270 bai toan BDHSG lop 9.doc

8/10/2019 270 bai toan BDHSG lop 9.doc

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TON BI DNG HS GII THCS

PHN I: BI1.Chng minh 7 l s v t.2. a)Chng minh : (ac + bd)2+ (ad bc)2= (a2+ b2)(c2+ d2) b)Chng minh bt dng thc Bnhiac!"#i : (ac + bd)2 (a2+ b2)(c2+ d2)

3.Ch$ " + % = 2. &'m gi t* nh nht c,a bi- thc : = "2+ %2.4. a)Ch$ a /0 b /. Chng minh bt 1ng thc Cach% :

a bab

2

+ .

b)Ch$ a0 b0 c /. Chng minh 3ng :bc ca ab

a b ca b c

+ + + +

c)Ch$ a0 b / v 4a + 5b = 62. &'m gi t* ln nht c,a t8ch 9 = ab.5.Ch$ a + b = 6. &'m gi t* nh nht c,a bi- thc : = a4+ b4.6.Ch$ a4+ b4= 2. &'m gi t* ln nht c,a bi- thc : ; = a + b.7.Ch$ a0 b0 c l cc s d


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22.Chng minh 3ng : ;A s tO nhi>n a #hng !hTi l s ch8nh !h


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39.Chng minh 3ng [ ]2" b3ng [ ]2 " h$[c [ ]2 " 6+40.Ch$ s ng%>n dnl NL.

41.&'m cc gi t* c,a " 1- cc bi- thc sa cH ngh^a :2

2 2

6 6 6 2M= " 4 B C _ ` " 2"

"" " 5 6 " 4" 2" 6 = = = = + +

+ 2S 4" 6 5" 4 " " 6= + + +

42. a)Chng minh 3ng : D M + B D D M D + D B D . _ = "T% a #hi n$ G b)&'m gi t* nh nht c,a bi- thc sa : 2 2 " " " L" N= + + + + .

c)SiTi !h


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TON BI DNG HS GII THCS2 2 2 2 2a) " " 2 " 2 / b) " 6 6 " c) " " " " 2 / = + = + + =

2 2d) " " 2" 6 6 Z) " " " / g) " 2 " 4 5 + = + + + = + = 2 2 2h) " 2" 6 " L" N 6 i) " 5 2 " " 25 + + + = + + =

#) " 4 " 6 " L " 6 6 l) " 6 4" 5 7" 2" 2+ + + = + + = + + 55.Ch$ hai s thOc " v % tha mKn cc 1iU #i?n : "% = 6 v " %. CI:2 2" %

2 2" %

+

.

56.It gn cc bi- thc :

a) 64 4/ 2 N 2 b) m 2 m 6 m 2 m 6

c) 2 4. 2 2 4 . 2 2 2 4 . 2 2 2 4 d) 227 4/ 2 624 22

+ + + + +

+ + + + + + + + + +

57.Chng minh 3ng L 22 42 2

+ = + .

58.It gn cc bi- thc :

( ) ( )L 2 L 4 2 L 2 L 4 2 N L 2 La) C b) _

2 4

+ + + + = =

.59.$ snh :

a) L 2/ v 6+ L b) 67 62 2 v 2 6 c) 2 6L 4 v 4 2+ + +

60.Ch$ bi- thc :2

M " " " = +a) &'m t! "c 1*nh c,a bi- thc M.b) It gn bi- thc M.

61.It gn cc bi- thc sa : a) 66 2 6/ b) N 2 6

4 66 L 2 5 2 Lc)

2 L 2 5 7 2 6/

+ + +

+ + +62.Ch$ a + b + c = / W a0 b0 c /. Chng minh 1ng thc :

2 2 2

6 6 6 6 6 6

a b c a b c+ + = + + 63.SiTi bt !h


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TON BI DNG HS GII THCSb) It gn bi- thc M. c) &'m gi t* c,a " 1- M 2.68.&'m 2/ ch@ s th! !hen 1] ti>n c,a s : /0NNNN....N (2/ ch@ s N)

69.&'m gi t* nh nht0 gi t* ln nht c,a : M = | " f 2 | + | % 6|vi | "| +| %| = 570.&'m gi t* nh nht c,a M = "+ %+ JbiAt 3ng "% + %J + J" = 671.&$ng hai s : n n 2 v 2 n+6+ + (n l s ng%>n d


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90.&8nh : M 4 5 4 5= + + b3ng hai cch.

91.$ snh : a)4 7 5 2

v L0N b) 64 62 v 7 L5

+

92.&8nh :

2 4 2 49

2 2 4 2 2 4

+ = +

+ + .93.SiTi !h


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a b" a b"b) B

a b" a b"

+ + =

+ vi ( )2

2am" 0 m 6

b 6 m= n " 1- bi- thc M l mRts ng%>n.104.&'m gi t* ln nht (nA cH) h$[c gi t* nh nht (nA cH) c,a cc bi-thc sa:

2a) N " b) " " (" /) c) 6 2 " d) " 5 > +

2 2 6Z) 6 2 6 4" g) 2" 2" 5 h) 6 " 2" 5 i)2" " 4

+ + + +

105.It gn bi- thc : M " 2" 6 " 2" 6= + 0 b3ng ba cch G

106.It gn cc bi- thc sa : a) 5 4 5 6/ 7 4+ +

b) 6/ 2 5 6/ 2 5 c) N 2 5 N 2 5+ + + + + .

107.Chng minh cc h3ng 1ng thc vi b / W a ba) ( )2a b a b 2 a a b+ = b)

2 2a a b a a ba b

2 2

+ =

108.It gn bi- thc : M " 2 2" " 2 2" = + +

109.&'m " v % sa$ ch$ : " % 2 " % 2+ = +

110.Chng minh bt 1ng thc : ( ) ( )2 22 2 2 2a b c d a c b d+ + + + + + .

111.Ch$ a0 b0 c /. Chng minh :2 2 2a b c a b c

b c c a a b 2

+ ++ +

+ + +.

112.Ch$ a0 b0 c / W a + b + c = 6. Chng minh :a) a 6 b 6 c 6 405 b) a b b c c a L+ + + + + < + + + + + .

113.C : ( ) ( ) ( ) ( )2 2 2 2 2 2 2 2a c b c a d b d (a b)(c d)+ + + + + + + vi a0 b0 c0 d /.

114.&'m gi t* nh nht c,a : M " "= + .

115.&'m gi t* nh nht c,a : (" a)(" b)M"

+ += .

116.&'m gi t* nh nht0 gi t* ln nht c,a M = 2" + 4%biAt 2"2+ 4%2= 5.

