50 de on Tap Toan8 Co Ban

7/23/2019 50 de on Tap Toan8 Co Ban

1/48

1

1 (43)

Cu 1:

Cho x =2 2 2

2

b c a

bc

; y =

2 2

2 2

( )

( )

a b c

b c a

Tnh gi tr P = x + y + xy

Cu 2:Gii phng trnh:

a, 1

a b x =

1

a+

1

b+

1

x(x l n s)

b,2

2

( )(1 )b c a

x a

+

2

2

( )(1 )c a b

x b

+

2

2

( )(1 )a b c

x c

= 0

(a, b, c l hng s v i mt khc nhau)

Cu 3:

Xc nh cc s a, b bit:

3

(3 1)

( 1)

x

x

=

3( 1)

a

x +

2( 1)

b

x

Cu 4:

Chng minh phng trnh: 2x2 4y = 10 khng c nghim nguyn.

Cu 5:

Cho ABC; AB = 3AC

Tnh t s ng cao xut pht t B v C


2/48

2

2 (44)

Cu 1:

Cho a, b, c tho mn: a b c

c

=

b c a

a

=

c a b

b

Tnh gi tr M = (1 +b

a)(1 +

c

b)(1 +

a

c)

Cu 2:

Xc nh a, b f(x) = 6x4 7x3 + ax2 + 3x +2 chia ht cho y(x) = x2 x + b

Cu 3:

Gii PT:

a, (x-4) (x-5) (x-6) (x-7) = 1680.

b, 4x2 + 4y 4xy +5y2 + 1 = 0

Cu 4:

Tm gi tr ln nht ca phn s m t s l mt s c 3 ch s m mu l tng cc ch s ca n.

Cu 5:

Cho ABC cn ti A, trn AB ly D, trn AC ly E sao cho: AD = EC = DE = CB.

a, Nu AB > 2BC. Tnh gc A ca ABC

b, Nu AB < BC. Tnh gc A ca HBC .


3/48

3

3 (45)

Cu 1:

Phn tch thnh nhn t:

a, a3 + b3 + c3 3abc

b, (x-y)3 +(y-z)3 + (z-x)3

Cu 2:

Cho A =2 2

2

(1 )

1

x x

x

:

3 31 1( )( )

1 1

x xx x

x x

a, Rt gn A

b, Tm A khi x= -1

2

c, Tm x 2A = 1

Cu 3:a, Cho x + y + z = 3. Tm gi tr nh nht ca M = x2 + y2 + z2

b, Tm gi tr ln nht ca P =2( 10)

x

x

Cu 4:

a, Cho a,b,c > 0, CMR:

1 < a

a b +

b

b c +

c

c a < 2

b, Cho x,y 0 CMR:

2

2

x

y+

2

2

y

x

x

y+

y

x

Cu 5:

Cho ABC u c di cnh l a, ko di BC mt on CM = a

a, Tnh s o cc gc ACM

b, CMR: AM AB

c, Ko di CA on AN = a, ko di AB on BP = a. CMR MNP u.


4/48

4

4 (46)

Cu 1:

Phn tch thnh nhn t:

a, a8 + a4 +1

b, a10 + a5 +1

Cu 2:a, Cho a+b+c = 0, Tnh gi tr ca biu thc:

A =2 2 2

1

b c a +

2 2 2

1

c a b +

2 2 2

1

a b c

b, Cho biu thc: M =2

2 3

2 15

x

x x

+ Rt gn M

+ Tm x Z M t gi tr nguyn.

Cu 3:

a, Cho abc = 1 v a3 > 36,

CMR:2

3

a+ b2 + c2 > ab + bc + ca

b, CMR: a2 + b2 +1 ab + a + b

Cu 4:

a, Tm gi tr nh nht ca A = 2x2 + 2xy + y2 - 2x + 2y +1

b, Cho a+b+c= 1, Tm gi tr nh nht

P = a3 + b3 + c3 + a2(b+c) + b2(c+a) + c2(a+b)

Cu 5:

a, Tm x,y,x Z bit: x2 + 2y2 + z2 - 2xy 2y + 2z +2 = 0

b, Tm nghim nguyn ca PT: 6x + 15y + 10z = 3

Cu 6:

Cho ABC, H l trc tm, ng thng vung gc vi AB ti B, vi AC ti C ct nhau ti D.a, CMR: T gic BDCH l hnh bnh hnh.

b, Nhn xt mi quan h gia gc A v D ca t gic ABDC.


5/48

5

5 (47)

Cu 1:

Phn tch thnh nhn t:

a, (x2 x +2)2 + (x-2)2

b, 6x5 +15x4 + 20x3 +15x2 + 6x +1

Cu 2:a, Cho a, b, c tho mn: a+b+c = 0 v a2 + b2 + c2= 14.

Tnh gi tr ca A = a4+ b4+ c4

b, Cho a, b, c 0. Tnh gi tr ca D = x2003 + y2003 + z2003

Bit x,y,z tho mn:2 2 2

2 2 2

x y z

a b c

=

2

2

x

a+

2

2

y

b+

2

2

z

c

Cu 3:

a, Cho a,b > 0, CMR: 1a + 1b 4a b

b, Cho a,b,c,d > 0

CMR: a d

d b

+

d b

b c

+

b c

c a

+

c a

a d

0

Cu 4:

a, Tm gi tr ln nht: E =2 2

2 2

x xy y

x xy y

vi x,y > 0

b, Tm gi tr ln nht: M =2( 1995)

x

x vi x > 0

Cu 5:

a, Tm nghim Z ca PT: xy 4x = 35 5y

b, Tm nghim Z ca PT: x2 + x + 6 = y2

Cu 6:

Cho ABC M l mt immin trong ca ABC . D, E, F l trung im AB, AC, BC; A, B, C

im i xng ca M qua F, E, D.

a, CMR: ABAB l hnh bnh hnh.

b, CMR: CC i qua trung im ca AA


6/48

6

6 (48)

Cu 1:

Cho a

x y =

13

x z v

2

169

( )x z =

27

( )(2 )z y x y z

Tnh gi tr ca biu thc A =3 22 12 17 2

2

a a a

a

Cu 2:

Cho x2 x = 3, Tnh gi tr ca biu thc M = x4 - 2x3 + 3x2 - 2x + 2

Cu 3:

a, Tm gi tr nh nht ca M = x(x+1)(x+2)(x+3)

b, Cho x,y > 0 v x + y = 0, Tm gi tr nh nht ca N = 1

x+

1

y

Cu 4:

a, Cho 0 a, b, c 1

CMR: a2 + b2 + c2 1+ a2b + b2c + c2a

b, Cho 0


7/48

7

7 (49)

Cu 1:

Cho a, b, c khc nhau tho mn:

2 2 2

2

b c a

bc

+

2 2 2

2

c a b

ac

+

2 2 2

2

a b c

ab

= 1

Th hai phn thc c gi tr l 1 v 1 phn thc c gi tr l -1.

