Chuyfin c i r B D H S G Tojn gia trj I6n nhS't v.i gia tri nh6 nliat - Phan Huy

KITST

Cty TMHH M!V DVVH KhanglZiH"

VAIBAITOAN KHAC VE

G^^yd-.

hay u^ = 2 + 2 V 3 - ( x ^ - 6 x + l l ) .
Ttf do ta di den phU'dng trinh he qua sau:

GIA TRj idiN NHAT VA NHO NHAT CUA HAM SO

= 2 + 2 N / 3 - U <=> U^ - 2 = 2 ^ 3 - u

\J2

ce

2o

ok

Theo bat d^ng thtfc Cosi suy ra 2 < f^(x) < 2 + f(x - 2) + (4 - x ) ]

L a i CO f(x) =

2. V 3 x ^ + 6 x + 7+V5x2+10x + 14 = 4 - 2 x - x ^

/g

Ta tha'y m i e n xac djnh cua phifdng trinh la D = {x: 2 < X < 4 } .

=> 2 < f ' ( x )

J

Bai 2. Giai cac phifdng trinh sau:

,

Ta CO f^(x) = 2 + 2 7 ( x - 2 ) ( 4 - x ) .

- x ^ + 6x - 9 = 0 o X = 3.

duy nha't cua phifdng trinh da cho.


ff(x)-3
g(x) = 3

fx = 3
x =3

o x

= 3.

Do vay X = 3 la n g h i e m duy nha't cua (1)
297


Chuyen de BDHSG Toan gia trj I6n nhjt vk gia tr| nhi nha'l - Phan Huy Khii
Cty TNHH M T V D W H Khang Vigt

2. Xet phurdng trinh Vsx^ + 6x + 7 + Vsx^ + lOx + 14 = 4 - 2x Ta

CO

'

' (2)
p^i 4. Giai phUOng trinh 2^^ ' - 2"

f(x) = V3x^ + 6x + 7 + Vsx^ + lOx +14

,
5; f(5) = 5 o x = - 1 .
TCf do ta CO minf(x) = 5 o x = - 1 .

Ta

' '

xeK
CO

g(x) = (x - 1)^ > 0 Vx

Vay ming(x) = 0 <=>x= 1.

g(x) = 4 - 2x - x^ = 5 - (x + 1)^ => g(x) < 5 , Vx e M ;

Do(x - l ) ' > O o x ^ - 2 x + 1 >()

g(x) = 5 c > x = - l .


.,

,.

0.

, ,

om

"

' ' "* *^

o x=1

x=

/

Hudng ddn giai

r
Ap dung bat dang thiJc Cosi, ta c6

.c

'

rx = i
3 => log,(4x^ - 4x + 4) > 1.

\

+ I>3? 1 + ^x2
V
2
)

2

bo

d,


g(x) = 8 o x = - .

A ' : t • .;.x>1
i

=>

1-

j;

Jil + \^? < - - x ^ Vx e

v.

-.u

u
(1)

2

Vay maxg(x) = 8 o X = - .

Lai ap dung bat dang thtfc Cosi cho 4 so, ta c6:

2

1


Tir (1) (2) suy ra f(x) = 3x'' - 4x' - (1 f(x) = O o x = 0

298

X <=> X =

Nhir the' (1) CO nghiem duy nhat x = 1.

ro

up

f(x) = 2'"+' + 2^-^^ > l^I^^^Kl'-^' = 24¥ = 8,

Vay minf(x) = 8 o x = - .
xeR
2

g(x) = 0

Ta

.

s/

Theo bat dang thurc Cosi, ta CO

f(x) = 0

0 Vx e IR


Chuyen dg BDHSG ToAn gii tr| Idn nhaft va gii trj nh6 nhS't - Phan Huy Kh^i

Cty TNHH MTV DWH Khang Vigt

Vay minr(x) = () o x = 0.

Isin'x + cos'x = 32(sin"x + cos''x)

xel

^ f(x) = g(x)
Ttf cac ket qua tren suy ra

Ro rang phiMng Irinh da cho co the vict duTdi dang f(x) = 0
Ttr do suy ra phu'(tng trinh da cho c6 dang min f(x) = 0 o x = 0

,,

(1)

, '

s


( k 6

up

sin''x = sin^x

Z).

/g

ro

ir
o x =k cos X = cos X
2

1

Ta

•'

s/

=> f(x) = s i n \ cos^x < 1 Vx e R .

.c

.



,n-2

3
X "n-2' + ,x "„ n" -- X
l - x ) + ... + ( l - x )

Tijr do CO bang bien thien sau:

om

Tiirdotaco maxf(x) = l o x = k - , k e Z .
"
xeR
2
Ap dung ket qua sau day: Ne'u a, b > 0 va a + b = 1, thi vdi moi n nguyen
> 2 ta c6:

(1) v6 nghiem.