7


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TON BI DNG HS GII THCS117.&'m gi t* ln nht c,a M = " + 2 " .118.SiTi !h+ + +

vi a0 b0 c /.

129.Ch$ 2 2" 6 % % 6 " 6 + = . Chng minh 3ng "2+ %2= 6.

130.&'m gi t* nh nht c,a M " 2 " 6 " 2 " 6= + +

131.&'m S&;;0 S&q; c,a M 6 " 6 "= + + .132.&'m gi t* nh nht c,a 2 2M " 6 " 2" 5= + + +133.&'m gi t* nh nht c,a 2 2M " " 62 " 2" 4= + + + + .134.&'m S&;;0 S&q; c,a :

( )2 2a) M 2" 5 " b) M " NN 6/6 "= + = +

135.&'m S&;; c,a M = " + % biAt "0 % / tha mKna b

6" %

+ =

(a v b l h3ng s d


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TON BI DNG HS GII THCS140.&'m gi t* nh nht c,a M = 4"+ 4% vi " + % = .

141. &'m S&;; c,ab c

Mc d a b

= ++ +

vi b + c a + d W b0 c / W a0 d /.

142.SiTi cc !h


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153.&8nh :6 6 6 6

M ...2 6 6 2 4 2 2 4 4 4 6// NN NN 6//

= + + + ++ + + +

.

154.Chng minh :6 6 6

6 ... n2 4 n

+ + + + > .

155.Ch$ a 67 6=

. QK% t8nh gi t* c,a bi- thc: M = (a5+ 2a 67a4 a2+6a 67)2///.156.Chng minh : a a 6 a 2 a 4 < (a 4)

157.Chng minh : 2 6

" " /2

+ > (" /)

158.&'m gi t* ln nht c,a " 6 % 2= + 0 biAt " + % = .

159.&8nh gi t* c,a bi- thc sa vi4 6 2a 6 2a

a : M 6 6 2a 6 6 2a

+ = = +

+ + .

160.Chng minh cc 1ng thc sa :

( ) ( ) ( )a) 65 6/ L 65 2 b) 2 2 L 2 4 6+ = + = +

( ) ( ) ( )2

c) 4 5 4 5 6/ 2 d) 7 4 6 Z) 67 N 5 5 22

+ = + = + + =

161.Chng minh cc bt 1ng thc sa :5 5 5 5

a) 27 L b) 6/ /5 5 5 5

+ + > +

+ + + 2 4 6 2 4 4 4 6

d) 4 2 /2 L 2 L 2 L 2 L 2

+ + + + >

+ +

Z) 2 2 2 6 2 2 2 6 60N g) 67 62 2 2 4 6+ + > + >

( ) ( ) 2 2 4 2 2h) 4 5 7 4 5 7 4 i) /0+ +

+ + + + < n t n>n m 7. [t m =7# (# )0 ta cH m2= N#2(2). & (6) v (2) s% a 7n2= N#2n>n n2= 7#2(4). & (4) ta lFi cH n2 7 v v' 7 l s ng%>n t n>n n 7. m v n cPng chia

hAt ch$ 7 n>n !hen sm

n#hng ti giTn0 ti giT thiAt. E% 7 #hng !hTi l

s h@ tW d$ 1H 7 l s v t.2.hai ti-n vA ti v 1[t nhen t} chng0 ta 1kc vA !hTi. & a) b) v' (adbc)2 /.3.Cch 1: & " + % = 2 ta cH % = 2 f ". _$ 1H : = "2+ (2 f ")2= 2(" f 6)2+2 2.E% min = 2 " = % = 6.Cch 2: ! dong bt 1ng thc Bnhiac$!"#i vi a = "0 c = 60 b = %0 d = 60&a cH :(" + %)2 ("2+ %2)(6 + 6) .2("2+ %2) = 2 .2 mim = 2#hi " = % = 6

4.b) ! dong bt 1ng thc Cach% ch$ cc c[! s dngbc ca bc ab ca ab

v W v W va b a c b c

0 ta l]n lkt cH:

bc ca bc ca bc ab bc ab2 . 2cW 2 . 2b

a b a b a c a c+ = + = W

ca ab ca ab2 . 2a

b c b c+ = cRng

tng vA ta 1kc bt 1ng thc c]n chng minh. _ b3ng "T% a #hi a = b = c.c) Ei cc s d


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TON BI DNG HS GII THCS8.E' D a + b D / 0 D a b D / 0 n>n : D a + b D D a b D a2+ 2ab + b2 a2 2ab+ b2 ab / ab /. E% a v b l hai s cPng d.9. a) XYt hi? : (a + 6)2 a = a2+ 2a + 6 a = a2 2a + 6 = (a 6)2 /.b) &a cH : (a + 6)2 a W (b + 6)2 b W (c + 6)2 c v cc bt 1ng thc n% cHhai vA 1U dng0 n>n : (a + 6)(b + 6)(c + 6)2 Labc = L.6 = 2. E% (a + 6)

(b + 6)(c + 6) .10. a)&a cH : (a + b)2+ (a b)2= 2(a2+ b2). _$ (a b)2 /0 n>n (a + b) 2 2(a2

+ b2).b) XYt : (a + b + c)2+ (a b)2+ (a c)2+ (b c)2. hai ti-n v t gn0 ta 1kc :4(a2+ b2+ c2). E% : (a + b + c)2 4(a2+ b2+ c2).

11. a)

2" 4 6 " 4" "2" 4 6 " 4

2" 4 " 6 " 2" 2

= = = = = = =b) "2 " 5 (" 2)2 44 D " 2 D 4 f4 " 2 4 f6 " 5.c) 2"(2" 6) 2" 6 (2" 6)2 /. ;hng (2" 6)2 /0 n>n ch cH th- : 2" 6= /E% : " = .12. EiAt 1ng thc 1K ch$ d + + = + + = = > .c)

24 2 6N 24 2 6L 24 2.5 25 27

4 4 4

< = = = < .

d)SiT s}

( ) ( )2 2

4 2 2 4 4 2 2 4 4 2 2 4 6 62 6 62> > > > > .

Bt 1ng thc ci cPng 1ng0 n>n : 4 2 2 4> .

18. Cc s 1H cH th- l 602 v2 4

2

+

19.EiAt lFi !h


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TON BI DNG HS GII THCSEA ti c,a !hn

a2 0 d$ 1HD a D 2 (6). Bt 1ng thc !hTi chng minh t


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( ) 2 2 2 2 2 22 2 2

" J % " J " " J % " J % "%J/

" % J

+ + + + .

