ra nu n tin ti mt

im c th no th cng gii quyt tng t).

- Biu din tng s hng thnh dng sai phn ri rt gn.

- Tnh ton gii hn thu c ri kt lun.

Mt s bi ton tng t:

(1) Cho 0a v xt dy s ( )nu c xc nh bi 2

1 11,n

n n

uu u u

a vi 1,2,3,...n

Tnh gii hn ca tng 31 22 3 4 1

... nn

u uu uu u u u

.

(2) Cho dy s ( )nx xc nh bi 2

1 1 11

41 ,2 2

n n nn

x x xx x

vi mi 2.n

t 21

1nn

k k

yx

. Chng minh rng:

a. Dy s ( )ny c gii hn hu hn v tm gii hn .

b. Dy s nxn

hi t v 1.

Bi 3.

Cho s thc a v hm s ( )f x kh vi trn [0; ) tha mn cc iu kin (0) 0f v

( ) ( ) 0f x af x vi mi [0; )x .

Chng minh rng ( ) 0f x vi mi 0.x

Li gii.

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Xt hm s ( ) ( )axg x e f x vi [0; )x th d thy hm s ny kh vi trn [0; ) v

( ) ( ) ( ) 0axg x e f x af x vi mi [0; )x .

Do 0( ) (0) (0) 0g x g e f vi mi [0; )x .

T suy ra ( ) 0axe f x hay ( ) 0f x vi mi 0.x

Nhn xt.

Cu hi dng ny xut hin nhiu trc v y c gii thiu vi hnh thc kh n gin. Tuy nhin, cch t hm s nh th nu cha s dng gii ton ln no th ni chung cng khng phi d dng ngh ra.

Bi 4.

Cho hm s ( )f x kh vi lin tc trn [0;1] . Gi s rng

1 1

0 0( ) ( ) 1f x dx xf x dx .

Chng minh rng tn ti im (0;1)c sao cho ( ) 6.f c

Li gii.

Xt hm s ( ) 6 2g x x th d dng thy rng

1 1

0 0( ) ( ) 1g x dx xg x dx .

T suy ra 1

0( ) ( ) 0f x g x dx .

t ( ) ( ) ( )h x f x g x th ( )h x lin tc trn [0;1] v c tch phn 1

0( ) 0h x dx .

Do khng th c ( ) 0, (0;1)h x x hoc ( ) 0, (0;1)h x x . Suy ra phng trnh ( ) 0h x c t nht mt nghim trong (0;1).

Gi s ( ) 0h x ch c mt nghim l (0;1)x a . Ta c cc trng hp:

(1) Nu ( ) 0h x vi mi (0; )x a th ( ) 0h x vi mi ( ;1).x a Ta c

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45

1 1 1 1 1

0 0 0 0 01 1

0 0

( ) 1 ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) 0

a

aa

a

xf x dx xf x dx xg x dx xh x dx xh x dx xh x dx

ah x dx ah x dx ah x dx

Suy ra 1

0( ) 1xf x dx , mu thun vi gi thit.

(2) Nu ( ) 0h x vi mi (0; )x a th ( ) 0h x vi mi ( ;1).x a Chng minh tng t,

ta cng c 1

0( ) 1xf x dx , mu thun.

Suy ra phng trnh ( ) 0h x phi c t nht hai nghim trong (0;1) . Gi s hai nghim l , (0;1)a b vi a b .

Ta c ( ) ( ) 0h a h b nn ( ) ( ) ( ) ( )f b f a g b g a . Do tnh lin tc ca hm s ( ),h x theo

nh l Lagrange th tn ti ( ; ) (0;1)c a b sao cho

( ) ( ) ( ) ( )( ) 6f b f a g b g af cb a b a

.

T y ta c pcm.

Nhn xt.

Hm s ( )g x c a vo ng l kh bt ng v cng ng l n tha mn gi thit

1 11 1 1 12 2 3 2

0 0 0 00 0( ) (6 2) (3 2 ) 1, ( ) (6 2 ) (2 ) 1g x dx x dx x x xg x dx x x dx x x .

Lin h gia gi thit v kt lun rt kho v mt ci kh ca bi ton chnh l ( ) 6f c ,

ti sao li l s 6, s khc c c khng? Hy phn tch k thm bi ton c cu tr li thch hp.

Mt bi tng qut xut pht t tng ca bi ton ny:

Cho cc s dng , .a b Xt hm s ( )f x kh vi lin tc trn [0; ]a . Gi s rng

0 0( ) ( )

a af x dx xf x dx b .

Chng minh rng tn ti im (0; )c a sao cho 36 (2 )( ) .b af c

a

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Bi 5.