Cu 2:

Cho x, y, z > 0 v xyz = 1

Tm gi tr ln nht A =3 3

1

1x y +

3 3

1

1y z +

3 3

1

1z x

Cu 3:

Cho M = a5 5a3 +4a vi aZ

a, Phn tch M thnh nhn t.

b, CMR: M120 aZ

Cu 4:

Cho N 1, n N

a, CMR: 1+ 2+ 3+....+n = ( 1)

2

n n

b, CMR: 12 +22 + 32 +......+n2 = ( 1)(2 1)

6

n n n

Cu 5:

Tm nghim nguyn ca PT:

x2 = y(y+1)(y+2)(y+3)

Cu 6:

Gii BPT:2 2 2

1

x x

x

>

2 4 5

2

x x

x

- 1

Cu 7:

Cho 0 a, b, c 2 v a+b+c = 3

CMR: a2 + b2 + c2 5

Cu 8:

Cho hnh ch nht ABCD c chiu di BC gp 2 ln chiu rng CD, t C k Cx to vi CD mt g

150 ct AD ti E

CMR: BCE cn.


8/48

8

8 (50)

Cu 1:

Cho A =3 2

3 2

2 1

2 2 1

n n

n n n

a, Rt gn A

b, Nu nZ th A l phn s ti gin.

Cu 2:

Cho x, y > 0 v x + y = 1

Tm gi tr ln nht ca P = (1 -2

1

x)(1 -

2

1

y)

Cu 3:

a, Cho a, b ,c l di 3 cnh ca 1 tam gic

CMR: a2 + b2 + c2 < 2(ab+bc+ca)

b, Cho 0 a, b , c 1

CMR: a + b2 +c3 ab bc ca 1

Cu 4:

Tm x, y, z bit:

x+yz = y+z-x = z+x-y = xyz

Cu 5:

Cho nZ v n 1

CMR: 13 + 23 +33 +......+n3 =2 2( 1)

4

n n

Cu 6:

Gii bt phng trnh:

(x-1)(3x+2) > 3x(x+2) + 5

Cu 7:

Chia tp N thnh cc nhm: 1; (2,3); (4,5,6)....., nhm n gm n s hng. Tnh tng cc s trong nh94.

Cu 8:

Cho hnh vung ABCD. M, N l trung im AB, BC, K l giao im ca CM v DN

CMR: AK = BC


9/48

9

9 (51)

Cu 1:

Cho M = a

b c +

b

a c +

c

a b ; N =

2a

b c +

2b

a c +

2c

a b

a, CMR: Nu M = 1 th N = 0

b, Nu N = 0 th c nht thit M = 1 khng?

Cu 2:

Cho a, b, c > 0 v a+b+c = 2

CMR:2a

b c +

2b

a c +

2c

a b 1

Cu 3:

Cho x, y, z 0 v x + 5y = 1999; 2x + 3z = 9998

Tm gi tr ln nht ca M = x + y + z

Cu 4:

a, Tm cc s nguyn x x2 2x -14 l s chnh phng.

b, Tm cc s ab sao cho ab

a bl s nguyn t

Cu 5:

Cho a, b, c, d l cc s nguyn dng

CMR: A = aa b c + b

a b d + c

b c d + d

a c d khng phi l s nguyn.

Cu 6:

Cho ABC cn (AB=AC) trn AB ly im M, trn phn ko di ca AC v pha C ly im N sao

cho: BM = CN, v hnh bnh hnh BMNP

CMR: BC PC

Cu 7:

Cho x, y tho mn: 2x2 + 21x +

2

4y = 4 (x 0)

Tm x, y xy t gi tr nh nht


10/48

10

10 (52)

Cu 1:

Cho a, b, c > 0 v

P =3

2 2

a

a ab b +

3

2 2

b

b bc c +

3

2 2

c

c ac a

Q =

3

2 2

b

a ab b +

3

2 2

c

b bc c +

3

2 2

a

c ac a

a, CMR: P = Q

b, CMR: P 3

a b c

Cu 2:

Cho a, b, c tho mn a2 + b2 + c2 = 1

CMR: abc + 2(1+a+b+c+ab+bc+ca) 0

Cu 3:

CMR x, yZ th: A = (x+y)(x+2y)(x+3y)(x+4y) + y4 l s chnh phng.

Cu 4:

a, Tm s t nhin m, n sao cho: m2 + n2 = m + n + 8

b, Tm s nguyn nghim ng: 4x2y = (x2+1)(x2+y2)

Cu 5:

Tm gi tr ln nht, gi tr nh nht: A = 24 3

1

x

x

Cu 6:

Cho x =2 2 2

2

b c a

ab

; y =

2 2

2 2

( )

( )

a b c

b c a

Tnh gi tr: M =1

x y

xy

Cu 7:

Gii BPT: 1 x a x (x l n s)

Cu 8:

Cho ABC , trn BC ly M, N sao cho BM = MN = NC. Gi D, E l trung im ca AC, AB, P

giao ca AM v BD. Gi Q l giao ca AN v CE.

Tnh PQ theo BC


11/48

11

11 (53)

Cu 1:

Cho x = a b

a b

; y =

b c

b c

; z =

c a

c a

CMR: (1+x)(1+y)(1+z) = (1-x)(1-y)(1-z)

Cu 2:

Tm gi tr nh nht, ln nht ca A =4

2 2

1

( 1)

x

x

Cu 3:

a, Cho a, b, c > 0 v a+b+c = 1

CMR: b+c 16abc

b, Cho 0 < a, b, c, d < 1. CMR c t nht mt bt ng thc sai trong cc bt ng thc sau:

2a(1-b) > 1 8c(1-d) > 1

3b(1-c) > 2 32d(1-a) > 3

Cu 4:

Gii BPT: mx(x+1) > mx(x+m) + m2 1

Cu 5:

a, Tm nghim nguyn t ca PT: x2 + y2 + z2 = xyz

b, Tm s nguyn t p 4p + 1 l s chnh phng.

Cu 6:Tm s c 2 ch s m s y l bi s ca tch hai ch s ca n.

Cu 7:

Cho hnh thang ABCD (BCAD). Gi O l giao im ca hai ng cho AC, BD; Gi E, F l trun

im ca AD, BC

CMR: E, O, F thng hng.


12/48

12

12 (54)

Cu 1:

Tm a thc f(x) bit:

f(x) chia cho x+3 d 1

f(x) chia cho x-4 d 8

f(x) chia cho (x+3)(x-4) thng l 3x v dCu 2:

a, Phn tch thnh nhn t:

A = x4 + 2000x2 + 1999x + 2000

b, Cho:2 2 2x yz y zx z xy

a b c

CMR:2 2 2a bc b ca c ab

x y z

Cu 4:

CMR: 1

9+

1

25+.....+

2

1

(2 1)n 0 v 9b(b-a) = 4a2

Tnh M =

a b

a b

Cu 2:

a, Cho a, b, c > o

CMR:2a

b c +

2b

c a +

2c

a b

2

a b c

b, Cho ab 1

CMR:2

1

1a

+2

1

1b

2

1ab

Cu 3:

Tm x, y, z bit:

x+2y+3z = 56 v 1

1x =

2

2y =

3

3z

Cu 4:

a, Tm gi tr ln nht, gi tr nh nht ca M =2

2 1

2

x

x

b, Tm gi tr nh nht A =2

2

6 5 9x x

Cu 5:

Gii BPT: mx2 4 > 4x + m2 4m

Cu 6:

a, Tm s nguyn dng x tho mn: x(x+1) = k(k+2)

k l s nguyn dng cho trc.

b, Tm nghim nguyn ca PT: 2x-5y-6z =4.