Xet ham so h(x) = x" + (1 - x)" vcti 0 < x < 1
=> h'(x) = nx"~' - n ( l - x ) " " " ' = n x " - ' - ( l - x ) " - '

X€D

Bai 6. Giai phiTdng trinh sin^x + cos^x = 32(sin'^x + c o s ' \

Mat khac f(x) = 1

x = - + k - , k e Z (3)


g(x) = 1 o sin^x = cos^x = <=>x= — + k — , k G
4
2

(1)

De thay g(x) > 2 Vx e R (do sin^x > 0 Vx e R )
Mat khac g(x) = 2 o sinx = 0 o x = kn (k e Z ) .
Vay ta c6 ming(x) = 2 <=> x = kTt.
xeR

Ap dung bat ding thuTc Bunhiacopski, ta c6:

'

(2)


Cty TNMH MTV DVVH Khang Vigt
Chuyen dg BDHSG Toan gia tri Idn nha't va gia tri nh6 nhaft - Phan Huy KhSi

cos^ 3x + (2 - cos^ 3x) (1 +1) > |cos3x + V 2 - c o s ^ 3 x
do suy ra

fir

Vx e

2k7t

X =

1.

di den max f(x) =

' (3)

Bai 9. Giai phU'dng trinh

Tir do suy ra

+1 sin^ X +

COS^ X +

\J
HUdng ddn giai
1 .
Datg(y)=12+ - s i n y . y e R.
COS

ce

bo

ok

.c

0
Vie't lai phUcJng trinh da cho diTdi dang
\ / x ^ + x - l + V x - x ^ + l - ( x + l) = x ^ - 2 x + l . ( l )
Datg(x) = x ^ - 2 x + l;f(x)= V x ^ + x - l + V x - x ^ + l - ( x + l), vdi x G D .
Tac6g(x) = ( x - l ) ^ > O V x e D
g(x) = O o x = l
TiJf d6 suy ra min g(x) = 0 o X = 1.
xeD

,

1),

T a c 6 g ( y ) < 1 2 - Vy e

I'M

1 ,


= 12 + ^ s i n y .

^

sin^ X y

, // . f

Ta
s/

up

x = 2—.keZ
lgW = 2
X = kn, k e Z
3
_ 2kn
^
, X G i •
i
»x = k 2 7 c , k e Z X = k7l,k€Z
j^., '
Vay X = k27t, k e Z la nghiem cua phiitfng trinh da cho.
^'i'H>Bai 8. Giai phtfdng trinh Vx^+ x - l + \ / x - x ^ + 1 = x^ - x + 2.
^,
t.s,.\.:^^:^',:.
HUdngddngiai
Mien xac dinh cua phiTdng trinh la tap hcJp D gom nhiTng phan tijf x thoa man

fcosSx > 0
cos^ 3x = 2 - cos^ 3x
o c o s 3 x = 1.
Vay maxf(x) = 2 o c o s 3 x = 1
Tit


ChuySn de BDHSG Toan gia tri I6n nhat va gia tri nli6 nhat

Cty TNHH IVITV DWH Khang Vl§t

Phan iiuy Kh^i

( 4 s i n \ 2sin^x - Bsinx - 1)^ = 5 - sinx

f(x) =l2-<=> sin'2x = 1 c=> cos2x = ( ) « x = - + n ^ , n € Z .
2
4
2
V i phifdng trinh da cho c6 dang f(x) - g(y)
l-(x) =

<r> 16sinS + 16sin'*x - 20sin''x - 2 0 s i n \ 5 s i n \ 7sinx - 4 = 0

x, y e M

o

(3)


Hiidng ddn giai

<::>x = —+ n — ; v = — + k27t, n va k 6 Z
4
2
2

D a l f(x) =

B a i 1 0 . G i a i phiTdng Irinh (sin3x + cos2x)' = 5 - sinx.

x^

+2X

+ 10

G o i m la gia t r i l i j y y . K h i do phU'dng trinh sau (an x )

HUdng dan giai

4x^ +14X + 46
—^
=m

D a l f ( x ) = (sin3x + cos2x)\(x) = 5 - sinx, x e R
D o sinx < 1 V x e M ^

4X^



.

'

„• •

x^ + 2x + 10 > 0), nSn

(l)<::>4x^ + I 4 x + 46 = mx^ + 2 m x + 10m

TiJf do suy ra m i n g ( x ) = 4 c:>x = - + k27t.
xeM
2

s/

Ta

(1)

up

L a i CO lsin3x + cos2x|
xeR

bo

Tir do ta CO max l"(x) = 4 o x thoa man ( 2 )

, ,

V a y (1) CO n g h i e m <=> 3 < m < 5.

