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Cc chuynBi dng HSG Ton lp 7 1
DY CC S VIT THEO QUY LUT Bi 1: Tm s hng th n ca cc dy s sau: a)3, 8, 15, 24, 35, ...b)3, 24, 63, 120, 195, ... c)1, 3, 6, 10, 15, ... d)2, 5, 10, 17, 26, ... e)6, 14, 24, 36, 50, ... f)4, 28, 70, 130, 208, ... g)2, 5, 9, 14, 20, ... h)3, 6, 10, 15, 21, ... i)2, 8, 20, 40, 70, ... Hng dn: a)n(n+2) b)(3n-2)3n c) ( 1)2n n + d)1+n2 e)n(n+5) f)(3n-2)(3n+1) g) ( 3)2n n + h) ( 1)( 2)2n n + + i) + + ( 1)( 2)3n n n Bi 2: Tnh: a,A = 1+2+3++(n-1)+n b,A = 1.2+2.3+3.4+...+99.100 Hng dn: a,A = 1+2+3++(n-1)+n A = n (n+1):2 b,3A = 1.2.3+2.3(4-1)+3.4.(5-2)+...+99.100.(101-98) 3A = 1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+99.100.101-98.99.100 3A = 99.100.101 A = 333300 Tng qut:A = 1.2+2.3+3.4+. + (n - 1) n A = (n-1)n(n+1): 3 Bi 3: Tnh: A = 1.3+2.4+3.5+...+99.101 Hng dn: Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 2A = 1(2+1)+2(3+1)+3(4+1)+...+99(100+1) A = 1.2+1+2.3+2+3.4+3+...+99.100+99 A = (1.2+2.3+3.4+...+99.100)+(1+2+3+...+99) A =333300 + 4950 = 338250 Tng qut: A = 1.3+2.4+3.5+...+(n-1)(n+1) A= (n-1)n(n+1):3 + n(n-1):2A= (n-1)n(2n+1):6 Bi 4: Tnh: A = 1.4+2.5+3.6+...+99.102 Hng dn: A = 1(2+2)+2(3+2)+3(4+2)+...+99(100+2) A = 1.2+1.2+2.3+2.2+3.4+3.2+...+99.100+99.2 A = (1.2+2.3+3.4+...+99.100)+2(1+2+3+...+99) A = 333300 + 9900 A = 343200 Bi 5: Tnh: A = 4+12+24+40+...+19404+19800 Hng dn: 12A = 1.2+2.3+3.4+4.5+...+98.99+99.100 A= 666600 Bi 6: Tnh: A = 1+3+6+10+...+4851+4950 Hng dn: 2A = 1.2+2.3+3.4+...+99.100 A= 333300:2 A= 166650 Bi 7: Tnh: A = 6+16+30+48+...+19600+19998 Hng dn: 2A = 1.3+2.4+3.5+...+99.101 A = 338250:2 A = 169125 Bi 8: Tnh: A = 2+5+9+14+...+4949+5049 Hng dn: 2A = 1.4+2.5+3.6+...+99.102 A = 343200:2 A = 171600 Bi 9: Tnh: A = 1.2.3+2.3.4+3.4.5+...+98.99.100 Hng dn: 4A = 1.2.3.4+2.3.4(5-1)+3.4.5.(6-2)+...+98.99.100.(101-97) Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 34A = 1.2.3.4+2.3.4.5-1.2.3.4+3.4.5.6-2.3.4.5+...+98.99.100.101-97.98.99.100 4A = 98.99.100.101 A = 2449755 Tng qut:A = 1.2.3+2.3.4+3.4.5+...+(n-2)(n-1)n A = (n-2)(n-1)n(n+1):4 Bi 10: Tnh: A = 12+22+32+...+992+1002 Hng dn: A = 1+2(1+1)+3(2+1)+...+99(98+1)+100(99+1) A = 1+1.2+2+2.3+3+...+98.99+99+99.100+100 A = (1.2+2.3+3.4+...+99.100)+(1+2+3+...+99+100) A = 333300 + 5050 A = 338050 Tng qut: A = 12+22+32+...+(n-1)2+n2 A = (n-1) n (n+1):3 + n(n +1):2 A = n(n+1)(2n+1):6