Cu 7:

Cho hnh vung ABCD, V pha ngoi hnh vung trn cnh BC v BCF u, v pha trong hnh

vung trn cnh AB v ABE u.

CMR: D, E, F thng hng.


14/48

14

14 (56)

Cu 1:

Cho A = (2

2 2 3 2

1) : ( ) :

x x y y x

y xy x xy x xy x y y

a, Tm TX ca A

b, Tm x, y A > 1 v y < 0.Cu 2:

a, Gii PT: x4 + 2x3 2x2 + 2x - 3 = 0

b, Gii BPT: 3 mx < 2(x-m) (m+1)2

Cu 3:

Cho a, b, c > 0

CMR: 3

2

a b c

b c a c a b

Cu 4:

CM: A = n6 n4 +2n3 +2n2 khng l s chnh phng vi nN v n >1

Cu 5:

Cho f(x) = x2 + nx + b tho mn 1

( ) ; 12

f x x

Xc nh f(x)

Cu 6:

Cho x, y > 0 tho mn xy= 1

Tm gi tr ln nht A =4 2 2 4

x y

x y x y

Cu 7:

Cho hnh thang ABCD (AD//BC). M, N l trung im ca AD, BC. T O trn MN k ng thng

song song vi AD ct AB, CD ti E v F.

CMR: OE = OF


15/48

15

15 (57)

Cu 1:

Cho xyz = 1 v x+y+z = 1 1 1

x y z = 0

Tnh gi tr M =6 6 6

3 3 3

x y z

x y z

Cu 2:

Cho a 0 ; 1 v 1 21 2 31 2

1 11; ; .....

2 1 1

x xax x x

a x x

Tm a nu x1997= 3

Cu 3:

Tm m phng trnh c nghim m: ( 2) 3( 1)

11

m x m

x

Cu 4:

Vi nN v n >1

CMR: 1 1 1 1

.... 12 1 2 2n n n

Cu 5:

Cho M = 3x2 - 2x + 3y2 2y + 6x +1

Tm gi tr M bit: xy = 1 v x y t gi tr nh nht.

Cu 6:

Tm x, yN bit: 2x + 1 = y2

Cu 7:

Cho ABC (AB < AC). AD, AM l ng phn gic, ng trung tuyn ca ABC . ng thng

qua D v vung gc vi AD ct AC ti E

So snh S ADM v S CEM


16/48

16

16 (58)

Cu 1:

Cho (a2 + b2 + c2)( x2 + y2 + z2) = (ax + by + cz)2

CMR: x y z

a b c vi abc 0

Cu 2:

Cho abc 0 v2 2 4 4

x y z

a b c a b c a b c

CMR:2 2 4 4

a b c

x y z x y z x y z

Cu 3:

Cho a, b, c l 3 s dng v nh hn 1

CMR: Trong 3 s: (1-a)b; (1-b)c; v (1-c)a khng ng thi ln hn 1

4Cu 4:

Cho x3 + y3 + 3(x2+y2) + 4xy + 4 = 0 v xy > 0

Tm gi tr ln nht A = 1 1

x y

Cu 5:

a, CMR PT: 3x5 x3 + 6x2 18x = 2001 khng c nghim nguyn.

b, Tm 4 s nguyn dng sao cho tng ca chng bng tch ca chngCu 6:

Cho nN v n >1

CMR: 1 +2 2 2

1 1 1.... 2

2 3 n

Cu 7:

Cho ABC v pha ngoi ABC v tam gic vung cn ABE v CAF ti nh A.

CMR: Trung tuyn AI ca ABC vung gc vi EF v AI = 12EF

Cu 8:

CMR: 21 4

14 3

n

n

l phn s ti gin (vi nN).


17/48

17

17 (59)

Cu 1:

Phn tch ra tha s:

a, (x+1)(x+3)(x+5)(x+7) +15

b, x3 + 6x2 + 11x + 6

Cu 2:Cho x > 0 v x2 +

2

1

x= 7

Tnh gi tr ca M = x5 +5

1

x

Cu 3:

Cho x, y tho mn 5x2 + 8xy + 5y2 = 72

Tm gi tr ln nht, gi tr nh nht: A = x2 + y2

Cu 4:

a, Cho a, b, c > 0 v a+b+c 1

CMR:2 2 2

1 1 19

2 2 2a bc b ac c ab

b, Cho a, b, c tho mn a+b+c = 2; ab+bc+ca = 1.

CMR: 0 a, b, c 4

3

Cu 5:Tnh tng S = 1+2x+3x2+4x3+.....+ nxn-1 (x1)

Cu 6:


xy xz yz

z y x = 3

Cu 7:

Cho ABC bit ng cao AH v trung tuyn AM chia gcBACthnh 3 phn bng nhau.

Xc nh cc gc ca ABC


18/48

18

18 (60)

Cu 1:

Rt gn: M =2 2 2

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

a bc b ac c ab

a b a c b a b c a c a b

Cu 2:

Cho: x =2 2 2

( )( );2 ( )( )b c a a b c a c by

bc a b c b c a

Tnh gi tr P = (x+y+xy+1)3

Cu 3:

Cho 0 < a, b, c, d < 1. CMR c t nht mt bt ng thc sai trong cc bt ng thc sau:

2a(1-b) > 1 8c(1-d) > 1

3b(1-c) > 2 32d(1-a) > 3

Cu 4:Cho P = 5x+y+1; Q = 3x-y+4

CMR: Nu x = m; y = n Vi m, n N th P.Q l s chn.

Cu 5:

a, CMR PT: 2x2 4y2 = 10 khng c nghim nguyn.

b, Tm s t nhin nh nht n > 1 sao cho:

A = 12 + 22 +....+n2 l mt s chnh phng.

Cu 6:Cho ABC vung cn A, qua A v ng thng d sao cho B, C thuc cng na mt phng c b

d, v BH, CK cng vung gc vi d (H, K l chn ng vung gc).

a, CMR: AH = CK

b, Gi M l trung im BC. Xc nh dng MHK


19/48

19

19 (61)

Cu 1:

Cho a, b, c 0; a2 + 2bc 0; b2 + 2ca 0; c2 + 2ab 0

v a2 + b2 + c2 = (a+b+c)2

CMR: S =2 2 2

2 2 2 1

2 2 2

a b c

a bc b ac c ab

M =2 2 2

12 2 2

bc ca ab

a bc b ac c ab

Cu 2:

a, Cho a, b, c > 0

CMR:2 2 2 2 2 2

1 1 1a b b c a c

a b b c a c a b c

b, Cho 0 a, b, c 1

CMR: a+b+c+ 1abc

1 1 1a b c

+ abc

Cu 3:

a, Tm gi tr nh nht:

A = 1 2 5 3 8x x x

b, Tm gi tr ln nht:

M =

2 2

2 2

x xy y

x xy y

(x,y > 0)

Cu 4:

a,Tm nghim Z+ ca: 1 1 1

2x y z

b, Tm nghim Z ca: x4 + x2 + 4 = y2 y

Cu 5:

Cho ABC , t trn cc on ko di ca AB, AC cc on BD = CE. Gi M l trung im ca BC

N l trung im ca DE.