C]n chng minh t} #hng em0 tc l : "4J2(" %) + %4"2(% J) + J4%2(J ") /.(6)Bi- thc #hng 1Vi #hi h$n v* vng "%J" n>n cH th- giT s} " ls ln nht. XYt hai tn #hng vkt p " + % (6). &hZ$ 1*nh ngh^a !h]n ng%>n0 [ ]" %+ ls ng%>n ln nht #hng vkt p " + % (2). & (6) v (2) s% a : [ ]" + [ ]%

[ ]" %+ .

Cch 2: &hZ$ 1*nh ngh^a !h]n ng%>n : / " f [ ]" 6 W / % f [ ]% 6.

% a : / (" + %) ([ ]" + [ ]% ) 2. XYt hai tn

[ ]" %+ = [ ]" + [ ]% + 6 (2). &$ng cT hai t


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TON BI DNG HS GII THCS32. &a cH "2 L" + 67 = (" 4)2+ n>n t} v m c,a M l cc s dn 1- chng minh" % J

4% J "

+ + ta c]n chng minh:% J %

6J " "

+ (6)

(6) "% + J2 %J "J (nhen hai vA vi s dng "J) "% + J2 %J "J / %(" J) J(" J) / (" J)(% J) / (2)

(2) 1ng vi giT thiAt 3ng J l s nh nht t$ng 4 s "0 %0 J0 d$ 1H (6) 1ng.

& 1H t'm 1kc gi t* nh nht c,a" % J

% J "+ + .

34. &a cH " + % = "2

+ 2"% + %2

= 6L. &a lFi cH (" %)2

/ "2

2"% + %2

/. & 1H s% a 2("2+ %2) 6L "2+ %2 . min M = #hi ch #hi " = % = 2.35. ! dong bt 1ng thc Cach% ch$ ba s #hng em :

6 = " + % + J 4.4 "%J (6)

2 = (" + %) + (% + J) + (J + ") 4.4 (" %)(% J)(J ")+ + + (2);hen tng vA c,a (6) vi (2) (d$ hai vA 1U #hng em) : 2 N.4 M

M =4

2

N

ma" M =4

2

N

#hi v ch #hi " = % = J =6

4.

36. a) CH th-. b0 c) hng th-.37. Qi? c,a vA ti v vA !hTi b3ng (a b)2(a + b).

38. ! dong bt 1ng thc 26

"% (" %)

+vi "0 % / :

2 2 2 2

2

a c a ad bc c (a ad bc c )

b c d a (b c)(a d) (a b c d)

+ + + + + ++ =

+ + + + + + + (6)

&ng tO2 2

2

b d (b ab cd d )

c d a b (a b c d)

+ + ++

+ + + + + (2)

CRng (6) vi (2) 2 2 2 22

a b c d (a b c d ad bc ab cd)

b c c d d a a b (a b c d)

+ + + + + + ++ + +

+ + + + + + += B

22


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C]n chng minh B6

20 bt 1ng thc n% tn m0 ! sa$ ch$ :

14 2 43mchso0

96000...00 a + 65! 14 2 43

mchso0

97000...00

&c l NL +m ma 15p

10 10 N7 (6). Si a + 65 l s cH # ch@ s : 6/# 6 a + 65

6/#

+


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b) 5 64 4 5 (2 4 6) 2 4 4 6 + = + = = . E% hai s n% b3ng

nha.c) &a cH :

( ) ( ) ( ) ( )n 2 n 6 n 2 n 6 6 v n+6 n n 6 n 6+ + + + + = + + = .

n 2 n 6 n 6 n n>n n+2 n 6 n 6 n+ + + > + + + < + .49. M = 6 f D 6 4" D + D 4" 6 D2 = ( D 4" 6D f )2+ .

& 1H s% a : min M = " = h$[c " = 6L51. = 52. " = 6 W % = 2 W J = f4.

53. 9 = D 5" 2 D + D 4 5" D D 5" 2 + 4 5" D = 6. min 9 = 6 2 4

"5 5

.

54. C]n nh cch giTi mRt s !h


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_ 1ng thc "T% a #hiL 2 L 2

" W %2 2

+ = = h$[c

L 2 L 2" W %

2 2

+ = =

62.

2

2 2 2 2 2 21 1 1 1 1 1 1 1 1 1 1 1 2(c b a2a b c a b c ab bc ca a b c abc+ + + + = + + + + + = + + + =

= 2 2 21 1 1

a b c+ + . % a 1iU !hTi chng minh.

63. iU #i?n :2

x 6(x 6)(x 10) 0x 16x 60 0

x 10x 10x 6x 6 0

x 6

+

.

B'nh !h


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TON BI DNG HS GII THCScH : / a 6 a(a 6) / a2 a / a2 a. & a2 a 6 s% a a

a 6.

E%20chso9 20chso9

0,999...99 0,999...99=14 2 43 14 2 43 .

69. a) &'m gi t* ln nht. ! dong D a + b D D a D + D b D.

M D " D + 2 + D % D + 6 = L + 2 ma" M = L + 2 (#hi chng hFn " = f20 % = f 4)b) &'m gi t* nh nht. ! dong D a b D D a D f D b .M D " D f 2 D % D f 6 = f 2 min M = f 2 (#hi chng hFn " = 20 % = 4)70. &a cH : "+ % 2"2%2 W %+ J 2%2J2 W J+ " 2J2"2. % a :

"+ %+ J "2%2+ %2J2+ J2"2 (6)

[t #hc0 d dng chng minh 1kc : ;A a + b + c = 6 th' a2+ b2+ c21

3.

_$ 1H t giT thiAt s% a : "2%2+ %2J2+ J2"21

3 (2).

& (6) 0 (2) : min M =1

3 " = % = J = 3

3

71. qm nh bi c ( 2). &ha% v' s$ snh n n 2 v 2 n+6+ + ta s$ snh

n 2 n 6+ + v n 6 n+ . &a cH :n 2 n 6 n 6 n n n 2 2 n 6+ + < + + + < + .

72. Cch 1: EiAt cc bi- thc di d cn thnh b'nh !h + + > > . E% a b

l 1ng.b) B'nh !hn \i s$ snh.

76. Cch 1: [t M = 7 7+ 0 ng M / v M2= 2 M = 2

Cch 2: [t B = 7 7 2 2.B 2 7 2 7 2 /+ = + =

B =/.

2L


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TON BI DNG HS GII THCS77

( ) ( )2 4 2 2 4 2 4 2.4 2. 2 6 2

2 4 2 4

+ + + + ++ + + += = = +

+ + + +.