(2)

ok

cos2x = - l

(2)

o3
o x =2

V a y X = 2 la n g h i e m duy nhat can tim.
V a y X = ^ + k n , k e Z la nghiOm ciia phi/dng Irinh da cho.
Nhanxet:
K h o CO each giai nao khac gon gang hdn each giai trcn
Cac ban ciJ thijr ti/dng lu"dng sau khi siir dung cong thuTc sinSx = 3sinx - 4sin
cos2x = 1 - 2sin^x, la difa phiTi^ng trinh da cho ve dang:
304

5 i i iAy

-

A^Aa« jcef: X e t each giai khac sau day
4x^^14x^46 ^^^,_3^^^3

- •
"

'

;
l «

'•• ^ "

x ^ + 2 x + 10
o 4 x ^ + 1 4 x + 46 = ( 2 x ^ - 8 x +13)(x^ + 2 x + 1 0 )


ol+sinx

(6)

(75rri.i+7^.i)'< (75^71)'+(7771)'

cos(xy) + 2'^i = 0

HUotng ddn giai

(1^+1^)

=> 7x+T + 7y + 1 ^72(x + y + 2 ) < 4

Viet lai phtfcJng trinh dU'di dang sau:

max ( 7 ^ + 7 7 7 l ) ^ 4 « : / i l L . £ I i v a 7 ^ + 7 ^ = 4

iL
ie
uO
nT
hi
Da
iH
oc
01
/

2l''l-cos^(xy) = 0 .
1 Vy € M
cos^(xy) < 1 Vx, y e R .

hit ii.^'f

M

I*til

P = 0 o i 2'^l-cos^(xy) = 0 O '
cos ( x y ) - l

2''"''=cos(xy)

2M = i

[2''"''=1

o x = y = 3J^'^''^S*™i'«-'r'

Nhan xet: Neu khong sit diing phiTdng phap tim gia trj U^n nha't ci'ia hiim so de

"^ly^o

danh gia hai ve', ta c6 the giai ihuan tuy he phtfdng trinh trcn nhU'sau:

/g

Tir do suy ra nghiem cua (1) la x = kK, y = 0 vdi k e Z .

om

Bai 13. (De thi tuyen sinh Dai hoc Cao ddn^ khoi A)

ok
ce

t =3
35
t = --

y + l>0

(4)

xy>0

(5)

, ;

De thay neu - 1 < x < 0 va - 1 < y < 0 thi yfx + l+Jy + l

0 1 = 3.
if.'

M

^ f t

Khi do 7 x y = 3 <::> xy = 9

Tir(3)(4)suyrax>-I;y>-1
thoa man (2)

()

2''"" -cos(xy) = 0

X€i he phi/dng trinh

x=y

Tur do suy ra (2) o X = y = 3 va x = y =3 cung thoa man (1).

V i t h e P > 0 Vx,y G R

Giai he phifdng trinh

1

|x+y-6
Vay(l)(2)oj"-_^
" o x = y = 3.
jTa thu lai ket qua trcn!

!nfa^,(1i/
.,(

• • ,'1 f

' i

'
cos2a
ro

sin'(x-a)

^

Bai 2. T i m gia tri Idn nha't va nho nhii'l cua ham so:

{*'\ >^ 4 / n %

up

B i e n ddi r „ ( x ) ve dang sau day

4

2cot^ a , neu — < a < •
4
2

HUfing ddn gidi

( s i n 2 x + sin2a)^

. ,

p a u bang trong (4) xay ra <=> cos2x = - 1

thuoc tham so m. Tiay theo gia trj cua m hay khao sat cac linh chsi't cua cac



-i

i
i

-

1

8

ok

Luc nay

1

0

• "


.
a + b+ 2 c a - b
. , a+ b+ 2 c b - a
1
cos2x
1
cos2x +
2
2
2
2
Ham so f ( x ) xac djnh v d i m o i x e R k h i va chi k h i

bo

i

F,„(l)

+

s/

7

-1

a


a-b.

|i|0.
2
2 ~


cos^2x

2 J

ba kha nang sau:
a. Neu
2m

ce
w.

X

sin2x = - 1
cos2x = 0

deu C O nghiem, nen suy

ra max F(x) = max f(x) + max g(x) = J2(a + b + 2c) + m
x€R

312

xeR

x€R

I



cos2x = 0

= -

m < - - . Liic nay ta c6 bang bien thien
3
m + 1

4(x)

0

T t r d o t a c o m i n L , ( x ) = f„,(2) = 7 m + 4 ; m a x f „ ( x ) = f ^ ( l ) = 3m + 3.

Vay maxf^(x) = 2(a + b + 2c)
.c

J

[a-b]

bo

2

(10)

ok

a + b + 2c^

/g

ro

(9)

D o f ( x ) > O V x e M . n e n maxf(x)=- /maxf^(x)
xeR
V xeK
2

up

Khi m < 0 thi maxg(x) = - m <=> sin2x = - 1 .


/

g(x) < - m Vx € R neu m < 0

I

^

xeD

bang bien thien sau:
, j 4 , 5 . - ! ' f„
m + 1
X
1
2m

< m Vx€;R.