Cho a thc ( )P x bc n c h s thc sao cho ( 1) 0P v ( 1)( 1) 2

P nP

. Chng minh

rng ( )P x c t nht mt nghim 0x vi 0 1.x

Li gii.

Gi s 1 2, ,..., nx x x l cc nghim (thc hoc phc) ca ( )P x . Khi , theo nh l Bezout

th tn ti k sao cho 1

( ) ( )n

ii

P x k x x

. Ta c cc cng thc

1 1

( ) 1 ( 1) 1( ) ( 1) 1

n n

i ii i

P x PP x x x P x

.

Do 1 1

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

n ni

i ii i

xn PP x x

.

Ta c 2

2 2

11 ( 1)( 1) 1Re

1 11 1ii i i i

i ii i

xx x x xx xx x

vi mi 1,2,3,...,i n .

V ( 1)2 ( 1)n P

P

nn 2

21

10

1

ni

ii

x

x

.

T ng thc ny ra ( )P x phi c t nht mt nghim 0x m 0 1x .

Nhn xt.

Ci kh ca bi ny chnh l phi x l trn s phc cc nghim ca phng trnh. Tt nhin d l thc hay phc th a thc cng tha mn nh l Bezout, nh l Viete,... Thc ra nu thay a thc cho bng a thc c n nghim thc th li gii vn tng t nhng lp lun trn tp s thc s d dng hn, n on

2

21 1

1 1( 1) 1 1 02 ( 1) 2 1 2 ( 1)

n ni i

i ii i

x xn PP x x

th bi ton hon tt. Dng ton ny kh lu mi c xut hin li trong Olympic SV.

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47

Cc bi ton tng t:

(1) Gi s a thc 2008( ) 0P x x mx m m v c 2008 nghim thc. Chng minh rng trong cc nghim ca ( )P x , c t nht mt nghim 0x tho mn iu kin 0 2x .

(2) Cho a thc 6 5 4 3 2( ) 3 9 18 21 15 6 1, .P x x x x x x x x Chng minh rng ( )P x c 3 nghim , ,a b c thc hoc phc phn bit m 3a c b .

Bi 6.

a. Tm tt c cc hm s ( )f x kh vi lin tc trn [0;1] sao cho (1) (0)f ef v

21

0

( ) 1( )

f x dxf x

.

b. Tm tt c cc hm s :f lin tc tha mn (1) 2010f v

( ) 2010 ( ) 2010 ( )yxf x y f y f x vi mi , .x y

Li gii.

a. Ta c

2 21 1 0

0 0 0

2 2 21 1 11

00 0 0

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

( ) ( ) (1) ( )2 ln ( ) 1 2 ln 1 1( ) ( ) (0) ( )

f x f x f xdx dx dxf x f x f x

f x f x f f xdx f x dx dxf x f x f f x

T ta c 2

1

0

( ) 1( )

f x dxf x

.

Hn na, theo gi thit th 2

1

0

( ) 1( )

f x dxf x

nn ng thc phi xy ra, tc l

21

0

( ) 1 0( )

f x dxf x

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Do hm s ( )f x kh vi lin tc trn [0;1] nn ta c

( ) 1( )

f xf x

vi mi [0;1].x

Suy ra ( ) ( ), [0;1]f x f x x , do ( ) , 0xf x ce c .

Th li, ta thy hm s ny tha mn bi.

b. T gi thit, ta c

( )2010 ( ) 2010 ( ) 2010 ( )x y y xf x y f y f x vi mi , .x y

t 2010 ( ) ( )x f x g x . Ta c

( ) ( ) ( )g x y g x g y vi mi , .x y

y chnh l phng trnh hm Cauchy v nh tnh lin tc ca ( )g x , ta thu c

nghim l ( )g x ax , suy ra ( ) 2010xf x ax . Hn na, t iu kin (1) 2010f , ta c

1a v ( ) 2010xf x x .

Th li ta thy hm s ny tha mn.

Vy tt c cc hm s cn tm l ( ) 2010 , .xf x x x

Nhn xt.

Cu a ca bi ton ny a v mt phng trnh vi phn, ni chung t xut hin trong cc k thi Olympic SV. Nu khng cng nhn cc l thuyt v phn ny th kh c th

chng minh rng ( ) , 0xf x ce c m t ht cc nghim c th c. on lp lun

dn n ng thc xy ra kho lo s dng c gi thit. Cu b th d hn r rng

v ch cn mt thao tc chia hai v cho 2010x y l a c v phng trnh hm Cauchy. C l hu ht cc th sinh s u tin x l bi ny hn l chn cu a.

