8/6/2019 BDThay

1/5

Dng Vn Sn Gio vin tr ng THPT H Huy T p, Vinh Ngh An .

M T S PHNG PHP CH NG MINH B T NGTH C C CH A BI U TH C XYZ

Bi ton: Chng minh bt ng thc (BT)c cha biu thc xyz trong z y x ,, l cc s thc khng m, c vai tr bnh ng v BT tngng v i ),,()( z y xP xyz n v i * N n ; ),,( z y xP l a thc th ng gy rtnhiu kh khn cho hc sinh v vic nh gi ),,()( z y xP xyz n l khngthun li.

Trong bi vit ny, tc gixin gi i thiu mt s k nng gii bi tondng ny.1. S dng BT: V i x, y, zl cc s th c khng m ty , ta c

( )( )( ) x y z x z y y z x xyz (1).Th d 1. Cho x, y, z l cc s thc khng m tha mn 1 z y x .Chng minh rng

27

720 xyz zx yz xy .

( thi IMO nm 1984) L i gii. p dng (1) v githit, ta c

xyz xyz zx yz xy z y x xyz z y x 8)(4)(21)21)(21)(21( .Suy ra

4

12

xyz xyz zx yz xy (2)

Mt khc, ta c 27133

z y x xyz (3)

T(2) v (3) suy ra27

72 xyz zx yz xy .

Ngoi ra, tgithit suy ra 1,,0 z y x .Do 0)1()1(2 zx x yz z xy xyz zx yz xy .2. S dng tnh ch t: Trong ba s z y x ,, lun t n t i t nh t hai s saocho chng cng khng l n hna hoc cng khng nh hn a , v i a l s thc ty (4). Th d 2. Cho z y x ,, l cc s thc khng m tho mn 4 xyz zx yz xy (*). Chng minh rng: zx yz xy z y x . ng thc xy ra khi no.

( thi hc sinh gii Quc gia nm 1996) Li gii. Theo tnh cht (4) vvai tr z y x ,, trong bi ton bnh ng nn

khng mt tnh tng qut ta c th gi s1

1

y

x hoc1

1

y

x .

Khi , ta c y x xy y x 10)1)(1( Suy ra zx yz xy z xy xyz yz xz z xyz y x z xy z )()1( (5)

Ta s chng minh: z xy xyz z y x (6)Tht vy: 4)1)((4)6( z y x z xy zx yz xy z y x (7)

ww.VIETMATHS.com

See on VIETMATHS.comWeb page VIETMATHS.com

8/6/2019 BDThay

2/5

Dng Vn Sn Gio vin tr ng THPT H Huy T p, Vinh Ngh An .

Nu 0 y x (*) tr thnh 0 =4 v l. Do 0 xy y x v t (*) ta c: z =

xy y x

xy4

V th : 4)41)(()7( xy y x

xy y x

)(4)4)(( xy y x y x y x (V 0 xy y x )0)( 2 y x , ng.

T (5) v (6) suy raiu phi chng minh. Trong trng hp ny ng thc xy ra khi

x = y = z = 1 hoc x = y = 2, z = 0.Do ng thc xy ra khi: x = y = z = 1 hoc x = y = 2, z = 0hoc x = z = 2 , y = 0 hoc z = y = 2 , x = 0.3. nh gi ri t n ph.Th d 3. Gi s z y x ,, l cc s thc tha mn 2222 z y x .Chng minh xyz z y x 2 .

(Poland 1991)Li gii. p dng BT Bunhiacpxki, ta c

222 )1(1.)()1(1).( xy z y x xy z y x )22)(2()( 222222 xy y x z y xy x xyz z y x

)22)(1(2)( 222 xy y x xy xyz z y x (8)

V 122.22222

z y x y x y x xy nn 11 xy .Do t xyt , ta c

)()22)(1()22)(1( 222 t f t t t xy y x xy , v i 11 t .D dng chng minh c .2)(max

]1;1[t f

Suy ra 2)22)(1( 22 xy y x xy (9)T(8) v (9) suy ra 4)( 2 xyz z y x .Vy 2 xyz z y x hay ta c iu phi chng minh.

4. t n ph xyzQ zx yz xyP z y xS ;; .Th d 4. Cho ba s thc khng m z y x ,, . Chng minh rng)(9)2)(2)(2( 222 zx yz xy z y x (10)

(Asian Pacific Math 2004)L i gii. Ta c

)(98)(4)(2)10( 222222222 zx yz xy z y x z y x z y x (11)t xyzQ zx yz xyP z y xS ;; , ta c BT (11) tr thnh

PPSSQPQ 98)2(4)2(2 222

039

3539

839

10

3222

2

PSPSQPS

Q (12)

ww.VIETMATHS.com

See on VIETMATHS.comWeb page VIETMATHS.com

8/6/2019 BDThay

3/5

Dng Vn Sn Gio vin tr ng THPT H Huy T p, Vinh Ngh An .