Ta c6f^(x) = (a + b + 2c) + 2,

j

, Khi m > 0. Tif f|„(x) = 2mx + m + 1 , va do m > 0, nen

I'lr!) ft

g(x) < m


a + b + 2c

,

jChi m = 0, la c6 f;,(x) = x + 2

Nc'u a < b, hoan loan ti/dng ti/ta CO

,

4(x)
fm(x)

1

2
+

m +1
2m
0


Cty TNHH MTV DWH Khang Vi^t

Chuygn di BDHSG loAn gia Iri I6n nhgt va gia Iri nUd nh3't - Phan Huy KhSi

Tfifdotaco

fLijc n a y ( I ) CO d a n g 2 t - - I + at > - 1 hay 2 t ' + at > 0

3m2+6m-l

2m

4m

'-^ii'-i^

^

3m + 3 nc'u

1

= m m 3m + 3 ; 7 m + 4

f.(t)

1

0 <=> a > - 2 .

minf„(x) =

s/
/g

ro

7m + 4 , ne'u m > - —
. ~ 4
^ ; maxf,„(x) =

up

4

m i n f.(t) = 4 ( - l ) = 2 - a - :

t

.1 i

'"il'

^

-l 0 o a < 2.

w.

xe R .

i

+

I n' >.

2m

m i n r . | ( t ) > ( ) . (2)
- Kill '

iL
ie
uO
nT
hi
Da
iH
oc
01
/

c. Nc'u 1 < —

\
1

fa(0

^

^

^

+
0^

a- < 0 <=> a = 0.

Ro rang he (5) (6) c6 nghiem (thi du x = — thoa man (5) (6))

Tom lai a = b = 0 la cac gia Iri can tim cua tham so' a va b.

T6m lai phiTdng trinh da cho CO nghiem o m = - 2

'*.

Bai 2. Tim m de phiCdng trinh sau:

2 ,
^/7V
2 ,t

cos4x = 1

2 0 x ' +10X + 3

mm

cos2x = — 1

£ f

Ri

3x^+2x + l

( i ) Vi; ''

Ta

ocos2x = - l .
cos4x = - 1
^

Ta

CO

(20 - 3a)x^ + 2(5 - a)x + 3 - a = 0 (2)

(1)

0 o 2a - 19a + 35 < 0

-

< CY

cos2x = 1

Chu y rang: A „ = |(m + if

•

3x^+2x + l

V a f ( x ) = 4 <;=>|cos4x-cos2x| = 2

€ir

"

0ai 3. Tim m de phtfctng trinh sau:

(cos4x - cos2x)^ = ( m ^ + 4 m + 3 ) ( m ^ + 4 m + 6) + 7 + sin3x c6 nghiem.

,Dat

s i n 3 x = = - l (6)

xeR

7 a

Vay (2) c6 nghiem
(2)

=>mingn,(x) = g n , ( 3 - 2 m ) = m ^ - 4 m + 11 = ( m - 2 ) ^ + 7.

•

Khi m 9^-2, thi ming,,,(x)>4.

(3)

Nhu vay ta c6:

xeR
xeM

Phu'dng trinh da cho c6 dang: l"(x) - gn,(x)
T i r ( l ) ( 2 ) (3)suy ra:

,'6 'I3t

a) Neu m = 2, thi ming_,(x) = 7
= mx^ + (m + 1 )x + m + 2 > 0

( 1 ) diing vdi

m

+

+

Lucnaytaco: minfn,(x) = f,„(l) = m + 3 m - 2 .

... m - x £ » : { x >

xeR

Dieu nay xay ra khi va chi khi:

min 1"„ (x) > 0.

(2)

s/

I
4
Vay m > — la cac gia tri can tim cua tham so m.
Bai 5. Tim m dc bat phiTdng trinh sau:
+ 2 | x - m | + m - + m - l < 0 C O nghiem.

-m-2

xeR

Ttjf do di den xet he sau:

W.;.v

w.


fa

"
™

(2)

min i",„ (x) < 0.

Lijcn^y tac6: minf„,(x) = fn,(m)=: 2m^ + m - l
xeR

319


Chuyen

BDHSG Toan gii tri I6n nha't vA gia trj nh6 nhat - Phan Huy Khdi

Cty TNHH MTV DWH Khang Vijt

- 1 < rn
m < —

4~

1

m

-

(1)

2"

-

(2)

3

< X

3
X
I

-

^\

+

+

1

ro

= m—

m

Nhu" do dan de'n xet he sau:

l

I -

• it-

/g

1. Neu — > 3 (tuTc la neu m > 6). Liic do ta c6 bang bien thien sau:
2
in

s/

Tif do suy ra xet cac kha nang sau:

m>6

0

L u c n a y t a c o : min f„,(x)=:f,„

m

Luc nay taco:

.i & - i : / ' „ . . ;

A

Tilfd6suyrahe(l)(2)c6nghiemkhivachikhi:
0.

"11 + 1 X ) I - 2 ^ 2 < m < 2N/2

(3)

£ i

=>4(l) = 0

. TOd6 x^t cac kha nang sau:

l

a cac gia tri ciin tim cua m.

iL
ie
uO
nT
hi
Da
iH
oc
01
/

Dieu nay xay ra khi va chi khi:

m +1


r

/

ban.