CMR: MN // ng phn gic trong ca gc A ca ABC

Cu 6:

Tm cc s nguyn dng n v s nguyn t P sao cho

P = ( 1)

12

n n


20/48

20

20 (62)

Cu 1:

a, Cho a+b+c = 1; a2 + b2 + c2 = 1 v x y z

a b c ; abc 0

CMR: xy + yz + xz = 0

b, Cho x, y, z > 0 v 2x2 + 3y2 2z2 = 0

CMR: z l s ln nht.

Cu 2:

a, Cho a, b, c 0

CMR:2 2 2

2 2 2

a b c a b c

b c a b c a

b, Cho nN, n > 1

CMR: 2 21 1 1 1

....5 13 ( 1) 2n n

Cu 4:

Tm gi tr nh nht vi a, b, c > 0

a, P = a b c a b c a b c

b c c a a b c b a

b, Q = a b c d

b c d a c d a b d a b c

Cu 5:Tm cc s chnh phng sao cho chia n cho 39 c thng s nguyn t v d 1

Cu 6:

Cho t gic ABCD, ng thng AB v CD ct nhau ti E. Gi F, G l trung im ca AC, BD.

a, CMR: S EFG = 1

4 ABCDS

b, Gi M l giao im ca AD, BC. Chng minh FG i qua trung im ME.


21/48

21

21 (63)

Cu 1:

Cho a, b, c tho mn a+b+c = abc

CMR: a(b2-1)( c2-1) + b(a2-1)( c2-1) + c(a2-1)( b2-1) = 4abc

Cu 2:

Cho n l s nguyn tCMR: A = n4 14n3 +71n2 154n + 120 chia ht cho 24.

Cu 3:

Tm nghim nguyn ca PT: 4x2y = (x2+1)(x2+y2)

Cu 4:

Tm a, b M = x4 - 6x3 +ax2 +bx + 1 l bnh phng ca mt a thc khc.

Cu 5:

Tm gi tr ln nht, gi tr nh nht ca PT:P = x2+y2 v bit x2+y2+xy = 4

Cu 6:

a, Cho a, b, c > 0 CMR: c t nht mt BT sai l ng.

a+b c+d (a+b)cd )( c+d)ab

(a+b)( c+d) ab+cd

b, Tm cc s a, b, c tho mn ng thi cc BT:

a b c ; b a c ; c a b

Cu 7:

Cho hnh thang ABCD (AD//BC), AD > BC. Cc ng cho AC v BD vung gc vi nhau ti I.

Trn AD ly im M sao cho AM c di bng di trung bnh ca hnh thang ABCD.

CMR: MAC cn ti M


22/48

22

22 (64)

Cu 1:

Cho x3 + x = 1.

Tnh A =4 3 2

5 2

2 3 5

2

x x x x

x x x

Cu 2:

Gii BPT: 2 21 4 3x x

Cu 3:

Cho 3 s dng x, y, z tho mn:

x = 1 - 1 2y

y = 1 - 1 2z

z = 1 - 1 2x

Tm s ln nht trong ba s x, y, z.

Cu 4:

Cho x, y tho mn: x+y=1

Tm gi tr nh nht ca M = x3+y3+xy

Cu 5:

CMR:2 2 2

1 1 1 5....

1 2 3n

Cu 6:

Tm nghim nguyn dng ca PT sau:

x+y+z+t = xyzt

Cu 7:

Cho hnh vung ABCD, ly im M nm trong hnh vung sao cho:

MAB =MBA= 150

CMR: MCA u


23/48

23

23 (65)

Cu 1:

a, Cho a2 + b2 + c2 = ab bc ca . CMR: a = b = c

b, Cho (a2 + b2)( x2 + y2) = (ax+by)2. CMR: a b

x y vi x, y 0

c, Rt gn:A = (x2-x+1)(x4-x2+1)(x8-x4+1)(x16-x8+1)(x32-x16+1)

Cu 2:

a, Tm s nguyn dng n n5+1 chia ht cho n3+1

b, Tm cc s a, b, c sao cho: ax3+bx2+c chia ht cho x+2 v chia cho x2-1 thi d x+5.

c, Nu n l tng 2 s chnh phng th n2 cng l tng 2 s chnh phng.

Cu 3:

a, Cho A = 11.....1 (n ch s 1), b = 100....05 (n-1 ch s 0)CMR: ab + 1 l s chnh phng.

b, Tm nghim t nhin ca PT: x+y+1 = xyz

Cu 4:

a, Cho x, y N Tm gi tr ln nht ca A =8 ( )

x y

x y x y

b, Cho x, y, z > 0 x+y+z = 1 Tm gi tr nh nht B = x y

xyz

Cu 5:

a, MCR: 7 1 1 1 1 1 5

1 ....12 2 3 4 99 100 6

b, MCR: 1 1 1 1

1 .... ( ; 0)2 3 4 2 1 2n

nn N n

Cu 6:

Cho ABC

vung ti A, cnh huyn BC = 2AB, D l im trn AC sao cho gc

ABD =

1

3

ABC, E

im trn AB sao cho gcACE= 1

3ACB . F l giao im ca BD v CE, K v H l im i xng

ca F qua BC, CA.

CMR: H, D, K thng hng.


24/48

24

24 (66)

Cu 1:

Cho M =2

3 2 2

25 2( ) : ( )

10 25 2

x y

x x y y

Tnh gi tr M bit: x2+9y2-4xy = 2xy- 3x

Cu 2:a, Cho a+b = ab. Tnh (a3+b3-a3b3)3 + 27a6b6.

b, Cho a, b tho mn: 2

2a b

a b a b

Tm cc gi tr c th ca N = 3

5

a b

a b

Cu 3:

a, Tm s t nhin n n4+4 l s nguyn t.

b, Tm s nguyn t p sao cho 2p+1 l lp phng ca s t nhin.

Cu 4:

a, Cho 1; 1999; 1 1999a a c b

CMR: 3998ab c

b, Chng t c t nht mt bt ng thc sau l sai:

a(a+b) < 0; 2a > b2+1

c, Chng t c t nht 1 BT sau l ng

a3b5(c-a)7(c-b)9 0; bc5(c-b)9(a-c)13 0; c9a7(b-c)5(b-a)3 0

Cu 5:

Tm gi tr nh nht: A = (x+5)4 + (x+1)4

Cu 6:

Cho ABC c 3 gc nhn, ng cao AH, BK, CL ct nhau ti I. Gi D,E,F l trung im ca BC,

CA, AB, Gi P, Q, R l trung im ca IA, IB, IC.

a, CM: PQRE, PEDQ l hnh ch nht.

b, CM: PD, QE, RF ct nhau ti trung im ca mi on thng.

c, CM: H,K,L,D,E,F,P,Q,R cng cch u mt im.