78. EiAt / 2 2.5 W 5L 2 2.7 W 6/ 2 5.7= = = . E% 9 = 2 5 7+ + .79. & giT thiAt ta cH : 2 2" 6 % 6 % 6 " = . B'nh !h .

83. ; L 4 2 6 62 4 L 2 2= + + + = + + + + + =

= ( ) ( ) ( )2 2

2 4 2 2 2 2 4 2 2 2 4 2 2 2 4 2 2+ + + + = + + = + + .

84. & " % J "% %J J"+ + = + + ( ) ( ) ( )

2 2 2

" % % J J " / + + = .

E% " = % = J.85. ! dong bt 1ng thc Cach% ch$ 6 v ai ( i = 60 20 40 n ).86. ! dong bt 1ng thc Cach% vi hai s a + b / v 2 ab /0 ta cH :

( )2

a b 2 ab 2 2(a b) ab ha% a b 2 2(a b) ab+ + + + + .

_ = "T% a #hi a = b.

87. SiT s} a b c /. &a cH b + c a n>n b + c + 2 bc a ha%( ) ( )

2 2

b c a+ >

_$ 1H : b c a+ > . E% ba 1$Fn thng a 0 b 0 c l! 1


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b) iU #i?n :

2(" 2) " /" /

" /" 2

2" /

"

+ > >

. Ei cc 1iU #i?n 1H th' :

2 2 " 2 . "(" 2) " (" 2) . "B

2 " 2 " 2""

+ = = =

.

;A / " 2 th' D " 2 D = f(" 2) v B = f " .;A " 2 th' D " 2 D = " 2 v B = "

89. &a cH : ( )2

22

2

2 2 2

a 6 6a 2 6a 6

a 6 a 6 a 6

+ ++= = + +

+ + +. ! dong bt 1ng thc

Cach%:2 2

2 2

6 6a 6 2 a 6. 2

a 6 a 6+ + + =

+ +. E%

2

2

a 22

a 6

+

+. ng thc "T% a

#hi :2

2

6a 6 a /

a 6+ = =

+.

93. ;hen 2 vA c,a !t vi 2 0 ta 1


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96. iU #i?n :2

" (" 6) /

6 " 2" (" 6) /" 2" (" 6) /

" 6 /

< >

XYt t>n hai #h$Tng 6 " 2 v " 2. At pT :2 2

M v M=6 " "f6

=

105. Cch 1: &8nh M 2 . Cch 2: &8nh M2

Cch 3 : [t 2x 1 = % /0 ta cH : 2" 6 = %2.2 2 y 1y 1 2y y 1 2y2x 2 2x 1 2x 2 2x 1 y 1

A2 2 2 2 2 2

+ + + + += = =

Ei % 6 (tc l " 6)0

1

A (y 1 y 1) 22= + + = .

Ei / % 6 (tc l1

2 " 6)0

1 2yA (y 1 y 1) y 2 4x 2

2 2= + + = = = .

108. ;A 2 " th' M = 2 2 . ;A " th' M = 2 x 2 .

109. BiAn 1Vi : x y 2 2 x y+ + = + . B'nh !h


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( ) ( ) ( )2 2 2

2 2 2a b c" b c c a a b

b c c a a b

+ + + + + + + + + +

2

a b c. b c . c a . a b

b c c a a b

+ + + + + + + +

[ ]2 2 2 2 2 2

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

+ ++ + + + + + + + + + + + + + .

112. a) &a nh'n tVng a + 6 d


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9hen t8ch sai l]m : a #hi chng minh |(") f1

40 chia ch a tn ta #hng ch a 1kc h3ng s m M2 . Be% gix0 ta viAt M2didFng :

M2= ( )2

2. 2x 3. 3y+ \i ! dong (6) ta cH :

( ) ( ) ( ) ( )2 2 2 2

2 2 2A 2 3 x 2 y 3 (2 3)(2x 3y ) 5.5 25 = + + = + + =

_$ M2 25 n>n f5 M 5. min M = f5 x y

x y 12x 3y 5

= = = + =

ma" M = 5 x y

x y 12x 3y 5

= = = + =

117. iU #i?n " 2. [t 2 x = % /0 ta cH : %2= 2 ".2

2 1 9 9 9 1 7a 2 y y y maxA * y x2 4 4 4 2 4

= + = + = =

118.iU #i?n " 6 W " 65 W " 24 " 6.Ch%-n vA0 \i b'nh !hm 1iU #i?n " 27.B'nh !hng hai vA : 2" + N"2= (65"2 64" + 2) 66"2 2" + = /

(66" 2)(" 2) = / "6= 266 W "2= 2.CT hai nghi?m 1U #hng tha mKn 1iU #i?n. E% !h


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x 1 1 x 1 1 2 x 1 x 1 1 1 + + = + =

y ;A " 2 th' : x 1 x 1 1 1 x 1 1x 2 + = = = 0 #hng thRc #h$Tng1ang "Yt.y ;A 6 " 2 th' : x 1 1 x 1 1 2 + + = . E s nghi?m 6 " 2

At ln : 6 " 2.120.iU #i?n : "2+ 7" + 7 /. [t 2x 7x 7+ + = % / "2+ 7" + 7 = %2.9hng t'nh 1K ch$ tr thnh : 4%2 4 + 2% = 2 4%2+ 2% 5 = / (% 6)(4% + 5) = / % = f 54 (l$Fi) W % = 6. Ei % = 6 ta cH 2x 7x 7+ + = 6 "2+ 7" + L =/ (" + 6)(" + L) = /. Cc gi t* " = f 60 " = f L tha mKn "2+ 7" + 7 / lnghi?m c,a (6).

121. EA ti : 2 23(x 1) 4 5(x 1) 9 4 9 5+ + + + + + = .EA !hTi : 2" "2= 5 (" + 6)2 5. E% hai vA 1U b3ng 50 #hi 1H " = f 6. Eigi t* n% cT hai bt 1ng thc n% 1U tr thnh 1ng thc. At ln : " = f 6

122. a)SiT s} 3 2 = a (a : h@ t) 5 f 2 6 = a2 25 a

62

= .

EA !hTi l s h@ t0 vA ti l s v t. E l8. E% 3 2 l s v t.b) SiTi tng tO ce a.123. [t x 2 = a0 4 x = b0 ta cH a2+ b = 2. chng minh a + b 2.

CRng tng vA bt 1ng thc :

2 2

a 1 b 1a ! b2 2+ + .124. [t cc 1$Fn thng BQ = a0 QC = c t>n mRt 1xng thng. QM BC vi MQ = b. _ th% MB.MC 2MBC= BC.MQ.125. B'nh !h


33/53


ab a b2

+ +

f ( )ab a b+ = 1ab a b a b2

+ +

= =

2 21 1

ab a b2 2

+

/

XT% a d 1ng thc : a = b = 14

h$[c a = b = /.