Bai 8. Cho ham so: r„,(x) = 4x" - 4mx + m ' - 2m. Xct trcn mien - 2 < x < 0.

Tim m dc

m +3> 0

m>-3

he vo nghipm. TiTdo loai kha nang n;iy.

m < 0 (tuTc lii khi m > 0). Liic nay ta c6 bang bicn thicn sau:
,4
r
m
0
I
~ T

&r

/
/
/

+
0
4(t)
/

\
/

0

+

/
Vay:

I

0).

-

/
/

w.
khi

•>

y
()

^

• Ncu — < - 2 (<=> m < - 4 ) .
2
Lijc nay ta c6 bang bie'n thien sau:
323


rgaii y i a u | l u i r i i i i j i vd g i d in iniu i i i i t l l -

m

/
/

0

'in

+



/
/

1

m -6m-6

min I„, (u) = i„

() 0, v > 0 => x + 1 =

va y + 2 = v'.
0; v > 0

(3)

r,„(u) = 2u- - 2mu + m- - 3m - 3 = 0 (4)

,

,

•

- l - m
X€E

> 1

Xet cac kha nang sau:
min f()(x) = 0. Luc do (2) co dang 0 > - 1

Vay m = 0 thoa man yeu cau de bai

()
Vithc:

:2 •

Ta

Bay gid ket h(1p vrJi (6), la co:

-2m

veil J J. C

= m - 3m - 3.

-2^m>-l-m^

m^ -2V2m + l > 0

m>0

m>0

mN/2 + l ^

m>0

(X

m < V2 - 1


m>-V2 + l ^

;

mN/2

m
3-m

. . . .

/

25

23 1

16

16

J .

Tif bang bien thien suy ra: m + ^ > m - 1 > m - 3.
T u do suy ra:

= max

-A.

A^Aan jcef; Ta chtfng minh (*) nhiT sau:

ww

ra:

1


23

/g

x = i.

= max 3 - m; m + 8

m

up

. ,

1. D a l gn,(x) = - 2 x ^ + X + m vc'ti - 1 < X < 1.

-1

8

m + - neu — < m < 3
8
16 ~

Ta

>,fyi'(&iA'iMtm

Hiiihig ddn gidi

= max 3 - m ; - m -

g N c u — - < m < 3, i h i m - 3 < 0 va m + - > 0, do do:
8
8

l-V2
01
/

2V2m>-l-ni^

m O l h i
-2.

oc
01
/

Viet lai bat phtfrtng Irinh da cho difdi dang lu"(ing du^dng sau:

l„,(-2)>-2

-2m^ - m + 4 > - 2

f,na)>-2

m^ + 2 m - 2 > - 2

-2

(1)
It

fm(7t)

,

+

minf_,(x)

min f„,(x) = L , ( l ) = m ^ + 2 m - 2 .
-2 1, y > 1 =^ 4x + 5y > 9. D o la

IT

dicu v6 l i .
i i v i A V ' sC'ik)
V i le do D = D , U D j , trong do:

• TT


I

VxeE

Cong lirng vc (6) (10) di den: (min 1„, (x)

+ max f,„ (x))

>2 . t • >

Ta
LLfONG GIAC

/g

HINH HQC,

ro

up

§3. GI6I THIEU MQT SO BAI TOAN GIA TRj L6N NHAT. NHO NHAT

s/

Do

TRONG SO HQC,

om

41 - 1 < 0
4

Ti^ (3) va do t = 0; - 1 ; - 2 ; . . . . nen suy ra ifng v d i t = 0, ta c6:
min

(4)

P=::12.

Khi (x; y ) e D2 i h i P = -5x - 3y.

khao khac. Tuy nhicn trong muc nay, chung toi muon gicKi thicu vc'Ji cac ban

D o x < 0 ; y > 0 , t i r ( * ) ta c6:

ce

bo

ok

hoc, hinh hoc, liTi.Jng giac so di/dc chiing loi trinh bay trong mot cuon chiiycn

fa

w.

ww

0
5

Z)

A. Vai bai toan ve gid tri Ida nhat, nho nhat trong so hoc

L u c n a y P = 1 3 t - 12.

Bai 1. Cho P = 5|x| -3|y|,d day x, y thuoc tap hdp D diTdc xac dinh nhiTsau:

Til (6) va do t = 1; 2;... suy ra tfng vdi t = 1, thi:

{ ( x ; y ) : x , y e Z va 4x + 5y = 7)

'

'

'

•

'

>



'a,a: ••• ak chi' g o m nhCTng ihiTa so nguyen to 2 va 3, va khong eo qua 2 ihiTa so

' ' '

nguyen to 2. T h a i vay:

HUifiig dan gidi

. . r , i J.y
i, (ij).;t/

, V i 12"'" la so chan, con 5"" la so tan cung hang 5, nen suy ra:

lhay lich aiaj ... a^ tang Icn. D i c u nay mau ihuan v d i u'eh da cho la Idn

r,*;

-

(3)

.

ojljir

Vdi tinh chat nhuf vay ihi de dam bao aiaj... a^ idn nha'l, ta can phan ti'eh

(mod 13)

(4)

s/

R o r a n g t a c o : 12'"" = ( - 1 ) " ' "

Ta

hoac la 12'"" - 5"" chia cho 13 thi diT 12.

