50 de on tap toan8 co ban
TRANSCRIPT
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7/23/2019 50 de on Tap Toan8 Co Ban
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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
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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 .
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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.
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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.
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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
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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
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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.
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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
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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
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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
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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.
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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.
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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
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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
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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).
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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:
Tm nghim nguyn ca PT:
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
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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
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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
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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.
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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
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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
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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.
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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.
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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
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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
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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.
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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.
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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?
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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
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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
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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
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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.
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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
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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:
Tm nghim nguyn ca PT:
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).
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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.
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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.
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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:
Tm nghim nguyn ca PT:
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.
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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:
Tm nghim nguyn ca PT:
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.
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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?
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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
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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?
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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.
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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.
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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:
Tm nghim nguyn ca PT:
a, xy 2 = x + y
b, 3xy + x y = 1
Cu 5:
Gii PT: x4+3x3+4x2+3x+1 = 0
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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
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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
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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.