25/48

25

25 (67)

Cu 1:

Cho A = 4x2+8x+3; B = 6x2+3x

a, Bin i S thnh tch bit S = A + B

b, Tm gi tr ca x A v B ly gi tr l s i nhau.

Cu 2:Cho 3 s x, y, z tho mn ng thi

x2+2y = -1

y2+2z = -1

z2+2x = -1

Tnh gi tr ca A = x2001 + y2002 + z2003

Cu 3:

CMR PT: 2x2-4y2 = 10 khng c nghim nguyn.Cu 4:

Cho 2 ng thng ox v oy vung gc vi nhau v ct nhau ti O, Trn ox ly v hai pha ca O h

on thng OA = 4cm; OB = 2cm. Gi M l mt im nm trn ng trung trc ca on AB. MA

MB ct nhau vi oy C v D. Gi E l trung im ca AC, F l trung im ca BD.

a, CMR: MF + ME = 1

2(AC+BD)

b, ng thng CF ct ox ti P. Chng minh P l mt im c nh khi M di chuyn trn ng trutrc ca AB.

Cu 5:

Tm gi tr ln nht ca phn s m t s l mt s c 3 ch s, Mu s l tng cc ch s ca t s


26/48

26

26 (68)

Cu 1:

Cho x, y > 0 sao cho: 9y(y-x) = 4x2 . Tnh: x y

x y

Cu 2:

Cho a, b, c tho mn: abc = 1 v

2 2 2

2 2 2

a b c a c b

b c a c b a

CMR: C t nht 1 phn s l bnh phng ca mt trong 2 s cn li.

Cu 3:

Tm cc nghim nguyn tho mn 2 BPT: 16+5x > 3+11 v 7 3

64 2 2

x x

Cu 4:

Cho A =

2 2 2( ) ( ) ( )

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

x a x b x c

a b a c b a b c c a c b

a, A thay i nh th no nu ta hon v 2 trong 3 s a, b, c.

b, Tm A nu x=a.

c, Tm A nu b = ;3 4

a ac

d, Nu a-b = b-c > 0. Tm x nu phn thc th nht bng phn thc th 3. Tm gi tr ca phn thc

th nht v phn thc th 3.

Cu 5:

Cho a b c > 0. CMR:2 2 2 2 2 2

3 4a b c b a c

a b cc a b

Cu 6:

Cho hnh ch nht ABCD, Ly P thuc BD, trn tia CP ly M sao cho

PM = CP, K MEAD; MFAB

a, CMR: AM // BD; EF // AC

b, CMR: E,F,P thng hng.

Cu 7:

Cho hnh vung ABCD c cnh l 1, trn AB, AD ly M,N sao cho

MCN = 450.

Tnh chu vi AMN


27/48

27

27 (69)

Cu 1:

Cho M = x3+x2-9x-9; N = (x-2)2 (x-4)2

a, Rt gn A = M

N

b, CMR: Nu x chn A ti gin.

Cu 2:

Tm s c 4 ch s abcd tha mn:

665(abcd +ab +ad +cd +1) = 738(bcd +b+ d)

Cu 3:

CMR: (x-1)(x-3)(x-4)(x-6) + 10 1

Cu 4:

Cho s chnh phng M gm 4 ch s. Nu ta thm vo mi s ca M mt n v th c mt s l s chnh phng.

Tm hai s M, N.

Cu 5:

So snh A, B bit:

A = 20+21+....+2100+9010

B = 2101+1020

Cu 6:CHo ABC , ng cao AF, BK, CL ct nhau ti H. T A k Ax AB, t C k Cy BC. Gi P l

giao ca Ax v Cy.

Ly O, D, E l trung im ca BP, BC, CA.

a, CMR: ODE ng dng vi HAB

b, Gi G l trng tm ca ABC CMR: O, G, H thng hng.


28/48

28

28 (70)

Cu 1:

Rt gn: A =2 2 2

2 2 2( ) ( ) ( )

x y z

x z z x x y

, vi x+y+z = 0

Cu 2:

a, CMR: M =7 2

8 11n nn n khng ti gin

n Z

b, CMR: Nu cc ch s a, b, c 0 tho mn: ab : bc = a:c

Th: abbb : bbbc = a:c

Cu 3:

a, Rt gn: P =4 4 4 4 4

4 4 4

(1 4)(5 4)(9 4)(13 4) .... (21 4)

(3 4)(7 4) .... (23 4)

b, Cho Q = 11,00....1 (mu c 99 ch s 0).

Tm gi tr ca Q vi 200 ch s thp phn.

Cu 4:

a, Cho a, b, c 0. CMR: a4+b4+c4 abc(a+b+c).

b, CMR: Nu a, b, c l s o 3 cnh ca 1 tam gic th:

a2+b2+c2 < 2(ab+ac+bc).

Cu 5:Cho x, y tho mn: x2+y2 = 4+xy.

Tm gi tr ln nht, gi tr nh nht ca A = x 2+y2

Cu 6:

Cho hnh vung ABCD c cnh l 1. Trn AB, AD ly P, Q sao cho APQ cn c chu vi l 2.

a, CMR: PQ + QD = PQ

b, CMR:PCQ = 450.


29/48

29

29 (71)

Cu 1:

Cho A =2 2 2

2 2 2

4 4 4; ; .

2 2 2

bc a ca b ab cB C

bc a ca b ab c

CMR: Nu a+b+c = 0 th:

a, ABC = 1

b, A + B + C = 3

Cu 2:

Cho nN, n > 0

CMR:2 2 2

1 2 11 .... 1, 65

2 3 n

Cu 3:

Cho a, b, c, d l cc s nguyn dng.

a, CMR: A = a b c d

a b c a b d b c d a c d

khng l s nguyn.

b, Tm 5 s t nhin lin tip sao cho lp phng ca s ny bng tng cc lp phng ca 4 s cn

li.

Cu 4:

Cho x, y, z tho mn. xyz = 1; 1 1 1

x y zx y z

CMR: C 1 trong 3 s x, y, z l ln hn 1.

Cu 5:

Cho ABC , ng thng d ct AB, AC, trung tuyn AM ti E, F, N.

a, CMR: 2AB AC AM

AE AF MN

b, Gi s d // BC. Trn tia i ca tia FB ly K, KN ct AB ti P, KM ct AC ti Q. CMR: PQ // BC

Cu 6:

Cho hnh thang c di hai ng cho l 3,5. di on thng ni trung im 2 y l 2.

Tm din tch hnh thang?


30/48

30

30 (72)

Cu 1:

CMR: ; 1n N n

2 2

1 1 1 1 9....

5 13 25 ( 1) 20n n

Cu 2:Cho: (x-y)2+(y-z)2+(z-x)2 = (x+y-2z)2+(y+z-2z)2+(x+z-2y)2

CMR: x = y = z.