(5)

Tim

gia trj Idn nhat va nho nhat cua P.

fa



Trong cac phan tich 100 ra tong cac so nguyen diTdng a i , aa,... ak ta quan tam

"? i

Do vai tro binh dang giDTa x,, x , , x , , ) la cd the khong giam long quat ma gia sif

ww

Bai 3. X e t tap hdp ta't ck cac so' nguyen di/dng a i , a2,... ak sao cho aj + 3 3 + .. +

x.,,, sao cho

HUdng ddn gidi

Mat khac neu chpn m = 1 ; n = 1, thi P = 7.

Theo dinh nghla ve gia trj Idn nhat suy ra: m i n P = 7.

I''

X i + X2 +... + x,„i = 2011. X e l d a i lu'ilng P = X | X : . . . x,„.

Tir (4) (5) suy ra k h i dem 12'"" - 5 " " chia cho 13 t h i khong the dir 1 hoac 12.
D i e u nay mau thuan v d i (3). V a y khong the c6 (2), «?c la (1) dung

.

"


.c

om

khac ta luon luon c6 the bieu dien:
n,, = 4k„ + z„ v d i k

hoiicla 1 2 ' " " - 5 " " c h i a c h o l 3 l h i d i r l

(mod

Gia suf long ed nhieu hdn hai so hang 2. Chii y rhng iicii lhay ba so 2 hang

hai .so 3 Ihi 2 + 2 + 2 = 3 + 3, nhirng 3.3 > 2.2.2, vay

Tif(3)tac6:

=12

Trong each phan tich da cho khong cd so hang b > 4, vi la ed ihe lhay b
bang hai so hang 2 va b - 2. Rd rang 2(b - 2) > 4 ( T h a i vay, vl 2(b - 2) ^-

K e t hdp iai cac d i c u tren, tuf (2) suy ra: |l2"'" - 5 " " 1=1

12'""

,

la lay mot so hang a tiiy y khiie, a > 1 ( d l nhien no ton tai). Luc nay la

That vay gia s i l r ( l ) khong diing, ttfc la ton tai hai so nguyen diTtJng m,,, n„ s,^,
" =-V:«' "''•.•.tfl- m

,

Trong each phan lich da chpn, khong cd so hang 1 v i neu eo so hang 1 i h i

khac:

- 1 ) + ^ + ... + x ^ + ( x ^ + 1 ) = 5^ + x^ +... + X j , , = 2011, nen

bp so m d i nay cung thoa man y e u cau de b a i . Cfng v d i bp so nay ta cd:

•L...

333


Chuyfin

BDHSG Toan gia Iri I6n nha't va gii tr| nh6 nhS't - Phan Huy KhSi

P = (^-l)5^...X^(x3o +l) = X2...X29(x3o-X,

Do x ^ > ( ) Vi = 2r29, va x 7 < x ^ ^ P > P .

P = 67^'".68.

(1)

T o m lai la co: max P = 67^*^68
K c t hop l a i ta co: m i n P = 1982 va max P = 67'''.68.

'"''^ '

p a i 5. Cho k la so nguyen di/Ong > 3.



trong do:

,

C3„-X,>2


z,, + 1. V i x„ + y„ + z„ = k nen ta co: x„ + (y„ - 1) + (z,, + 1) = k

up

Chu y rang do X3(, - x, > 2 ma x, > I => Xj,, - x, > 1

/g

ro

khac de thay: (x7 + l ) + x^ + ... + x ^ + ( x 3 o - l ) = x , + . . . + X30 = 2011,

om

nen bp so m d i cung thoa man yeu cau de bai. l?ng v d i bp so'n^y ta c6:


^

,f'

HUdng dan

29 thiira so'

Xet bo so' ( x , , X 2 , X 3 , , )

K i i i n g Vi^t

Nhu" vay chon day chting han 29 so' bang 67 vi mot so bang 68 t h i :

+l).

(1) chuTng to rang bo so (x,, X j , . . . , X 3 „ ) khong lam cho tich P dat gia trj nho

Tir do suy ra: m i n P =

[ivvil

Cty TNHH M I V

^

hoac X3(, = X|
hoac X3(, = X | + 1

Nhir the dieu kien can de P dat gia tri Idn nhat la trong 30 so thi khong diTdc

1 > 0 (do

X(, > Z() +

1) nen ta co:

l'(X() - 1, yo, z<, + 1) > x„y„z„ => f(x„ - 1, y„, z„ + 1) > f(x„, y,,, z,,).
Ba't dang thiirc nay cung mau thuan vdi (1).