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49


MN GII TCH NM 2011

Nhn xt chung.

thi ny khng c bi no qu kh, hu ht cc cu mc trung bnh v cng mang tnh suy lun da trn kin thc c bn v hm s, gii hn,... ch khng i hi phi c qu nhiu kinh nghim thi c. Cc cu 1, 2, 6 l d; 3,4 l trung bnh v cu 5 l mt bi kh mi ch khng kh ( i s ca nm ny cng c bi v nh thc ma trn nhng pht biu thng qua mt bi ton tr chi kh th v). thi nh vy r rng l hay nhng mt iu hi tic l cng cha c s xut hin nhiu ca cc yu t Gii tch cao cp nh: khai trin Taylor, nh l Lagrange, cc tnh cht ca tch phn cng cn kh m nht (mt hc sinh THPT b i cu 6b th cng c th lm trn vn ny).

Bi 1.

Cho hm s 2( ) ( 1)

xef xx

.

a. Chng minh rng ( )f x x c nghim duy nht trong 1 ;12

v ( )f x ng bin.

b. Chng minh dy s ( )nu xc nh bi 1 11, ( )n nu u f u tha mn 1 ;1 ,2n

u n

.

Li gii.

a. Xt hm s ( ) ( )g x f x x th

3

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

xx eg x f xx

vi mi 1 ;12

x

.

Do , ( )g x n iu gim trn 1 ;12

.

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50

Mt khc, ta cng c 1 4 1 8 9 02 9 2 18

e eg

v (1) 1 04eg , ( )g x lin tc trn

1 ;12

nn suy ra phng trnh ( ) 0g x hay ( )f x x c nghim duy nht trong 1 ;12

.

Ta c 3( 1)( )( 1)

xx ef xx

v 2

4

( 2 3)( ) 0( 1)

xx x ef xx

vi mi 1 ;12

x

.

Do hm s ( )f x ng bin trn 1 ;12

.

b. Ta s chng minh bng quy np rng 1 ;1 , 12n

u n

(*).

Tht vy,

- Vi 1n th 111 ;12

u

, tc l (*) ng vi 1.n

- Gi s (*) ng vi n k , ta c 1 ;12k

u

. Do tnh n iu ca hm s ( )f x trn

min 1 ;12

, ta c 11 4(1) ( )2 4 9k k

e ef f u f u

.

Do 14 1; ;1

4 9 2ke eu

. Ta c pcm.

Nhn xt.

Cc k thut s dng trong bi ny u quen thuc, ch cn nm vng tnh cht ca hm s l c th x l tt. tng quy np cho cu b cng kh t nhin.

Thc ra y l mt bi ton c v y khng thy s xut hin cu hi quan trng nht ca bi ton gc l: Chng minh dy s ( )nu hi t. Cc a, b nhm phc v cho

vic chng minh ny.

Bi 2.

Tnh tch phn 1

0 2 4 21 3 1dx

x x x x .

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51

Li gii.

Xt hm s 2 4 2

1( )1 3 1

g xx x x x

vi [ 1;1]x . Ta thy rng

1(0)2

g v 21( ) ( )

1g x g x

x

.

Ta cng c 2 2 4

3

1 1 3( )2( )

x x x xg xx x

.

Do

1 0 1 0 1

1 1 0 1 0

1 1

20 0

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

( ) ( ) .41

I g x dx g x dx g x dx g x d x g x dx

dxg x g x dxx

.

Vy tch phn cn tnh l .4

Nhn xt.

Vic t thm hm ( )g x v nhn lin hp nh trn l hon ton t nhin, k c tnh

cht 21( ) ( )

1g x g x

x

cng c chng minh hon ton d dng. Tuy vy, i vi

nhiu bn cha nm vng v tch phn suy rng th s khng dm nhn lin hp kiu nh vy do min cn tnh tch phn c cha s 0 v s kh c th tm c mt li gii th hai thay th.

Bi 3.

Cho hai dy s ( )nx v ( )ny tha mn cc iu kin

1 2n n

n

x yx

v

2 2

1 2n n

n

x yy

vi n .

a. Chng minh rng cc dy ( )n nx y , ( )n nx y l nhng dy n iu tng.

b. Gi s cc dy s ( ),( )n nx y b chn. Chng minh rng hai dy ny cng hi t v mt

im no .

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52

Li gii.

a. D thy rng cc dy ( ),( )n nx y dng vi 2.n

t ,n n n n n nx y s x y p . Ta c

1 2 2n n n

n n

x y sx p

v

2 2

1 2 2 2n n n n n

n n

x y x y sy p

.

Suy ra 1 1 1n n n ns x y s v 2

1 1 1 4n

n n n n

sp x y p , tc l ( )n nx y , ( )n nx y l nhng

dy n iu tng.

b. Do cc dy ( )nx v ( )ny b chn nn dy ( )n nx y b chn, hn na n l dy n

iu tng nn c gii hn.