128. &hZ$ bt 1ng thc Cach% :b c b c b c a

.1 1 + 2a a 2a

+ + + + + =

.

_$ 1H :a 2a

b c a b c

+ + +. &nM /.

XYt : ( ) 22A (x 2)(6 x) (x 1)(3 x)= + + . Qi-n nhi>n M2 / nh


34/53

TON BI DNG HS GII THCSM2= (" + 2)(L ") + (" + 6)(4 ") f 2 (x 2)(6 x)(x 1)(3 x)+ + =

= (" + 6)(L ") + (L ") + (" + 2)(4 ") (4 ") f 2 (x 2)(6 x)(x 1)(3 x)+ +

= (" + 6)(L ") + (" + 2)(4 ") f 2 (x 2)(6 x)(x 1)(3 x)+ + + 4

= ( )2

(x 1)(6 x) (x 2)(3 x) 3+ + + .

M2 4. _$ M / n>n min M = 3vi " = /.134. a) iU #i?n : "2 5.y &'m gi t* ln nht : ! dong bt 1ng thc Bnhiac!"#i :

M2= (2" + 6. 25 x )2 (22+ 66)("2+ 5 "2) = 25 M2 25.

2

2 2 2

2 2

x 0x5 x

A 25 x 4(5 x ) x 22

x 5 x 5

= = = =

.

Ei " = 2 th' M = 5. E% ma" M = 5 vi " = 2.y &'m gi t* nh nht : Ch 3ng t% t M2 250 ta cH 5 " 50 nh


35/53


( )2

min M a b= + vi

a% b"

" %

" a aba b6

" % % b ab

"0 % /

=

= + + = = +

>

Cch 2: _Png bt 1ng thc Bnhiac!"#i :

( )2

2a b a bM (" %).6 (" %) ". %. a b

" % " %

= + = + + + = +

.

& 1H t'm 1kc gi t* nh nht c,a M.136. M = (" + %)(" + J) = "2+ "J + "% + %J = "(" + % + J) + %J

2 "%J(" % J) 2 + + =

min M = 2 #hi chng hFn % = J = 6 0 " = 2 f 6.137. &hZ$ bt 1ng thc Cach% :

"% %J "% %J2 . 2%

J " J "+ = .

&ng tO :%J J" J" "%

2J W 2"" % % J

+ + . % a 2M 2(" + % + J) = 2.

min M = 6 vi " = % = J =6

4.

138. &hZ$ bi t! 2 :2 2 2" % J " % J

" % % J J " 2

+ ++ +

+ + +. &hZ$ bt 1ng thc

Cach% :"% %J J"" % % J J " "+%+J 6

"% W %J W J" n>n2 2 2 2 2 2

+ ++ + + = .

min M =6

2

6" % J

4 = = = .

139. a) ( ) ( ) ( )2 2 2

M a b a b a b 2a 2b 2= + + + = + .

6a bma" M 2 a b

2a b 6

== = =+ =

b) &a cH : ( ) ( ) ( )

2 2a b a b a b 2(a b Lab)+ + + = + +

&ng tO :

( ) ( )

( ) ( )

( )

2 2 2 2

2 2 2 2

2 2

a c 2(a c Lac) W a d 2(a d Lad)

b c 2(b c Lbc) W b d 2(b d Lbd)

c d 2(c d Lcd)

+ + + + + +

+ + + + + +

+ + +

% a : B L(a2+ b2+ c2+ d2+ 2ab + 2ac + 2ad + 2bc + 2bd + 2cd) = L(a + b

+ c + d)2 L

45


36/53


6a b c dma" B L a b c d

a b c d 6

= = == = = = =+ + + =

140. " % " % " % M 4 4 2. 4 .4 2 4 2. 4 6+= + = = = . min M = 6 vi " = % = 2.141. hng mt t8nh tVng pt0 giT s} a + b c + d. & giT thiAt s% a :

a b c db c2

+ + ++ .

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

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

+ + + + + + = + = + + + + + + + + [t a + b = " W c + d = % vi " % /0 ta cH :

" % % % " 6 % " % 6 " % 6 6M 6 2. . 2

2% % " 2% 2 " 2% " 2 2% " 2 2

+ + = + + = + =

6

min M 2 d / 0 " % 2 0 b c a d2

= = = + + W chng hFn #hi

a 2 60 b 2 60c 2 0 d /= + = = =

142. a) 2 2(" 4) ( " 4) / + = . ! s : " = 4.b) B'nh !hng hai vA0 1a vU : ("2+ )("2 " + ) = /. ! s : " = + 2 2.c) ! s : " = 2/.d) " 6 2 " 6 = + + . EA !hTi ln hn vA ti. E nghi?m.

') Ch%-n vA : " 2 " 6 6 " 6 = + . B'nh !h


37/53


) [t 2" 4 " 2 % W 2" 2 " 2 J+ + + = + + = (6). &a cH :2 2% J 6 2 " 2 W % J 6 2 " 2 = + + + = + + . % a % J = 6.

& 1H J " 2= + (2). & (6) v (2) t8nh 1kc ". ! s : " = 2 (ch l$Fi " =f 6).

) [t 2"2 N" + = a / W 2" 6 b /. 9h + + + + + == 2( n 6 6)+ (1!cm).

150. a cc bi- thc di d cn vU dFng cc b'nh !h = .

155. &a cH a + 6 = 67 . BiAn 1Vi 1a thc t$ng ng$[c thnh tVng cc l% thac s a + 6

M = (a + 6)5 4(a + 6) 65(a + 6)4+ 52(a + 6)2 6(a + 6)2///

= (25N 67 f 225 67 f 4 67 f 6)2///= 6.

156. BiAn 1Vi :6 6

a a 6 W a 2 a 4a a 6 a 2 a 4

= =

+ +

.

157.2 2

2 26 6 6 6 6" " " " " " " " /2 2 2

+ = + + + = +

.

_ = #hng "T% a v' #hng th- cH 1\ng thxi :6 6

" v "2 2

= = .

168. &c hAt ta chng minh : 2 2a b 2(a b )+ + (y) (a + b /)

! dong (y) ta cH : " 6 % 2 2(" 6 % 2) 2= + + =

47


38/53


"" 6 % 2 2

ma" 2" % 5

%2

= = = + = =

CH th- t8nh 2\i ! dong bt 1ng thc Cach%.