Bi 11: Tnh: A = 22+42+62+...+982+1002 Hng dn: A = 22(12+22+32+...+492+502) Bi 12: Tnh: A = 12+32+52+...+972+992 Hng dn: A = (12+22+32+...+992+1002)-(22+42+62+...+982+1002) A = (12+22+32+...+992+1002)-22(12+22+32+...+492+502) Bi 13: Tnh: A = 12-22+32-42+...+992-1002 Hng dn: A = (12+22+32+...+992+1002)-2(22+42+62+...+982+1002) Bi 14: Tnh: A = 1.22+2.32+3.42+...+98.992 Hng dn: A = 1.2(3-1)+2.3(4-1)+3.4(5-1)+...+98.99(100-1) A = 1.2.3-1.2+2.3.4-2.3+3.4.5-3.4+...+98.99.100-98.99 A = (1.2.3+2.3.4+3.4.5+...+98.99.100)-(1.2+2.3+3.4+...+98.99) Bi 15: Tnh: A = 1.3+3.5+5.7+...+97.99+99.101 Hng dn: A = 1(1+2)+3(3+2)+5(5+2)+...+97(97+2)+99(99+2) A = (12+32+52+...+972+992)+2(1+3+5+...+97+99) Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 4Bi 16: Tnh: A = 2.4+4.6+6.8+...+98.100+100.102 Hng dn: A = 2(2+2)+4(4+2)+6(6+2)+...+98(98+2)+100(100+2) A = (22+42+62+...+982+1002)+4(1+2+3+...+49+50) Bi 17: Tnh: A = 13+23+33+...+993+1003 Hng dn: A = 12(1+0)+22(1+1)+32(2+1)+...+992(98+1)+1002(99+1) A = (1.22+2.32+3.42+...+98.992+99.1002)+(12+22+32+...+992+1002) A = [1.2(3-1)+2.3(4-1)+3.4(5-1)+...+98.99(100-1)] +(12+22+32+...+992+1002) A = 1.2.3-1.2+2.3.4-2.3+3.4.5-3.4+...+98.99.100-98.99+(12+22+32+...+992+1002) A = (1.2.3+2.3.4+3.4.5+...+98.99.100)-(1.2+2.3+3.4+...+98.99) (12+22+32+...+992+1002) Bi 18: Tnh: A = 23+43+63+...+983+1003 Hng dn: Bi 19: Tnh: A = 13+33+53+...+973+993 Hng dn: Bi 20: Tnh: A = 13-23+33-43+...+993-1003 Hng dn: Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 5 Chuyn : T L THC-TNH CHT CA DY T S BNG NHAU A. C S L THUYT I. T L THC 1. nh ngha: T l thc l mt ng thc ca hai t s dcba= (hoc a : b = c : d). Cc s a, b, c, d c gi l cc s hng ca t l thc; a v d l cc s hng ngoi hay ngoi t, b v c l cc s hng trong hay trung t. 2. Tnh cht: Tnh cht 1: Nu dcba=thbc ad =Tnh cht 2: Nubc ad =v a, b, c, d0 =th ta c cc t l thc sau: dcba=,dbca= ,acbd=,abcd=Nhn xt: T mt trong nm ng thc trn ta c th suy ra cc ng thc cn li. II.TNH CHT CA DY T S BNG NHAU-Tnh cht:T dcba=suy ra: d bc ad bc adcba=++= =-Tnh cht trn cn m rng cho dy t s bng nhau:
fedcba= = suy ra:... =+ + =+ ++ += = =f d bc b af d bc b afedcba (gi thit cc t s trn u c ngha). * Ch : Khi c dy t s 5 3 2c b a= = ta ni cc s a, b, c t l vi cc s 2, 3, 5. Ta cng vit a : b : c = 2 : 3 : 5 Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 6 B.CC DNG TONV PHNG PHP GII DNG I: TM GI TR CA BIN TRONG CC T L THC. V d 1: Tm hai s x v y bit3 2y x= v20 = + y xGii: Cch 1: (t n ph) tky x= =3 2, suy ra:k x 2 = ,k y 3 =Theo gi thit:4 20 5 20 3 2 20 = = = + = + k k k k y x Do :8 4 . 