,1 i ,

Vay trong trU'dng hdp b . khong the xay ra.
Ne'u X() - 1 > y,, > Zo. L a p luan nhiT tren cung suy ra mau thuan

' '

T o m lai gia thiet Xo - z,, > 1 la sai, vay (2) dung
Tir (2) suy ra chi cd the xay ra hai kha nang sau:

De thoa man (*) c6 the chon a = 67, t = 29 (vi 30.67 + 1 = 2011)
335


Chuygn d l BDHSG Toan gia tri Idn nha't va gia tri nho nhat - Phan Huy Khii

i. Neu x„ -

Z|) =

Cty TNHH MTV DWH Khang Vigt


k +1

max f(x,y,z) neu sijf dung baft dang thtfc

(x.y,/)eD

,v,

Cosi.

:

iL
ie
uO
nT
hi
Da
iH
oc
01
/

;:

k k k

k—1



khi k = 0(mod3)

Khi do phan tur con lai cua X bao gom cac phan tuT 1, 45, 46,..., 2011. '

(k + 2 ) ( k - l ) '

khi k = l(mod3)

Vdi cac phan tuT con lai nay chi xay ra hai kha nang sau:

Ta

27

(x.y,/.)eD

a. Xet tich hai phan

(k + l ) ^ ( k - 2 )

khi k = 2 (mod 3)

s/

max f(x,y,z) =

up

27


khac.

thoa man btft d^ng thurc:

n>43

(1)

That vijy, gia s u r ( l ) khong dung, tuTc la ton tai so nguyen du'dng no < 43 ma

ww

Do k > 0 va k 13 nen

la 1. Tich ay chinh bang

Ta se chiJng minh rang moi so' nguyen duTdng n thoa man yeu cau de bai

w.

Dau " = " xay ra khi va chi khi x = y = z = - .

tiif

hdp con thoa man yeu cau de ra.

ce

27

si

Loai di 43 phan tur sau: 2, 3, 4 , 4 4 .

27

van thoa man yeu cau dau bai.
Xet 43 bo ba sau: (2, 87, 2.87); (3, 86, 3.86);... (44, 45,44.45).
'

27

2. Tuy nhien neu k / 3 thi khong the iip dung difde ba't dang thufc Co-Si de g ' ^
bai toan nay.
L i do d cho da'u " = " xay ra trong bat dang thuTc x y z < — khi va chi khi:

Dat l'(x) = x(89 - x) vdi 2 < X < 44. Ta eo: f (x) = 89 - 2x > 0 khi 2 < x < 44
=> f(x) la ham dong bien tren [2; 44].

' ~ •

Tir do ta eo: 2.87 < 3.86
nhat thoa m a n y e u cau dau bai la: m i n n = 43.
2,... T i m

B a i 7 , X e t t a p h d p la't ca 7 so n g u y e n t o k h a c n h a u c 6 c a c t i n h c h a t s a u :

m a x i"(n)
l
ce

bo

c a c so c h 5 n , m a c h i i n g l a i k h d c n h a u . NhiT v a y t a c o n h i e u h d n 2 so nguyen

fa

w.

ww

'

,j

X e t k h i n = k + 1. C o h a i tru'dng h d p x a y r a :

-

H o a c la k c h i n ( k = 2 m ) . K h i d o : l ( k + 1) = f ( 2 m + 1) = l(m) + 1.
D o m < k , n e n t h e o g i a thic't q u y n a p suy r a l ( k + 1) c h i n h b a n g so c a c chu"
sd^ 1 t r o n g b i e u d i e n n h i p h a n c u a m c o n g t h e m 1.
M a t khac, k+ 1 = 2 m + 1 m i i :

2 m + 1 2_
1

TiJfc l a : 2 m + 1 = (\|a2...(Vpl



G i a su" b d d e d a d u n g d e n n = k > i , i i k la v d i m o i / < k , ("(/) c h i n h b a n g

.c

d i j n g m o t t r o n g b a so a, b , c l a 2 , c o n h a i so' c o n l a i l a h a i so n g u y e n to Ic

to chS^n k h a c n h a u t r o n g 7 so n o i t r e n . D o l a d i e u v 6 l i .

'

sd^cac chi? so' 1 t r o n g b i c u d i e n n h i p h a n c u a s o / .

R o r a n g a + b + c la so Idtn nha't t r o n g 7 so n g u y e n t o n o i t r e n

a + b + c = 8 0 0 + c < 8 0 0 + 7 9 7 => a + b + c < 1 5 9 7 .

'

Ta

ij i ,

So n g u y e n t o kUn nha't d i r d i 8 0 0 l a 7 9 7 , v i t h e :

' '

Mat k h a c : 1 = I / 2 d d a y ta d u n g k i h i e u a,a2...a„ 12 d e c h i so g h i t h e o h e c d

tdng quat)


la b i e u d i e n c u a

i &/ i
. stS,

'* >


Cty TNHH MTV DWH Khang Vi^t

Chuyen 66 BDHSG Toan gii tr| Mn nhS't va gi^ trj nh6 nhat - Phan Huy Kh^i

Tom

l a i , bo de dung k h i n = k + 1.