180.&a !hTi cH |M | 4 . _ th% M /. &a "Yt bi- thc : 26

B 2 4 "M

= =

. &a cH :2 2 2/ 4 " 4 4 4 " / 2 4 2 4 " 2 .

2min B 2 4 4 4 " " /= = = . hi 1H

6ma" M 2 4

2 4= = +

2ma" B 2 4 " / " 4= = = . hi 1H min M =6

2

181. - ! dong bt 1ng thc Cach%0 ta "Yt bi- thc : 2" 6 "B6 " "

= + .

hi 1H :2" 6 "

(6)2" 6 "B 2 . 2 2 . B 2 2 6 " "

6 " "/ " 6 (2)

= = = <


39/53

TON BI DNG HS GII THCS&a "Zm cc bi- thc " 6 0 % 2 l cc t8ch :

2(% 2)" 6 6.(" 6) 0 % 2

2

= =

&hZ$ bt 1ng thc Cach% :" 6 6.(" 6) 6 " 6 6

" " 2" 2

+ = =

% 2 2.(% 2) 2 % 2 6 2

% % 2 2% 2 2 2

+ = = =

" 6 6 " 26 2 2 2ma" B

% 2 2 % 2

= = += + = = =

183.6 6

a 0 b6NN7 6NNL 6NN 6NN7

= =+ +

. &a th%

6NN7 6NNL 6NN 6NN7+ < +

;>n a b.184. a) min M = 5 f 2 L vi " = /. ma" M =

6

5vi " = L .

b) min B = / vi " = 6 5 . ma" B = 5 vi " = 6

185. XYt 6 " / th' M /. XYt / " 6 th'2 2

2 2 " (6 " ) 6M " (6 " )2 2

+ = = .

2 2" 6 "6 2

ma" M "2 2" /

= = =

>

186. M = |" % | /0 d$ 1H M ln nht #hi v chi #hi M2

ln nht. &hZ$ b1tBnhiac!"#i :2

2 2 2 26 6 5M (" %) 6." .2% 6 (" % )

2

= = + + =

2 2

2 52% 6 "

5 5ma" M = " 2

2 5" % 6 %

6/

= = + = =

h$[c

2 5"

5

5%

6/

=

=

187. a) Tm gi tr ln nht: & giT thiAt :4 2

4 4 2 2

4 2

/ " 6 " "" % " % 6

/ % 6 % %

+ + = 4 2

4 2

" "ma" M 6 " /0 % 6 E " 60 % /

% %

== = = = ==

b) Tm gi tr nh nht: (" + %)2 2("2+ %2) = 2 " + %" %

2 62

+ .

_$ 1H :

4N


40/53


( )( )4 44 4

" % " %" %

2

+ ++ . &hZ$ bt 1ng thc Bnhiac!"#i :

( ) ( ) ( ) ( ) ( )2 2 22 2

4 4 4 4 4 4(" % )(" %) " % " % " . " % . % + + = + + +

= ("2

+ %2

) = 66 2min M " %

22= = =

188. [t " a W % b= = 0 ta cH a0 b /0 a + b = 6.M = a4+ b4= (a + b)(a2 ab + b2) = a2 ab + b2= (a + b)2 4ab = 6 4ab.

_$ ab / n>n M 6. ma" M = 6 a = / h$[c b = / " = / h$[c " = 60 % =/.

&a cH2(a b) 6 6 6 6 6

ab ab 6 4ab . min M " %

+ = = = =

189. iU #i?n : 6 " / 0 2 " / n>n " 6. &a cH :" 6

6 " (" 6)(" 2) " 2 4" 2

+ =

6 " (" 6)(" 2) (" 6)(" 2) 4 6 " 4 " + = = = .

190. &a cH : L + " + 2"2= 2("2+ 2" + 6) + = 2(" + 6)2+ / vi mi ".E% !hng t'nh "c 1*nh vi mi gi t* c,a ". [t 2" 2" 4+ + = % /0 !hngt'nh cH dFng :

%2f % 2 f 62 = / (% f 4 2 )(% + 2 2 ) = / % 4 2

% 2 2 (l$ai v' % /

= =

_$ 1H 2" 2" 4+ + = 4 2 "2+ 2" + 4 = 6 (" 4)(" + 5) = / " =4 W " = f5 .191. &a cH :

6 6 6 6 6 6 6 6#. # #

(# 6)# # # 6(# 6) # # # 6 # # 6

= = = + + ++ + +

=# 6 6

6

# 6 # # 6

+ + +

. _$ 1H :6 6 6

2

(# 6) # # # 6

< + +

.

E% :6 6 6 6 6 6 6 6 6

... 2 6 2 ... 22 4 2 4 (n 6) n 2 2 4 n n 6

+ + + + < + + + + +

=6

2 6 2n 6

+ (a0 b / W a /).

193. [t " % = a 0 " + % = b (6) th' a0 b .a) ;A b = / th' " = % = /0 d$ 1H " 0 % .

/


41/53


b) ;A b / th'" % a a

" %b b" %

= =

+ (2).

& (6) v (2) :6 a 6 a

" b W % b 2 b 2 b

= + =

.

199. ;hn "Yt : ( )( )2 2 2 2 2

" a " " a " a+ + + = . _$ 1H :

( ) ( ) ( )( )2 2 2 22

2 2 2 2

2 2 2 2

5 " a " " a "5a2 " " a (6) 2 " " a

" a " a

+ + + + + + +

+ +

_$ a / n>n : 2 2 2" a " " " " " /+ + > + = + . % a : 2 2" a " /+ + > 0 ".E' v% : (6)

( )2 2 2 2 2 22 2 2

" /

" /2 " a 5 " a " 5" 4 " a

25" N" Na

>+ + + +

" /4

" a4/ " a

<

.

207. c) &


42/53

TON BI DNG HS GII THCSB ( a ; a. CH hai tn !hTi cH b + 2 = / d$ 1H 2a + c = /. &ha% b = f 2 0 c = f 2av$ !h : / /? @A *( cCa D:

( )6 2 2

2 # 6 # # # # # 6 #

= > = + + + +

.

_$ 1H ( ) ( ) ( )M 2 2 4 4 ... n n 6 > + + + + + + + = ( )2 n 6 2 2 n 6 2 2 2 n 6 4 2 n 4= + = + > + > .

b) Ch(ng minh M 2 n 2< : n a6// L 2. ;h v% 2 a6// 40 d$ 1H a6// = 2.

214. a) Cch 1(t8nh tOc tiA!) : a2= (2 + 4 )2= 7 + 4 .&a cH 4 = n>n L 4 7 64 a2 6. E% a2 = 64.Cch 2(t8nh gin tiA!) : [t " = (2 + 4 )2th' " = 7 + 4 .XYt bi- thc % = (2 f 4 )2th' % = 7 f 4 . % a " + % = 6._ th% / 2 f 4 6 n>n / (2f 4 )2 60 tc l / % 6. _$ 1H 64 " 6.