Cu 3:

a, Phn tch thnh nhn t:

A = x3(x2-7)2-36x.

b, CMR: A210 vi mi x N

Cu 4:Cho: 0 , , 1a b c .

Tm gi tr ln nht, gi tr nh nht ca: P = a+b+c-ab-bc-ca

Cu 5:

Cho ABC vung ti B, trn tia i tia BA ly D sao cho: AD = 3AB. ng thng vung gc vi

CD ti D ct ng thng vung gc vi AC ti E.

CMR: BDE cn


31/48

31

31 (73)

Cu 1:

Cho a+b+c = 0

CMR: ( )( ) 9a b b c c a c a b

c a b a b b c c a

Cu 2:

Tm x, y, z bit: 2 2 2x y z xy+3y+2z -4

Cu 3:

Cho a, b, c l di 3 cnh ca 1 tam gic.

CMR: 1a b b c c a

a b b c c a

Cu 4:

a, Cho a, b, c > 0 v a+b+c = 27Tm a, b, c sao cho: ab+bc+ca t gi tr ln nht.

b, Tm 4 s t nhin lin tip sao cho lp phng ca 1 s bng tng cc lp phng ca 3 s cn l

Cu 5:

Tm nghim nguyn dng ca PT:

x2 + (x+y)2 = (x+9)2

Cu 6:

Cho lc gic li ABCDEF, cc ng thng AB, EF ct nhau ti P, EF v CD ct nhau ti Q, CD vAB ct nhau ti R. Cc ng thng BC v DE; DE v FA; FA v BC ct nhau ti S,T,U.

CMR: Nu AB CD EF

PR QR QP th

BC DE FA

US TT TU


32/48

32

32 (74)

Cu 1:

a, CMR: 62k-1+1 chia ht cho 7 vi ; 0K N n

b, CMR: S a = 11.....1 + 44.....4 + 1 l bnh phng ca mt s t nhin.

(Trong c 2k ch s 1 v k ch s 4).

Cu 2:a, Tm s d ca php chia: x2002+x+1 chia cho x2-1

b, Tm s nguyn dng x, y sao cho:

3(x3-y3) = 2001.

Cu 3:

a, Cho a, b, c > o.

CMR: 1 1 1 9

2( )a b b c c a a b c

b, Tm gi tr nh nht, gi tr ln nht:

y = x3-6x2+21x+18

Vi 1

12

x .

Cu 4:

Cho ABC (AB = AC). BitBAC = 200, v AB = AC = b; BC = a

CMR: a3 + b3 = 3ab2


33/48

33

33 (75)

Cu 1:

Cho a, b, c tho mn: a+b+c = 0 v ab+bc+ca = 0

Tm gi tr ca: M = (a-1)1999+ b2000 + (c+1)2001

Cu 2:

Cho x, y, z l cc s nguyn khc 0. CMR:Nu : x2 yz = a

y2 zx = b

z2 xy = c

Th ax+by+cz chia ht cho a+b+c

Cu 3:

a, Cho nN, CMR: A = 10n + 18n 1 chia ht cho 27.

b, CMR: n5m nm5 chia ht cho 30 vi mi m,n Z.Cu 4:

a, Tm gi tr nh nht, gi tr ln nht ca M =2

4 3

1

x

x

b, Tm gi tr ln nht ca: N =2

2 2

8 6x xy

x y

Cu 5:

Cho a, b, c l s o 3 cnh ca 1 tam gic. Xc nh dng ca tam gic :A =

a b c

b c a a c b a b c

t gi tr nh nht.

Cu 6:

Cho hnh vung ABCD. T gic MNPQ c 4 nh thuc 4 cnh ca hnh vung (MAB; NBC;

PCD; QDA)

a, CMR: ( )4ABCD

ACS MN MP PQ QM

b, Xc nh M, N, P, Q chu vi MNPQ t gi tr nh nht.

c, Xc nh M, N, P, Q MNPQS t gi tr nh nht.


34/48

34

34 (76)

Cu 1:

Phn tch s 1328 thnh tng ca 2 s nguyn x, y sao cho:

x chia ht cho 23, y chia ht cho 29. Tnh x, y khi x-y = 52.

Cu 2:

Cho f(x) =5 3

230 6 15x x x

a, Phn tch f(x) thnh tch.

b, Chng t f(x) nhn gi tr nguyn khc 17 vi mi x Z.

Cu 3:

C bao nhiu s abc vi1 6;1 6;1 6a b c tho mn abc l s chn.

Cu 4:

Cho ABC , trung tuyn AM. Gi E, F l cc im ln lt thuc AB, AC sao cho ME = MF.CMR: ABC l tam gic cn ti nh A trong cc trng hp:

a, ME, MF l phn gic trong ca ;AMB AMC

b, ME, MF l trung tuyn ca ;AMB AMC


35/48

35

35 (77)

Cu 1:

a, Cho cc s a, b, c l 3 s khc nhau.

CMR: 2 2 2

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

b a c a a b

a b a c b c b a c a c b a b b c c a

b, Tm x, y, z bit:

x+y-z = y+z-x = z+x-y = xyz.

Cu 2:

Gii PT:

1 2 3 4

58 57 56 55

x x x x

Cu 3:

Tm gi tr ln nht.

A =3 3 3 3 3 3

1 1 1

1 1 1x y y z z x

(x, y, z > 0; xyz = 1).

Cu 4:


x(x2+x+1) = 4y(y+1)

Cu 5:

Cho hnh vung ABCD cnh l a. Ly M

AC, k ME AB, MF BC. Tm v tr ca M DEFSnh nht.

Cu 6:

Cho ABC c A = 500; B = 200. Trn phn gic BE caABCly F sao cho FAB = 200. Gi I l

trung im AF, ni EI ct AB ti K v CK ct EB ti M.

CMR: AI2 + EI2 = EA + (MF +2

EK).


36/48

36

36 (78)

Cu 1:

a, Cho a+b+c = 0 v a2 + b2 + c2 = 14. Tm gi tr B = a4+b4+c4.

b, Cho x > 0 v x2+2

1

x= 7.

CMR: x5

+ 51

x l s nguyn.

Cu 2:

Cho a, b, c > 0. CMR:3 3 3a b c

ab bc cab c a

Cu 3:

Cho a, b, c > 0 v a+b+c = 1.

Tm gi tr nh nht: A = 2 2 21 1 1

( ) ( ) ( )a b ca b c

Cu 4:

Xc nh a, b sao cho f(x) = ax4+bx3+1 chia ht cho g(x) = (x-1)2.

Cu 5:

Tm nghim nguyn ca PT: 1 1 1

1x y z

Cu 6:

CHo ABC , trung tuyn AM. Qua D thuc BC v ng song song vi AM ct AB, AC ti E, F.a, CMR: Khi D di ng trn BC th DE + DF c gi tr khng i.

b, Qua A v ng thng song song vi BC ct EF ti K.

CMR: K l trung tuyn ca EF.


37/48

37

37 (79)

Cu 1:

Cho S = (n+1)(n+2).....(n+n)

CMR: Vi mi n N th S chia ht cho 2n.