\i the: a-^ + 3 = f25a2 + {5r2 + 3) ^ + 3

Theo nguyen l i quy nap suy ra bd dc dung v d i m o i n (dpcm).
,1

>a, + 3 = ((5r2 + 3 ) ' ( m o d 2 5 ) .

Ta lha'y so' nho nhii't c6 11 chiy so' 1 trong bicu d i c n d\idi he nhi phan

1_1__1

(5r2 + 3 ^ + 3 j ; 2 5

a = 125a,+ 38.

Do phat hicn ra da'u bang trong (1). la lict kc ra cac so la nho nha't, g;1n nh6


lai thu djnh nghla vc gia trj be nhat ta c6:

max

t'(n) = 10 o

n thda man (2).

T i r ( l ) s u y ra, n o i r i e n g ( a - V 3 ) ; 5 . V l the tif (2) c6
Do r,

G

| 0 ; 2; 3; 4} nen suy ra: r, = 3 => a = 5ai + 3.

D;)t a, = 5a: + f:, v d i rj e {0; 2; 3; 4 } , ta c6:
a = 5(5a:

+ r . ) + 3 = 25a2 + Srj + 3.

V

Ta lha'y a cd: 9 + 2.51 = 111 chCT so. Do vay sau khi xda di 100 chu" so tuy y
(1)

< 4, ta c6:

(mod 5) => (a^ + 3) =rl+5 (mod 5)


du-dng lir 1 den 60 Iheo thiir lit tir nho den Idn, lu-c la: A = 1234...5960

w.

Hii(ing dan gidi

,11-a,

Bai 10. So nguyen A du'dc tao lhanh bling each viet lien nhau cac so' nguyen

fa

b. n viet trong he thap phan c6 4 so tan cung la 1997.

„

^^..^

dau bai.

IJai 9. T u n n so' nguyen du'dng n nho nhat thoa man cac tinh chat sau:
a. n la lap phu'dng cua mot so' nguyen dtfdng.

i^^A"^.wMf^

Nhir vay n = 1413 = 2821151997 la so tir nhiC-n nho nhat thda man yeu cau

ok

l

Tir (5) (6) di den: 125a, + 38 = 16l + 5 => 33 = 16l - 125a3

= 1919

.c

Vay

9chiIso' 1

Ta

11110

Tir (a- + 3 ) = { r U 3 )

= 1535

s/

0 ( . JjciVf!

1_LJ

up

OLJ
9 chiJsiY



(2)

b d t c a c chcr so k h a c 9 i r o n g c a c so l i f 1 d e n 4 9 ) . T a se x a c d i n h t h e m 6 c h i j sg-

f

c o n l a i c i i a n tiT d a y so:

Tir(l)(2) suyra:

5 0 51 52 53 54 55 5 6 57 58 5 9 6 0
Dcthay6sod6

'

X

;


dinh cua hinh

D i r i c h l e t c o i t nhat hai d i n h co cung m o t b o ba m a u l i e n

ok

Cac

om

v u o n g , n e n c a c l a m c i i a c h i i n g n ; l m t r o n g h i n h v u o n g A | B | C i D | c 6 t a r n 0 vii

a-2

n h m trong

mot

hinh

vuong

nhc

i

xanh ( X ) , va m a u d o (D)), ma m o i m a u d i i n g de to d i i n g ba canh.

, nen lai c6:
343


\3
^

c2

(5)


13

f
i

Bai 4.

M o t lam giac deu diftfc chia thanh n^ tam giac deu b ^ n g nhau. M o t so' tam

V d i n = 6, xet each to mau sau day: Do (D), Xanh ( X ) ,

giac do di/dc dcinh so bc'li cac so' 1, 2, 3,... , m sao cho cac tam giac v d i cdc so'

Vang ( V ) , T i m ( T ) , Nau ( N ) , L a m ( L ) .

hSn tie'p thi phai c6 canh chung.

Cach to mau nay thoa man m o i yeu cau dc ra.

T i m gia trj Idn nhat c6 the c6 ciia m.

Ta

T 6 m l a i , neu n = 5, ta luon di den mau thuan. Do do n > 5 1 / ,

s/

Hi((ing din gidi

Chia cac canh tam giac deu thanh n phan bang nhau. Tir cac d i e m chia ve cac
A .
Tir (2) va (3) suy ra:

To mau cac tam giac thanh cac tam giac den,

Theo each danh s6' tam giac thi hai tam giac

>.

ww

V a y tong so'cac cap 6 do l a : A =

=

••\: ,

w.

6 hang thi? i , so cac cSp 6 de c6 the c6 la: C^

.c

^ ,•
, ,
HUdngddngidi
'

thuoc (xem hinh ve, du'c^ng net lien dam ( —

13

m