E% " = 64 tc l a2 = 64.b) ! s : a4 = 56.

215. [t " % = a W " % b+ = (6) th' a v b l s h@ t. XYt hai t


43/53


a);A b / th'" % a a

" %b b" %

= =

+ l s h@ t (2). & (6) v (2)

&a cH : 6 a

" b2 b

= +

l s h@ t W6 a

% b2 b

=

l s h@ t.

b) ;A b = / th' " = % = /0 hi-n nhi>n " 0 % l s h@ t.216. &a cH6 n 6 6 6 6 6 6

n nn(n 6) n n 6(n 6) n n n 6 n n 6

= = = + = + ++ + + n 6 6 6 6

6 2n 6 n n 6 n n 6

= + < + + + . & 1H ta giTi 1kc bi t$n.

217. Chng minh b3ng !hTn chng. SiT s} t$ng 25 s tO nhi>n 1K ch$0#hng cH hai s n$ b3ng nha. hng mt t8nh tVng pt0 giT s} a6 a2 .

a25. % a : a6 6 0 a2 2 0 ja25 25. &hA th' :

6 2 25

6 6 6 6 6 6.... ....

a a a 6 2 25+ + + + + + (6). &a lFi cH :

6 6 6 6 2 2 2.... .... 6

25 2 2 6 25 25 2 2 2 2+ + + + = + + + + n n>n M cH bT% ch@ s N liUn sa d

!hw%.Ch : 6/f 7= /0//////6.

b) SiTi tng tO nh ce a.222. &a th% vi n l s ch8nh !hn n #hng cH dFng ....05 . _$ 1H ng vimui s n ;ycH d% nht mRt s ng%>n ang]n n nht.&a th% 3ng0 vi n b3ng 60 20 40 0 50 L0 70 th' anb3ng 60 60 20 20 20 20 40 &a schng minh 3ng anl]n lkt nhn cc gi t* : hai s 60 bn s 20 s s 4 ;Hicch #hc ta s chng minh bt !hn l : #

2

# + 6 W #

2

# + 2 W W #

2

+ #. _$ 1H :


45/53


,

+ + + = + + + + + + + + + + = =

1 442 4 43 1 4 442 4 4 431 2 1980

2 so 4 so 88 so

1 1 1 1 1 1 1 1 1 1 1 1... ... ... 2.44 88

a a a 1 1 2 2 2 2 44 44 44

.

223. SiTi tng tO bi 2.a) 6 an 2. E% an = 6. b) 2 an 4. E% an =2.c) &a th% : 2= 6N4L 6NNL 2/25 = 520 cn L2= 266L.

a6= 1996 = a6 5.QK% chng t vi n 2 th' 5 an L.

;h v% vi n = 6 th' an = 0 vi n 2 th' an = 5.224. C]n t'm s tO nhi>n B sa$ ch$ B M B + 6. qm giTm v lm tRi M1- 1kc hai s tO nhi>n li>n tiA!.

&a cH : (n + 6)2 6Ln2+ n + 4 (n + 2)2 n + 6 216n 8n 3+ + n + 2

n2+ n + 6 n2+ 216n 8n 3+ + n2+ n + 2 n2+ n + (2n + 6)2 n2+ 216n 8n 3+ + (2n + 2)2.

q% cn bc hai : 2n + 6 M 2n + 2. E% M = 2n + 6.225. - chng minh bi t$n0 ta ch a s % tha mKn hai 1iU #i?n : / % /06 F1).

" + % l mRt s tO nhi>n cH tn cPng b3ng 2 F2).

&a chn % = ( )200

3 2 . &a cH / 3 2 /04 n>n / % /06.iU #i?n (6) 1kc chng minh.Be% gix ta chng minh " + % l mRt s tO nhi>n cH tn cPng b3ng 2. &a cH :

( ) ( ) ( ) ( )200 200 100 100

x y 3 2 3 2 5 2 6 5 2 6+ = + + = + + .

XYt bi- thc tVng pt n= an+ bnvi a = 5 + 2 6 0 b = 5 f 2 6 .n= (5 + 2 6 )n= (5 f 2 6 )n

M v b cH tVng b3ng 6/0 t8ch b3ng 6 n>n chng l nghi?m c,a !h


46/53

TON BI DNG HS GII THCS(SiTi tn0 nhHm 6 cH 4 s0 nhHm 2 cH 5 s0 nhHm 4 cH 7

s0 nhHm cH N s. Cc s thRc nhHm 6 b3ng 60 cc s thRc nhHm 2 b3ng 20cc s thRc nhHm 4 b3ng 40 cc s thRc nhHm b3ng .E% M = 6.4 + 2.5 + 4.7 + .N = 7/

228. a) XYt / " 4. EiAt M di dFng : M = .x

2.

x

2.(4 "). ! dong bt 1ng

thc Cach% ch$ 4 s #hng emx

20

x

20 (4 ") ta 1kc :

x

2.

x

2.(4 ")

3x x

3 x2 2 1

3

+ + =

.

_$ 1H M (6)b) XYt " 40 #hi 1H M / (2). $ snh (6) v (2) ta 1i 1An #At ln :

x3 x

maxA 4 x 22

x 0

= = =

.

229. a) q! !hng hai vA0 ! dong h3ng 1ng thc (a + b)4= a4+ b4+ 4ab(a+ b)0 ta 1kc :

3x 1 7 x 3. (x 1)(7 x).2 8 (x 1)(7 x) 0+ + + + = + = " = f 6 W " = 7(tha)

b) iU #i?n : " f 6 (6). [t 3 x 2 y ! x 1 - = + = . hi 1H " 2 = %2W " +6 = J2n>n J2 %4= 4. 9h


47/53

TON BI DNG HS GII THCSB'nh !hng 2 vA : ab = 2 + (a + b)2 2(a + b) 2 2(a + b) 2 = 2 + (a +

b)2 abEA !hTi l s h@ t0 vA ti l s v t (v' a + b /)0 me thwn.