Cu 2:

Cho f(x) = x2+nx+b tho mn: 1( ) 2f x khi 1x .

Xc nh f(x).

Cu 3:

Cho: 2 , , , 3a b c d

CMR: 2 ( ) 3 3

3 ( ) 3 2

a c d d

b d c c

Cu 4:

Tm s A c 2 ch s sao cho 4 mnh sau c 2 mnh ng, 2 mnh sai:

a, A chia ht cho 5 c, A + 7 l s chnh phng

b,A chia ht cho 23 d, A 10 l s chnh phng

Cu 5:

Cho t gic li ABCD. CMR: AD.BC + DC.AB AC.BD

Cu 6:

Cho ABC , O l im nm trong tam gic ABC, ng thng AO, BO, CO ct cc cnh ca ABCti A1, B1, C1.

Tm v tr ca O : P =1 1 1

OA OB OC

OA OB OC t gi tr nh nht.


38/48

38

38 (80)

Cu 1:

a, Gii PT: 4

1a b x a c x b c x x

c b a a b c

b, Tm cc s a, b, c, d, e bit:

2a2+b2+c2+d2+e2 = a(b+c+d+e)

Cu 2:


1+x+x2+x3 = y3

Cu 3:

a, Vi iu kin no ca x th A ti gin, khng ti gin.

A =3 2

2 2

9 9

( 2) ( 4)

x x x

x x

b, CMR: Nu a2-bc = x; b2-ac = y; c2-ab = z;

Th ax + by + cz chia ht cho x+y+z

Cu 4:

Cho gc vung xEy quay quanh nh E cu hnh vung EFGH. Ex ct FG, GH ti M, N; Ey ct FG

GH ti P, Q

a, CMR: ,NEP MMQ vung cn

b, Gi R l giao ca PN, QM. Gi I, K l trung im ca NP. QM. T gic EKRI l hnh g?c, CMR: F, H, K, I thng hng.

Cu 5:

Cho ABC c din tch l S. Trn AB ly BB1= AB. Trn BC ly CC1= BC, trn AC ly AA1= A

Tm t s1 1 1A B C

S

v ABCS theo S.


39/48

39

39 (81)

Cu 1:

a, Tm cc s a, b, c, d bit:

a2+b2+c2+d2-ab-bc-cd- d+2

5 = 0.

b, CMR: Vi mi n N; n > 0 th :

A = n4 + 2n3 + 2n2 + 2n + 1 khng l s chnh phng

Cu 2:


x7 x5 +x4 x3 x2 + x = 1992.

Cu 3:

Cho x, y, z, t > 0 Tm gi tr nh nht ca:

A = x y z t y z t x z t x y t x y z

y z t x z t x y t x y z x y z t

Cu 4:

a, Cho a, b, c i mt khc nhau. CMR: Trong cc BT sau c t nht mt BT l sai.

(a+b+c)2 9ab; (a+b+c)2 9bc; (a+b+c)2 9ac.

b, Cho n N; n > 0.

CMR: 1 1 1 1 1 1 1

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

Cu 5:

Cho ABC , t D trn AB k Dx//BC ct AC ti E, t C k Cy//AB ct Dx ti F. AC ct BF ti I.

a, Chng t ta c th chn v tr D BF l phn gic gc B

b, CMR: Nu D l trung im ca AB th CI = 2IE.

c, Vi D l im bt k trn AB. CMR: IC 2 = IE.IA.


40/48

40

40 (82)

Cu 1:

Tm tng Sn= 7 + 77 +....+ 77......7

(n ch s)

Cu 2:

CMR: S = 1+2+3+....+n (nN) c tn cng l 0, 1, 3, 5, 6 hoc 8.Cu 3:

a, CMR: 12 + 22 + .... + n2 = ( 1)(2 1)

6

n n n

b, CMR: Vi n N th: ( 1)(2 1)

6

n n n l s nguyn.

Cu 4:

CMR: Nu n Z th:

5 3 7

5 3 15

n n n l s nguyn t.

Cu 5:

Cho a, b, c > 0

CMR:2 2 2

2 2 2 2 2 2

a b c a b c

b c c a a b b c c a a b

Cu 6:

Cho ABC vung cn ti A, M l trung im BC. T M v gc 450, hai cnh ca gc ct AB, AC t

E, F.

a, Xc nh v tr ca E, F MEFS t gi tr ln nht.

b, MEFS ln nht l bao nhiu?


41/48

41

41 (83)

Cu 1:

a, Cho a+b+c = 0.

CMR: ( )( ) 0a b b c c a c a b

c a b a b b c c a

b, CMR vi mi x, y Z

A = (x+y)(x+2y)(x+3y)(x+4y) + y4 l s chnh phng.

Cu 2:

Tm s nguyn x, y, z tho mn:

x2 + y2 + z2 < xy + 3y -3

Cu 3:

Tm gi tr ln nht, gi tr nh nht: y =2

4 3

1

x

x

Cu 4:

Tm x, y Z+ : x2 + (x+y)2 = (x+9)2

Cu 5:

CMR: A = 10n + 18n -1 chia ht cho 27 (n N)

Cu 6:

Cho ABC , trn BC, CA, AB ly M, N, P sao cho:

; (0 1)BM CN AP

k kMC NA PM v k cc on AM, BN, CP.

Tm din tch tam gic to bi cc on AM, BN, CP. Bit ABCS S

Cu 7:

Tm s nguyn x, y : 2 3 5x y


42/48

42

42 (84)

Cu 1:

Cho 3 s x, y, z: xyz = 1; v 1 1 1

x y zx y z

CMR: C ng 1 trong 3 s ln hn 1.

Cu 2:

Tm gi tr nguyn x, y tho mn ng thi:

x+y 25

y 2x+18

y x2+4x

Cu 3:

Gii PT: 2 3

3 4 1x x

Cu 4:Cho 3 s a, b, c tho mn: a4+b4+c4 < 2(a2b2+ b2c2+ a2c2)

Chng minh rng: Tn ti tam gic m c di 3 cnh l a, b, c.

Cu 5:

Cho 2 ng thng ox, v oy vung gc vi nhau, ct nhau ti O. Trn Ox ly v 2 pha ca im O

hai on OA = 4cm; OB = 2cm. Gi M l im nm trn ng trung trc ca AB. MA, MB ct Oy

C, v D. Gi E l trung im CA; F l trung im ca DB.

a, CMR: , ,MA BFO OEA ng dng v tm t s ng dng.

b, CMR: OEFM l hnh bnh hnh.

c, ng thng EF ct Ox ti P. CMR: P l im c nh khi M di chuyn trn ng thng trung tr

AB.

d, Cho MH = 3cm, t gic OFME l hnh g?


43/48

43

43 (85)

Cu 1:

Cho a, b, c l ba s phn bit tho mn: 0a b c

b c c a a b

CMR:2 2 2

0( ) ( ) ( )

a b c

b c c a a b

Cu 2:

Cho a, b, c 0 v 0x y z

a b c x y z a b c

CMR: xa2 + yb2 + zc2 = 0.