Gi s lim( ) lim 0n n nn nx y s s . Ta cng c 2

4n

n

sp v ( )ns tng nn

2

4nsp , tc l

dy ( )np cng b chn v dy ny cng n iu tng nn n c gii hn, t l p vi 2

0 .4sp Mt khc, ta c

2

1 4n

n

sp nn

2 2

lim lim4 4n

nn n

s sp p

.

T suy ra 2

4sp hay 2 4s p .

Theo nh l Viete th ,n nx y l cc nghim dng ca phng trnh 2 0n nt s t p v

cc nghim ca phng trnh ny l

2 21 14 , 42 2n n n n n ns s p s s p .

Suy ra 2 21 1lim lim lim 4 lim 42 2 2n n n n n n n nn n n nsx y s s p s s p

.

Nhn xt.

Cc dy s trong bi ton chnh l cc i lng chuyn i gi tr trung bnh. Cu a c gii d dng bng quy np nhng cu b thc s khng n gin v rt d b ng nhn. Nhiu bn s chng minh c hai dy hi t nhng chng minh c chng

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53

cng hi t v mt im qua khng t thm hai dy ,n ns p th s rt d b ng nhn (tt

nhin l c mt s cch khc thc hin iu ny). Di y l mt s bi tng t c cng dng:

(1) Cho hai dy s ( ),( )n nx y tha mn 1 10, 0x a y b v

1 11 1,2

n nn n n n

x yx y x y

vi 2,3,4,...n

Chng minh rng hai dy cho hi t v lim limn nn nx y .

(2) Cho hai dy s ( ),( )n nx y tha mn 1 10, 0x a y b v

1 1

1 1

2,1 12

n nn n

n n

x yx y

x y

vi 2,3,4,...n

Chng minh rng hai dy cho hi t v lim limn nn nx y .

Bi 4.

Cho , tha mn iu kin *1 11 1 ,n n

e nn n

Tm min .

Li gii.

T gi thit, ta c 11ln 1

n

n

vi mi .n

Xt hm s 1( )1ln 1

f x x

x

vi 1x th d thy f tng. Suy ra dy 11ln 1

n

n

tng. Do , ta c nh gi sau:

1 1 1 1min min sup 1 limln 21 1 1ln 1 ln 1 ln 1

nn n n

n n n

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Theo quy tc LHospital, ta c

0 0

1 1 1 ln(1 ) 1lim lim limln(1 ) ln(1 ) 21ln 1

n t t

t tnt t t t

n

.

Do 1 1 3 1min 1

ln 2 2 2 ln 2 .

Nhn xt.

bi ton ny, ta ch cn tm chn trn v chn di cho hm s 1( )1ln 1

f n n

n

t xc nh c khong cch ln nht gia , l xong. Hm s c chn y c th thay bng mt hm ty v cu hi nh trn kh ph bin trong cc bi ton v kho st min gi tr ca hm s.

Tuy nhin, c nhiu th sinh s ng nhn do bit trc kt qu quen thuc sau

1*1 11 1 ,

n n

e nn n

V t kt lun 0, 1 dn n sai lm ng tic.

Bi 5.

Ta gi on thng [ , ] l on thng tt nu vi mi b s , ,a b c tha mn iu kin

2 3 6 0a b c th phng trnh 2 0ax bx c c nghim thc thuc on [ , ] . Trong tt c cc on thng tt, tm on c di nh nht.

Li gii.

Vi 0a th 3 6 0b c v phng trnh 2 0ax bx c c nghim l 12

x , suy ra on

tt phi cha s 1 .2

Do 12

.

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Xt 0a , khi khng mt tnh tng qut, ta c th gi s 3a v 2 2b c . Phng trnh cho vit li thnh

23 2(1 ) 0x c x c .

Phng trnh ny c hai nghim l 2 2

1 21 1 1 1,

3 3c c c c c cx x .

D thy rng 1 2,x x l cc hm s lin tc v tng theo bin c ; hn na, ta c

1 21lim lim2c c

x x

.

Gi s [ , ] l mt on thng tt. Khi , r rng tn ti duy nht gi tr 0c sao cho

1 0( )x c . Vi 0c c th 1( )x c , tc l 1( )x c nm ngoi on tt, suy ra 2( )x c .

Cho 0( )c c th 2 0( ) .x c T , ta c

20 0

1 0 2 0

2 1 3( ) ( )3 3c c

x c x c

.

Do , di on thng tt khng th b hn 33

. Ta s ch ra tn ti duy nht on

thng nh vy.

Tht vy, nu chn 3 3 3 3,6 6

th

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56

- Nu 12

c , ta c 21, [ , ]2

x

.