231. a)SiT s} 3 5 l s h@ tm

n (!hen s ti giTn). % a 5 =

3

3

m

n. QK%

chng minh 3ng cT m ln n 1U chia hAt ch$ 50 ti giT thiAt mn

l !hen s

ti giTn.

b)SiT s} 3 32 4+ l s h@ tm

n(!hen s ti giTn). % a :

( )3 3

3 3 2 33 3 33

m m 6m2 4 6 3. 8. 6 m 6n 6mn (1) m 2 m 2

n n n= + = + = + = + M M

&ha% m = 2# (# ) v$ (6) : #4= Ln4+ 62#n2 #4= 4n4+ L#n2. %

a 4n

4

chia hAt ch$ 2 n4

chia hAt ch$ 2 n chia hAt ch$ 2. ;h v% m v ncPng chia hAt ch$ 20 ti vi giT thiAt

m

nl !hen s ti giTn.

232. Cch 1: [t a = "40 b = %40 c = J4. Bt 1ng thc c]n chng minh3a b c abc

3

+ + tng 1ng vi

3 3 3x y -xy- hay

3

+ + "4+ %4+ J4 4"%J /.

&a cH h3ng 1ng thc :

"4+ %4+ J4 4"%J =1

2(" + % + J)(" %)2+ (% J)2+ (J ")2. (bi t! sbt)

_$ a0 b0 c / n>n "0 %0 J /0 d$ 1H "4

+ %4

+ J4

4"%J /. ;h v% :3a b c abc

3

+ +

XT% a d 1ng thc #hi v ch #hi a = b = c.Cch 2: &c hAt ta chng minh bt 1ng thc Cach% ch$ bn s #hng em.&a cH :

( ) 4a b c 1 a b c 1

ab c ab. c abc4 2 2 2 2

+ + + + + = + + =

&$ng bt 1ng thc

4a b c

abc4

+ + +

0 1[t

a b c

3

+ +

= ta 1kc :4

4a b c

a b ca b c a b c a b c3 abc. abc.

4 3 3 3

+ + + + + + + + + + +

.

Chia hai vA ch$ s d


48/53


XT% a 1ng thc : a = b = c =a b c

3

+ + a = b = c = 6

233. & giT thiAt s% a :b c a 1

1b 1 c 1 1 a 1 a 1

+ + =+ + + + +

. ! dong bt

1ng thc Cach% ch$ 4 s d


49/53


=1 1 1 1 1 1

1 ... 1 12 2 3 n 1 n n

+ + + = 42 22.

Ei n 40 ta chng minh n n 1n n 1+> + (2). &ht v% :

( ) ( )nnn(n 1) n(n 1)

n n 1n 1 nn

(n 1) 1(2) n 1 n (n 1) n n 1 n

n n

+ +++ + + < + < < +


50/53

TON BI DNG HS GII THCS= (" + 2 2 )("2f 2 2" + 2) + (" + 2 2 ).L L" f 6L 2

= (" + 2 2 )(" f 2 )2f 2 f 2 .min M = f 2 vi " = 2 .

Cch 2: ! dong bt 1ng thc Cach% vi 4 s #hng em :

"4

+ 2 2 + 2 2 4. 33

x .2 2.2 2 = L".% a "4 L" f 2 . min M = f 2 vi " = 2 .241. Si " l cFnh c,a h'nh vng nh0 E l th- t8ch c,a h'nh hR!.C]n t'm gi t* ln nht c,a E = "(4 2")2.&hZ$ bt 1ng thc Cach% vi ba s dng :

E = "(4 2")(4 2")3

4x 3 2x 3 2x

3

+ +

=

ma" E = 2 " = 4 2" " =1

2

&h- t8ch ln nht c,a h'nh hR! l 2 dm4#hi cFnh h'nh vng nh b3ng 12

dm.

242. a) ! s : 2 W f 66. b) [t 3 2 x a! x 1 b = = . ! s :6 W 2 W 6/.

c) q! !hng hai vA. ! s : / W5

2

d) [t 3 2x 1 = %. SiTi h? : "4+ 6 = 2% 0 %4+ 6 = 2"0 1kc (" %)("2+ "% +%2+ 2) = /

" = %. ! s : 6 W 1 52

.

') It gn vA ti 1kc : ( )21

x x 42

. ! s : " = .

() [t 3 37 x a ! x 5 b = = . &a cH : a4+ b4= 20 a4 b4= 62 2"0 d$ 1H vA

!hTi c,a !hn3 3

3 3

a b a b

a b a b

=

+ + (a b)(a4+ b4) = (a + b)(a4 b4)

_$ a + b / n>n : (a b)(a2 ab + b2= (a b)(a2+ ab + b2).& a = b ta 1kc " = L. & ab = / ta 1kc " = 7 W " = 5.*) [t 3 3x 1 a! x 1 b+ = = . &a cH : a2+ b2+ ab = 6 (6) W a4 b4= 2 (2).& (6) v (2) : a b = 2. &ha% b = a 2 v$ (6) ta 1kc a = 6. ! s : " = /.) Cch 1: " = f 2 nghi?m 1ng !h


51/53


Cch 2 : [t 3 x 2+ = %. Ch%-n vA : 3 33 3y 1 y 1 y + + = . q! !h


52/53

TON BI DNG HS GII THCS246. &a cH :4(6 + 4 )4+ a(6 + 4 )2+ b(6 + 4 ) + 62 = /.

a #hi thOc hi?n cc !hY! biAn 1Vi0 ta 1kc bi- thc th gn :(a + b + 2) + (2a + b + 6) 4 = /.

E' a0 b n>n ! = a + b + 2 v p = 2a + b + 6 . &a !hTi t'm ccs ng%>n a0 bsa$ ch$ ! + p 4 = /.

;A p / th' 4 = f!

p0 v l8. _$ 1H p = / v t ! + p 4 = / ta s% a ! = /.

E% 6 + 4 l mRt nghi?m c,a !h . At pT " = 7.

254. a bi- thc vU dFng : 4 4M " 6 6 " 6 6= + + + + . ! dong D M D + D B

D = D M + B D min M = 2 f6 " /.255. ! dong bt 1ng thc Cach% hai l]n.256. [t 4 2 24 4" % th' " % 9 2 " 2= = = +

52


53/53


258. &a cH : ( ) ( )2 2

9 " a " b= + = D " a D + D " b D D " a + b " D = b a

(a b)._ 1ng thc "T% a #hi (" a)(" b) / a " b. E% min 9 = b a a "

b.

259. E' a + b c W b + c a W c + a b. ! dong bt 1ng thc Cach% ch$tng c[! s dn 1U dn "% L. _$ 1H : ma" "% = L "= % = .266. Ei mi a0 b ta ln cH : a2+ b2 2ab. ;hn : c2 2ab 2c2 a2+b2+ 2ab 2c2 (a + b)2 c 2

a + b ca b

2

+.

_ 1ng thc "T% a #hi v ch #hi a = b.

267. BiAn 1Vi ta 1

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