D dng chng minh c PSSQP 3,3 22 suy ra (12) ng.Vy ta c iu phi chng minh.

Vn dng cc phng php trn Th d 5. Cho z y x ,, l cc sthc khng m tha mn 3333 z y x .Chng minh rng 2 xyz zx yz xy .L i gii 1. p dng (1) v githit, ta c

xyz x z y y z x z y x ))()((

)1()1()1(63

3222222

222222333

x z zx yz z y xy y x xyz

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

M 3 333 333 33222222 333)1()1()1( x z z y y x x z zx yz z y xy y x Suy ra )(363 zx yz xy xyz Vy ta c iu phi chng minh.L i gii 2. Theo tnh cht (4) vvai tr z y x ,, trong bi ton bnh ng nn

khng mt tnh tng qut ta c th gi s1

1

y

x hoc1

1

y

x .

Khi , ta c 10)1)(1( xy y x y x Suy ra z xy xyz zx yz xy z xy y x z )( (13)Mt khc, ta c

3

111.1.;

3

11..

33 3

333 33 z z z

y x y x xy .

Suy ra 23

3333 z y x z xy (14)

T (13) v (14) suy ra iu phi chng minh.L i gii 3. V vai tr ca z y x ,, trong bi ton bnh ng nn khng mt tnhtng qut ta c thgis z y x z ;;min . Khi :

3333 33 z z y x (V 0,, z y x )1013 z z ;

)()1( x y z z xy xyz zx yz xy .M

.3

111.1.;

3

111.1.;

3

11..

33 3

33 3

333 33 y y y

x x x

y x y x xy

Suy ra

3

)7(

3

)1)(4(

3

2

3

2

3

)1)(1( 333333 z z z z y x z

z y x xyz zx yz xy

Do 3

433 z z xyz zx yz xy (15)

t 343

)(

3 z z z f , v i 10 z

ww.VIETMATHS.com

See on VIETMATHS.comWeb page VIETMATHS.com

8/6/2019 BDThay

4/5

Dng Vn Sn Gio vin tr ng THPT H Huy T p, Vinh Ngh An .

Ddng chng minh c 2)(max]1;0[

z f .Suy ra :v i 10 z ta c 2)( z f (16)T (15) v (16) suy ra iu phi chng minh.L i gii 4. Ta c ))((3 222333 zx yz xy z y x z y x xyz z y x (17).t xyzQ zx yz xyP z y xS ;; , t (17) v gi thit ta c

)3(33 2 PSSQ (18)

M3

11

3

11

3

111.1.1.1.1.1.

3333 33 33 3 z y x z y x z y x

33

6333 z y xS (19)

T(18) v (19) suy ra)3(333 2 PSQ (V PS 32 )

122 PSQP (20)Mt khc, ta c

3 333 333 332222 1..1..1..2 z z y y x x z y xPS

3

1

3

1

3

1 333333 z z y y x x

Suy ra:322 PS (21)

T (20) v (21) suy ra iu phi chng minh.Cc bi t p t luyn

1. Cho z y x ,, l cc s thc khng m tha mn 1 z y x .Chng minh rng 19)(4 xyz zx yz xy .2. Cho z y x ,, l cc s thc khng m tha mn 3222 z y x .Chng minh rng 2 xyz zx yz xy .3. Cho ba s thc cba ,, bt k. Chng minh rng

)1)(1)(1(32222 cbaabccba .(Marian Tetiva, Mircea Lascu, Gabriel Dospinescu)

4. Cho tam gic ABC c chu vi bng 1. Chng minh rng

4

13

9

2 333 abccba .(Bi T 5 / 353Tp ch Ton hc v tui tr thng 3 nm 2007)

5. Chng minh rng nu z y x ,, l cc s thc khng m tho mn iu kin: 4222 xyz z y x th ta c 20 xyz zx yz xy .

( thi USAMO-2001)6. Cho )1;0(,, z y x , tho mn )1)(1)(1( z y x xyz .

ww.VIETMATHS.com

See on VIETMATHS.comWeb page VIETMATHS.com

8/6/2019 BDThay

5/5

Dng Vn Sn Gio vin tr ng THPT H Huy T p, Vinh Ngh An .

Chng minh rng:4

3222 z y x .7. Cho ba s thc bt k z y x ,, . Chng minh rng

)1(4)3)(3)(3( 222 z y x z y x .

(Tp ch Ton hc v tui tr) 8. Cho ba s thc z y x ,, tha mn 1 xyz .Chng minh rng )(23222222 z y x x z z y y x .

ww.VIETMATHS.com

See on VIETMATHS.comWeb page VIETMATHS com