Cu 3:

Gii PT:

a, (x-4)(x-5)(x-6)(x-7) = 1680.

b,2

22

2 72 4

2 3

x xx x

x x

Cu 4:

Cho a, b, c tho mn: 1 1 1

21 1 1a b c

CMR: abc 1

8 .

Cu 5:

Cho a, y, z 0 v x, y , z Z tho mn: a + by 36 v 2x+3z 72.

CMR: Nu b 3 th x+y+z nhn gi tr ln nht l 36.

Cu 6:

Cho hnh vung OCID c cnh l a. AB l ng thng bt k i qua I ct tia OC, OD ti A, v B.

a, CMR: CA.DB c gi tr khng i (theo a).

b,2

2

CA OA

DB OB

c, Xc nh v tr A, B sao cho DB = 4CA.

d, Cho28

3AOB

aS

. Tnh CA + DB theo a.


44/48

44

44 (86)

Cu 1:

Cho a > b > 0. So snh A, B:

A =2 1 2 1

2 2

1 .... 1 ....;

1 .... 1 ....

n n

n n

a a a b b bB

a a a b b b

Cu 2:

a, Cho x+y+z = 0

CMR: 2(x5+y5+z5) = 5xyz(x2+y2+z2)

b, Cho a, b, c 0.

Tnh gi tr M = x2003+y2003+z2003. Bit z, y, z:2 2 2 2 2 2

2 2 2 2 2 2

x y z x y z

a b c a b c

Cu 3:

a, Cho a, y, z 0

CMR: a(x-y)(x-z) + y(y-z)(y-x) + z(z-x)(z-y) 0

b, Cho a, b, c tho mn a+b+c > 0; ab+bc+ca > 0; v abc > 0.

CMR: C 3 s u dng.

Cu 4:

Tm gi tr nh nht: A = x100 10x10 +10.

Cu 5:

Vi gi tr no ca A th PT:2 1 3x a x c nghim duy nht.

Cu 6:

Cho ABC ng thng d//BC ct AB, AC ti D, E.

a, CMR: Vi mi im F trn BC lun c DEFS khng ln hn 1

4 ABCS

b, Xc nh v tr D, E DEFS ln nht.


45/48

45

45 (88).

Cu 1:

a, Cho 1 1 1 1

a b c abc

CMR: 1 1 1 1

n n n n n na b c a b c

(vi n l s nguyn dng l; a, b, c 0)

b, Cho abcd = 1. Tnh gi tr:

M = 1 1 1 1

1 1 1 1abc ab a bcd bc b acb cd c abd ad d

Cu 2:

Cho a, b > 0.

Tm gi tr nh nht: P =2 2

2 2

ab a b

a b ab

Cu 3:a, Cho a, b Q v a, b khng ng thi bng khng.

CMR:2 2 2

2 2 2 1

1 1 1

a b c

a b c

b, Cho a, b, c tha mn: a2 + b2 + c2 = 1

CMR: 1

12

ab bc ca

Cu 4:


a, xy 2 = x + y

b, 3xy + x y = 1

Cu 5:

Gii PT: x4+3x3+4x2+3x+1 = 0


46/48

46

47 (90)

Cu 1:

Cho a, b, c 0 ; a3+b3+c3 = 3abc

Tnh gi tr biu thc: P = (1 )(1 )(1 )a b c

b c a

Cu 2:

a, Tm gi tr ln nht ca M =2

2

3 6 10

2 3

x x

x x

b, Tm gi tr nh nht: A = x2+26y2-10xy+14x-76y +59.

Cu 3:

Cho a+b+c+d = 1

CMR: (a+c)(b+d) + 2ac +2bd 1

2

b, Cho 3 s dng a, b, c u nh hn 1. CMR: c t nht 1 mnh sau l sai:

a(1-b) > 1

4; b(1-c) >

1

4; c(1-a) >

1

4

Cu 4:

a, Tm x, y Z : x2 + (x+1) = y4 + (y+1)4

b, Cho N = 1.2.3 + 2.3.4 +.....+ n(n+1)(n+2)

CMR: 4N+1 l s chnh phng vi mi nZ+

c, Tm nghim nguyn dng ca PT: x2 (x+y)2 = -(x+y)2

Cu 5:

Xc nh a, b, c : f(x) = x4+ax2+bx+c chia ht cho g(x) = (x-3)3.

Cu 6:

Cho O l trc tm ca ABC (c 3 gc nhn). Trn OB, OC ly B1, C1sao cho:

1AB C =

01 90AC B .

CMR: AB1 = AC1


47/48

47

49 (92)

Cu 1:

a, CMR: Nu (y-z)2+(z-x)2+(x-y)2 = (y+z-2x)2+(z+x-2y)2+(y+x-2z)2

th x = y = z.

b, Cho x2-y = a; y2-z = b; z2-x = c.

Tnh P = x3

(z-y2

)+ y3

(x-z2

)+ z3

(y-x2

)+xyz(xyz-1)Cu 2:

Tm x : P =4 3 2

2

4 16 56 80 356

2 5

x x x x

x x

t gi tr nh nht.

Cu 3: CMR:2 2

1 1 1 1.... 1

1 1n n n n

vi n N ; n > 0.

Cu 4:

Tm nghim nguyn dng ca PT:

2(x+y+z) + y = 3xyz.

Cu 5:

Cho ABC , trung tuyn AD. Gi G l trng tm ABC , mt ct tuyn quay quanh G ct AB, AC t

M, N. CMR: 3AB AC

AM CM

Cu 6:

Cho ABC , mt hnh ch nht MNPQ thay i sao cho: MAB; NAC;

PBC, QBC.

Tm tp hp tm O ca hnh ch nht MNPQ


48/48

50 (93)

Cu 1:

a, Cho x+y=a; x2+y2=b; x3+y3= c.

CMR: a3- 3ab + 2c = 0.

b, Xc nh a, b, c, d ng thc sau ng vi mi x.3

4 221 1 1 1x x a b cx d

x x x x

Cu 2:

Cho a, b, c 0 . Gii PT: 1 1 1

2( )x a x b x c

bc ac ab a b c

Cu 3:

a, Cho a, b, c l di cc cnh ca 1 tam gic.

CMR: 2a b c

b c c a a b

b, Cho a, b, c l s t nhin khng nh hn 1. CMR:2 2 2

1 1 1 3

1 1 1 1a b c abc

Cu 4:

Cho x, y, z tho mn: xy+yz+zx = 1

Tm gi tr nh nht: M = x4+y4+z4

Cu 5:Tm nghim nguyn ca PT: 5x 3y = 2xy 11

Cu 6:

Cho hnh thang ABCD (AB//CD). Giao im ca AC, BD l O, ng thng qua O v song song A

ct AD, BC ti M, N.

a, CMR: 1 1 2

AB CD MN

b, Cho 2 2; ;AOB CODS a S b Tnh ABCDS

c, Tm im K trn BD sao cho ng thng qua K v song song AB b hai cnh bn v 2 ng ch

chia thnh 3 on bng nhau.

50 de on Tap Toan8 Co Ban

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