- Nu 12

c , ta c 11 ; [ , ]2

x

.

Vy on thng tt c di nh nht l 3 3 3 3;6 6

.

Nhn xt.

y l mt bi ton rt th v vi pht biu kh l mt, xut pht t mt bi ton c hn l vi iu 2 3 6 0a b c cho, chng minh phng trnh bc hai 2 0ax bx c lun c nghim. Cn hiu r bn cht ca cu hi l: on tt l on cha t nht mt nghim ca phng trnh cho ch khng phi cha ng thi c hai nghim.

Mt cu hi m rng kh hn t bi ton ny l:

Vi cc s , nh nh ngha, chng minh rng 223

.

Bi 6.

a. Tm hm s :f tha mn iu kin

2 2( ) ( ) ( ) ( ) 4 ( )x y f x y x y f x y xy x y vi mi ,x y .

b. Cho hm s f lin tc trn 1 ; 22

v tha mn iu kin

1 1( ) 2xf x fx x

vi mi 1 ; 2

2x

.

Chng minh rng 212

( ) 2 ln 2f x dx .

Li gii.

a. t ,u x y v x y th ,2 2

u v v ux y nn ta c

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57

2 21( ) ( ) ( )( ) ( ) ( )4

uf v vf u u v u v u v u v vi ,u v .

hay 3 3 3 3( ) ( ) ( ) ( )uf v vf u u v uv v f u u u f v v .

Nu 0uv th ta vit ng thc trn thnh

3 3( ) ( )f u u f v vu v

.

Do tn ti c sao cho 3

3( ) , 0 ( ) , 0f u u c u f u u cu uu

.

Nu 0uv th d thy (0) 0f , nhng hm s 3( )f u u cu cng tha mn iu ny.

Th li ta thy hm s 3( )f x x cx tha mn.

Vy tt c cc hm s cn tm l 3( ) , .f x x cx x

b. Xt 212

( )I f x dx . t 2tx th 2 ln 2tdx dt , thay vo tch phn ny, ta c

1

1ln 2 (2 )2t tI f dt

.

Tng t, t 2 tx , ta c 1

1ln 2 (2 )2t tI f dt

.

Vi [ 1;1]t th 12 ; 22

t

nn theo gi thit, ta c

(2 )2 (2 )2 2t t t tf f .

Do ,

1 11 12 ln 2 (2 )2 (2 )2 ln 2 2 4 ln 2t t t tI f f dt dt

.

Suy ra 2 ln 2I . Ta c pcm.

Nhn xt.

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C l khi gp phn t chn ny, hu ht cc bn th sinh u s chn cu a v n quen thuc v d x l. Ch cn vi php i bin n gin l c th chuyn v i lng c

dng 3 3( ) ( )f u u f v v

u v

th mi chuyn xem nh hon tt (n phi l hng s), t

i lng ny l c ri t thay ngc dn ln ra kt qu.

Mt bi ton tng t:

Tm hm s :f tha mn iu kin

4 4( ) ( ) ( ) ( ) ( )x y f x y x y f x y xy x y vi mi ,x y .

cu b ca bi ton, ngoi cch i bin thnh dng 2t nh trn, ta c th lm trc tip nh sau. Ta c

2 1 21 1 12 2

( ) ( ) ( )I f x dx f x dx f x dx , trong tch phn th hai, t 21 dtx dxt t

th

12 1 1 1 12

1 1 1 1 12 2 212 2 2 2 2

1 1 1 1 1( ) ( ) ( ) ( )dtI f x dx f x dx f f x dx f dx f x f dxt x xt x x

.

T gi thit, ta c 21 1 2( )f x f

x xx

vi mi 1 ; 22

x

nn 1 1

1122

2 2 ln 2 ln 2.dxI xx

Mt cu hi tng t cho cu b ca bi ny.

Cho hm s f lin tc trn 1 ; 22

v tha mn iu kin

( ) 2xf x yf y vi mi 1, ; 22

x y

.

Tm gi tr ln nht ca 212

( ) .f x dx

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59


MN GII TCH NM 2012

Nhn xt chung.

C th ni nu khng c li trong khu bin son , dn n sai hai cu 2 v 5 th y l mt hay, c tnh phn loi ln v p ca bi ton. Cc cu d l 1 v 3, trung bnh l 4 v 5, cu kh l 2 v 6. Cc bi ton 5, 6 rt hay, mi m v p. c th x l trn vn c thi ny th cn phi nm vng kin thc v gii tch v hiu r bn cht vn . C l cc th sinh thc s gii trong k thi ny cha hn l nhng ngi t gii cao nht m chnh l cc bn bit cu no ng, cu no sai v cho phn v d cho chng, cn l ngi dm chn cu 6a (thay v lm mt v d hn cu 6b) v gii quyt trn vn bng khai trin Taylor.

