de thi va bai giai olympic toan giai tich 20062012)
TRANSCRIPT
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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
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Phn A.
CC THI
CHNH THC
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THI OLYMPIC TON SINH VIN
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
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THI OLYMPIC TON SINH VIN
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.
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THI OLYMPIC TON SINH VIN
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
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THI OLYMPIC TON SINH VIN
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 .
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THI OLYMPIC TON SINH VIN
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
Bi 6. (Th sinh chn mt trong hai cu)
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
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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.
Bi 6. (Th sinh chn mt trong hai cu)
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 .
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THI OLYMPIC TON SINH VIN
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
.
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Phn B.
LI GII CHI TIT V BNH LUN
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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.
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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 .
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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 .
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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.
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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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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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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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LI GII THI OLYMPIC TON SINH VIN
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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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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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
LI GII THI OLYMPIC TON SINH VIN
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
LI GII THI OLYMPIC TON SINH VIN
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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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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40
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
LI GII THI OLYMPIC TON SINH VIN
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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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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44
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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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
LI GII THI OLYMPIC TON SINH VIN
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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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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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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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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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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- 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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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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LI GII THI OLYMPIC TON SINH VIN
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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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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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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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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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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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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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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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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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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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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