Bi 1.

Cho dy s ( )na tha mn iu kin

1a v 11 2

n nna a

n n

vi 1,2,3,...n

Tm dy ( )na hi t.

Li gii.

Ta cng thc xc nh dy, ta c 12n

n n

aa a

n

vi mi 1.n

t 2n nx a th ta c 1 2x v

111n nx xn

v 1 1 11

11 ( 1)n

nk

x x n xk

.

Suy ra ( 1)( 2) 2na n vi mi .n

Nu 2 th d thy na khi n nn khng tha.

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60

Nu 2 th ta c 2na vi mi n nn dy ( )na hi t.

Vy gi tr cn tm ca dy cho hi t l 2.

Nhn xt.

Ngoi cch gii trn s dng trong p n, c mt bin i t nhin hn l a v

1 21 ( 1)

n na an n n n

.

t dy s mi nna

bn

th c ngay 12

( 1)n nb b

n n

, ch rng biu thc 2

( 1)n n c

th vit thnh sai phn 2 2 2( 1) 1n n n n

nn bi ton c gii quyt nhanh chng.

Bi 2.

Cho a thc ( )P x c bc khng nh hn 1 c h s thc v a thc ( )Q x xc nh bi

2 2 2( ) (2012 1) ( ) ( ) 2012( ( )) ( ( ))Q x x P x P x x P x P x .

Chng minh rng nu phng trnh ( ) 0P x c ng n nghim thc phn bit trong

khong 1 ;2

th phng trnh ( ) 0Q x c t nht 2 1n nghim thc phn bit.

Li gii.

Ta c 2 21006 1006( ) 2012 ( ) ( ) ( ) ( ) ( ) ( )x xQ x xP x P x xP x P x e e P x xP x .

Xt ,a b l hai nghim lin tip trong dy n nghim phn bit ca ( ) 0P x th theo nh

l Rolle, cc phng trnh 21006 ( )xe P x v ( )xP x s ln lt c cc nghim l 1 2,r r sao cho 1 2,a r r b . Ta s chng minh rng 1 2r r , nu khng gi s 1 2r r r th

( ) 2012 ( ) 0( ) ( ) 0

P r rP rP r rP r

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61

Do 0r nn ta nhn ng thc th nht vi r v tr cho ng thc th hai, ta c

2(2012 1) ( ) 0r P r .

V 12

r a nn 22012 1 0r , suy ra ( ) 0P r , cng mu thun.

Do , nu gi 1 21 ...2 n

c c c l cc nghim ca phng trnh ( ) 0P x th ( ) 0Q x

c t nht 2 2n nghim nm gia cc khong 1 2 2 3 1( , ),( , ),...,( , )n nc c c c c c .

ng thi, ta thy rng phng trnh ( ) 0xP x c nghim 0x nn gia khong 1(0; )c

th phng trnh ( ) ( ) 0P x xP x c thm mt nghim na.

Vy a thc ( )Q x c t nht (2 2) 1 2 1n n nghim thc phn bit. Ta c pcm.

Nhn xt.

Bi ton gc ca bi ny c l l:

Chng minh rng nu a thc ( )P x bc n c n nghim thc phn bit ln hn 1 th 2 2 2( ) ( 1) ( ) ( ) ( ) ( ( ))Q x x P x P x xP x P x cng c t nht 2 1n nghim thc phn bit.

Dng cu hi ny c xut hin trong i s nm 2011 trc nhng tng i d hn. bi ton ang xt, ta s dng kho lo tnh cht ca nh l Rolle ch ra chn di cho s nghim ca phng trnh. Ci kh chnh l phn tch c a thc cho thnh nhn t c cc dng phng trnh n gin hn. Nu a thc ( )Q x khng phn tch c thnh nhn t th bi ny thc s kh v khng d kim tra c tnh ng n ca n. ny c chnh sa li cho ng v gc ng tic b sai v trong p n trnh by kh vn tt, n ni khng hiu cho thm iu kin cc nghim

khng nh hn 12

lm g (s dng on lp lun cc nghim phn bit).

Bi 3.

Tnh tch phn 1

21 (2012 1)(1 )xdx

x .

Li gii.

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62

Ta c

1 0 1

2 2 21 1 0

0 1

2 21 0

1

2 20

1

20

(2012 1)(1 ) (2012 1)(1 ) (2012 1)(1 )( )

(2012 1)(1 ( ) ) (2012 1)(1 )1 1

(2012 1)(1 ) (2012 1)(1 )

1 2012 11 2012 1 2012 1

11

x x x

x x

x x

x

x x

dx dx dxx x x

d x dxx x

dxx x

dxx

1

20 4dx

x

Vy tch phn cn tnh l .4

Nhn xt.

Dng tng qut ca bi ny l: Cho hm s ( )f x chn v cc s thc , 0a b , khi ta

c ng thc tch phn sau

0

( ) ( )1

a a

xa

f x dx f x dxb

.

Ni chung dng ny qu quen thuc ngay t thi THPT. Hm s ( ) 1xg x b trn c

c im l ( ) ( ) 1g x g x nn ta hon ton c th thay bng hm s khc c tnh cht

( ) ( )g x g x bng mt hng s no , chng hn 2( ) ln sin 2 sing x x x .

Ch rng trong trng hp hm ( )f x l th ( ) 0a

af x dx

.

Bi 4.

Tm tt c cc hm s :f tha mn iu kin

12012 2 2013 2014x y yxf f f

vi mi ,x y .

Li gii.

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63

t 2014 2014,2012 2013

a b th r rng 0, 1b b .

Trong ng thc cho, thay ,x y ln lt bi 2014 2014,2012 2012

x y th

2014 1 2014( ) ( )2012 2 2013

f x y f x f y

hay 1( ) ( ) ( )2

f a x y f bx f y .

t ( ) ( ) (0)g x f x f , ta c

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

g a x y f a x y f f bx f y f g bx g y .

Ta cng c

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

g a x y g a x y g b x y g g b x y .

So snh hai ng thc trn, ta c

( ) ( ) ( )g b x y g bx g y vi mi ,x y .

Chn y sao cho ( )1

bb x y y y xb

th ( ) 0g bx vi mi .x

T suy ra ( ) (0)f x f c vi mi x , tc l ( )f x l hm hng.

Th li ta thy tha.

Vy tt c cc hm s cn tm l ( )f x c vi mi .x

Nhn xt.

y l mt bi phng trnh hm i hi k thut bin i i s ch khng mang tnh gii tch no. D dng on ra c ch c th l hm hng mi tha mn; tuy nhin, mun chng minh c iu ny th cn phi kho lo dng php th v la chn thch hp. Nhiu bn b ng nhn, d khng cho lin tc nhng vn p dng tnh cht

0 0

lim ( ) (lim )x x x x

f x f x

, gii ra nhanh chng v r rng y l mt sai lm nghim trng.

Ni chung, khi c thm gi thit lin tc th ch c trng hp a b th mi a v

dng ( ) ( )2 2

x y f x f yf

vi mi ,x y dn n hm s tuyn tnh dng ( )f x ux v

v kh hi m thi.

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64

Bi 5.

Gi s hm s ( )f x lin tc trn on [0; 2012] v tha mn iu kin

( ) (2012 ) 0f x f x vi mi [0; 2012]x .

Chng minh rng 2012

0( ) 0f x dx v phng trnh

2012

0(2012 ) ( ) 2012 ( )

xx f x f u du

c

nghim trong khong (0; 2012).

Li gii.

Theo gi thit th ( ) (2012 )f x f x vi mi [0; 2012]x , do

2012 2012 0 2012

0 0 2012 0( ) (2012 ) (2012 ) (2012 ) ( )f x dx f x dx f x d x f x dx

T suy ra 2012

0( ) 0f x dx .

t 20122012

0( ) (2012 ) ( )

xg x x f u du

th d thy rng hm s ny kh vi trn [0; 2012] v

c (0) (2012) 0g g . Theo nh l Rolle th tn ti (0; 2012)c sao cho

20122011 2012

0( ) 0 2012(2012 ) ( ) (2012 ) (2012 ) 0

cg c c f u du c f c

.

T y suy ra

2012

02012 ( ) (2012 ) ( ) 0

cf u du c f c

hay phng trnh 2012

0(2012 ) ( ) 2012 ( )

xx f x f u du

c nghim. Ta c pcm.

Nhn xt.

Trong bi ny, on chng minh gi tr ca tch phn xc nh bng 0 ch i hi k thut i bin v tn dng gi thit nn khng c vn g.

Cn phn chng minh phng trnh c nghim th cng nh kch bn c, ta s phi chn mt hm s thch hp m o hm ca n c dng nh th ri dng nh l Lagrange/Rolle. Tuy nhin, hm s nh th trong trng hp ny l khng d tm, ta c gng phn tch nh sau:

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65

Hm s cn tm s c dng 2012

0( ) ( ) ( )

xg x h x f u du

khi dng o hm ca hm c

dng tch th mi hy vng ra c dng trn. Ch rng ta cng cn thm (0) (2012) 0h h (iu ny c sn). Ta c

2012 2012

0 0( ) ( ) ( ) ( ) (2012 ) ( ) ( ) ( ) ( )

x xg x h x f u du h x f x h x f u du h x f x

.

Suy ra 2012

0( ) 0 ( ) ( ) ( ) ( )

xg x h x f x h x f u du

.

Do , so snh vi phng trnh ang quan tm, ta cn c ( ) ( )2012 2012

h x h xx

.

n y, d dng chn c 2012( ) (2012 )h x x .

Bi 6.

a. Cho hm s ( )f x kh vi lin tc cp 2 trn . Gi s (1) 0f v 10

( ) 0f x dx . Chng minh rng vi mi (0;1) , ta c

0 0 1

2( ) max ( )81 x

f x dx f x

.

b. Cho hm s : [0;1]f l hm lm (cn gi l li ln pha trn), kh vi lin tc tha mn (0) (1) 0.f f Chng minh rng

1 22

00 1 0 11 4 max ( ) 1 ( ) 1 2 max ( )

x xf x f x dx f x

.

Li gii.

a. Ta c

1 1 11

00 0 0( ) ( ) ( ) (1) ( ) 0xf x dx xf x f x dx f f x dx .

Do

1 1 12 2

0 0 0 0

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

ff x dx f x dx f x f x xf x dx x dx

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66

Suy ra 21 12 2 2

0 0 0

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

f xf x dx x dx dx f

.

Ta c 12 31

00

12 6 6x xdx v

32 1 1 2 1 1 4( 1) 2 (1 )(1 )

2 2 3 27

.

T , ta c c 0 0 1

1 4 2( ) ( ) max ( )6 27 81 x

f x dx f f x

. y chnh l pcm.

b. Gi 0x l im cc i v 0y l gi tr cc i ca ( )f x trn min [0;1] . Ta c

00

1 1

0 0 00 0( ) ( ) ( ) ( ) (0) (1) ( ) 2 ( )

x

xf x dx f x dx f x dx f x f f f x f x nn

1

0 0

1max ( ) ( ) ( )2

f x f x f x dx .

Bt ng thc th nht tng ng vi 221 1 2

0 01 ( ) 1 ( )f x dx f x dx

(*)

Ta c

2 21 12

0 0

1 12 2

0 0

1 1 12

0 0 02

1 ( ) ( )

1 ( ) ( ) 1 ( ) ( )

11 ( ) ( ) 11 ( ) ( )

f x dx f x dx

f x f x dx f x f x dx

f x f x dx dx dxf x f x

T suy ra bt ng thc (*) ng.

Bt ng thc th hai tng ng vi 1 12

0 01 ( ) 1 ( )f x dx f x dx (**)

Ta c

1 12

0 0

1 1 12

0 0 02

1 ( ) ( )

11 ( ) ( ) 11 ( ) ( )

f x dx f x dx

f x f x dx x dxf x f x

T suy bt ng thc (**) cng ng. Vy ta c pcm.

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67

Nhn xt. C 2 cu ca bi ton ny u kh nhng trn thc t, hu ht cc th sinh u chn cu b (c mt d x l hn). Cu a i hi phi chng minh ng thc

12 2

0 0

( )( ) ( 1)2

ff x dx x dx

.

Ni chung y l mt kt qu khng d dng c th khai thc c t gi thit nu khng nm vng khai trin Taylor. Nu hon tt vic chng minh c ng thc trn th cng vic cn li hon ton t nhin.

i vi cu b, li gii nu trn chng minh c mt nh gi p 1

0 0

1( ) ( )2

f x f x dx , trong khi li gii ca p n chnh thc kh rc ri v hi thiu t

nhin. D vy, li gii bng hnh hc di y s cho ta thy r bn cht vn hn.

Ta bit rng i lng 1 2

01 ( )l f x dx chnh l di ca ng cong ( )y f x

trn min [0;1] . Ta c th minh ha hnh hc cho bi ton ny nh sau

Chn ta cc im 0 0 0 0(0; 0), (1; 0), (1; ), (0; ), ( , )A B C y D y E x y nh hnh trn.

Bt ng thc cn chng minh tng ng vi

20 01 4 1 2y l y .

Do th ca hm s ny li ln pha trn nn

01 2l AD DE BC CE CD AD BC y .

Hn na 2 2 2 2 2 2 20( ) ( ) 1 4l AE BE AD DE BC CE AD BC DE CE y .

Bi ton c gii quyt hon ton.

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68

TI LIU THAM KHO

[1] Din n http://forum.mathscope.org/

[2] Din n http://diendantoanhoc.net/

[3] Din n http://www.artofproblemsolving.com/Forum/portal.php?ml=1

[4] Din n http://math.net.vn/forum.php

[5] Cc thi Olympic Ton Sinh vin ton quc, NXB GD, 2005.

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