2 = = x 12 4 . 3 = = yKL:12 , 8 = = y xCch 2: (s dng tnh cht ca dy t s bng nhau): p dng tnh cht ca dy t s bng nhau ta c:45203 2 3 2= =++= =y x y x Do :8 42= = xx 12 43= = yy KL:12 , 8 = = y xCch 3: (phng php th) T gi thit 323 2yxy x= =m12 60 5 203220 = = = + = + y y yyy xDo :8312 . 2= = xGenerated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 7KL:12 , 8 = = y xV d 2: Tm x, y, z bit: 4 3y x=,5 3z y= v6 3 2 = + z y xGii: T gi thit: 12 9 4 3y x y x= = (1)
20 12 5 3z y z y= =(2) T (1) v (2) suy ra: 20 12 9z y x= =(*) Ta c:32620 36 183 220 36318220 12 9= =+ + = = = = = =z y x z y x z y x Do :27 39= = xx 36 312= = yy 60 320= = zz KL:60 , 36 , 27 = = = z y xCch2:Saukhilmn(*)tatkz y x= = =20 12 9(saugiinhcch1ca VD1). Cch 3: (phng php th: ta tnh x, y theo z)T gi thit:
535 3zyz y= =
209453. 3434 3zzyxy x= = = = m60 6010653. 3209. 2 6 3 2 = = = + = + zzzz zz y xSuy ra:36560 . 3= = y , 272060 . 9= = xKL:60 , 36 , 27 = = = z y xGenerated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 8V d 3: Tm hai s x, y bit rng: 5 2y x= v40 . = y xGii:Cch 1: (t n ph) tky x= =5 2 , suy rak x 2 = ,k y 5 = Theo gi thit:2 4 40 10 40 5 . 2 40 .2 2 = = = = = k k k k k y x+ Vi2 = kta c:4 2 . 2 = = x10 2 . 5 = = y+ Vi2 = kta c:4 ) 2 .( 2 = = x10 ) 2 .( 5 = = yKL:10 , 4 = = y xhoc10 , 4 = = y xCch 2: ( s dng tnh cht ca dy t s bng nhau) Hin nhin x 0 =Nhn c hai v ca 5 2y x=vi x ta c: 85405 22= = = xy x
4162 = = xx + Vi4 = xta c1025 . 45 24= = = yy + Vi4 = xta c1025 . 45 24 == =yy KL:10 , 4 = = y xhoc10 , 4 = = y xCch 3: (phng php th) lm tng t cch 3 ca v d 1. BI TP VN DNG: Bi 1: Tm cc s x, y, z bit rng:a) 21 6 10z y x= =v28 2 5 = + z y x b) 4 3y x=,7 5z y= v124 3 2 = + z y x c) 544332 z y x= =v49 = + + z y x d) 3 2y x=v54 = xy Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 9 e) 3 5y x=v42 2= y xf)z y xy xzx zyz yx+ + = +=+ +=+ + 2 1 1 Bi 2: Tm cc s x, y, z bit rng:a) 21 6 10z y x= =v28 2 5 = + z y x b) 4 3y x=,7 5z y= v124 3 2 = + z y x c) 544332 z y x= =v49 = + + z y x d) 3 2y x=v54 = xy e) 3 5y x=v42 2= y xf)z y xy xzx zyz yx+ + = +=+ +=+ + 2 1 1 Bi 3: Tm cc s x, y, z bit rng:a)z y y x 5 7 , 2 3 = =v32 = + z y x b) 433221 == z y x v50 3 2 = + z y xc)z y x 5 3 2 = =v95 = + z y xd) 5 3 2z y x= = v810 = xyze) z y x zy xyx zxz y+ += +=+ +=+ + 1 3 2 1 f)y x 6 10 =v28 22 2 = y xBi 4: Tm cc s x, y, z bit rng:a)z y y x 5 7 , 2 3 = =v32 = + z y x b) 433221 == z y x v50 3 2 = + z y xc)z y x 5 3 2 = =v95 = + z y xd) 5 3 2z y x= = v810 = xyze) z y x zy xyx zxz y+ += +=+ +=+ + 1 3 2 1 f)y x 6 10 =v28 22 2 = y x Bi 5: Tm x, y bit rng:
x y y y66 1244 1182 1 +=+=+ Bi 6: Tm x, y bit rng:
x y y y66 1244 1182 1 +=+=+ Bi 7: Cho0 = + + + d c b av c b add b acd c abd c ba+ +=+ +=+ +=+ + Tm gi tr ca: c ba db ad cd ac bd cb aA+++++++++++=Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 10Gii: 13( ) 3a b c d a b c db c d a c d a b d a b c a b c d+ ++= = = = =++ ++ + + + + + ++( V 0 = + + + d c b a ) =>3a = b+c+d;3b = a+c+d => 3a-3b= b- a => 3(a- b) = -(a-b) =>4(a-b) = 0 =>a=b Tng t =>a=b=c=d=>A=4 Bi 8: Tm cc s x; y; z bit rng: a) x 7y 3=v 5x 2y = 87; b) x y19 21=v 2x y = 34; b) 3 3 3x y z8 64 216= =v x2 + y2 + z2 = 14. c) 2x 1 3y 2 2x 3y 15 7 6x+ + = = Bi 9: Tm cc s a, b, c bit rng: 2a = 3b; 5b = 7c v 3a + 5c 7b = 30. Bi 10: Tm cc s x, y, z bit : a)x : y : z = 3 : 4 : 5v5z2 3x2 2y2 = 594; b)x + y = x : y = 3.(x y) Giaia) p s: x = 9; y = 12; z = 15 hoc x = - 9; y = - 12; z = - 15. b) T bi suy ra: 2y(2y x) = 0, m y khc 0 nn 2y x = 0, do : x = 2y. T tm c : x = 4/3; y = 2/3. Bi 11. Tm hai s hu t a v b bit rng hiu ca a v b bng thng ca a v b v bng hai ln tng ca a v b ? Giai. Rt ra c: a = - 3b, t suy ra : a = - 2,25; b = 0,75. Bi 12: Cho ba t s bng nhau: a b c, ,b c c a a b + + +. Bit a+b+c 0 = .Tm gi tr ca mi t s ? Bi13. Shc sinh khi6,7,8,9ca mt trngTHCS lnlttl vi9;10;11;8. Bit rng s hc sinh khi 6 nhiu hn s hc sinh khi 9 l 8 em. Tnh s hc sinh ca trng ? Bi 14: Chng minh rng nu c cc s a, b, c, d tha mn ng thc: ( ) | | ( ) | | 0 ) 1 ( 2 2 . 22 2= + + + ab ab ab d c cd ab abth chng lp thnh mt t l thc. Gii:( ) ( )2 22 . 2 2( 1) 0 abab cd c d abab ab( + + + =(
=>ab(ab-2cd)+c2d2=0(V ab(ab-2)+2(ab+1)=a2b2+1>0 vi mi a,b) =>a2b2-2abcd+ c2d2=0 =>(ab-cd)2=0 =>ab=cd =>pcm Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 11
DNG II: CHNG MINH T L THC chng minh t l thc: DCBA=ta thng dng mt s phng php sau: Phng php 1: Chng t rngA. D = B.CPhng php 2: Chng t rng hai t s BA v DC c cng gi tr. Phng php 3: S dng tnh cht ca t l thc. Mt s kin thc cn ch : +)) 0 ( = = nnbnaba +) n ndcbadcba|.|
\|= |.|
\| =Sau y l mt s v d minh ha: ( gi thit cc t s u c ngha) V d 1: Cho t l thc dcba=.Chng minh rng: d cd cb ab a+=+ Gii: Cch 1: (PP1) Ta c:bd bc ad ac d c b a + = + ) )( ((1) bd bc ad ac d c b a + = + ) )( ((2)T gi thit:bc addcba= =(3) T (1), (2), (3) suy ra:) )( ( ) )( ( d c b a d c b a + = + d cd cb ab a+=+(pcm) Cch 2: (PP2) t kdcba= = , suy radk c bk a = = ,Ta c: 11) 1 () 1 (+=+=+=+kkk b k bb kbb kbb ab a(1) Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 12 11) 1 () 1 (+=+=+=+kkk dk dd kdd kdd cd c (2) T (1) v (2) suy ra: d cd cb ab a+=+(pcm) Cch 3: (PP3) T gi thit: dbcadcba= =p dng tnh cht ca dy t s bng nhau ta c: d cb ad cb adbca=++= = d cd cb ab a+=+(pcm) Hi: o li c ng khng ? V d 2: Cho t l thc dcba=. Chng minh rng: 2 22 2d cb acdab=Gii: Cch 1: T gi thit:bc addcba= =(1) Ta c:( ) adbd acbc abd abc d c ab = = 2 2 2 2 (2) ( ) bd bc acad cd b cd a b a cd .2 2 2 2 = = (3) T (1), (2), (3) suy ra:( ) ( )2 2 2 2b a cd d c ab = 2 22 2d cb acdab= (pcm) Cch 2: t kdcba= = , suy radk c bk a = = ,Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 13Ta c: 2222..dbkdkbd dk b bkcdab= = =(1)
( )( )222 22 22 2 22 2 22 22 22 22 211) () (dbk dk bd k db k bd dkb bkd cb a====(2) T (1) v (2) suy ra: 2 22 2d cb acdab= (pcm) Cch 3: T gi thit: 2 22 22222d cb adbcacbabdbcadcba= = = = =2 22 2d cb acdab= (pcm) BI TP VN DNG: Bi 1: Cho t l thc: dcba= . Chng minh rng ta c cc t l thc sau: (vi gi thit cc t s u c ngha). 1) d cd cb ab a5 35 35 35 3+=+2) 2 22 22d cb ad cb a++= |.|
\|++ 3) d cd cb ab a+=+4) ( )( )22d cb acdab=5) d cd cb ab a4 35 24 35 2+=+ 6) b ad cd cb a2007 20062006 20052007 20062006 2005+=+ 7) d c cb a a+=+8) bd bbd bac aac a5 75 75 75 72222+=+ Bi 2: Cho t l thc: dcba= . Chng minh rng ta c cc t l thc sau: (vi gi thit cc t s u c ngha). Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 14a) d cd cb ab a5 35 35 35 3+=+b) 2 22 22d cb ad cb a++= |.|
\|++c) d cd cb ab a+=+ d) ( )( )22d cb acdab= e) d cd cb ab a4 35 24 35 2+=+ f)2008 2009 2008 20092009 2010 2009 2010a b c dc d a b =+ + g) d c cb a a+=+ h) bd bbd bac aac a5 75 75 75 72222+=+ i)2 22 2 2 27a 3ab 7c 3cd11a 8b 11c 8d+ += Bi 3: Cho dccbba= =. Chng minh rng: dad c bc b a= |.|
\|+ ++ +3 Bi 4: Cho dccbba= =. Chng minh rng: dad c bc b a= |.|
\|+ ++ +3 Bi 5: Cho 2005 2004 2003c b a= = Chng minh rng: 2) ( ) )( ( 4 a c c b b a = Bi 6: Cho dy t s bng nhau: 3 2008 1 22 3 4 2009a a a a...a a a a= = = =CMR: Ta c ng thc: 20081 2 3 2008 12009 2 3 4 2009a a a ... a aa a a a ... a| | + + + +=|+ + + +\ . Bi 7: Cho 19983221..... ..........aaaaaaaa= = = =v 0 ...9 2 1= + + + a a aChng minh rng: 9 2 1... a a a = = =Bi 8: Cho 2005 2004 2003c b a= = Chng minh rng: 2) ( ) )( ( 4 a c c b b a = Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 15Bi 9: Chng minh rng nu : dbba= thdad bb a=++2 22 2 Bi 10: Cho 19983221..... ..........aaaaaaaa= = = =v 0 ...9 2 1= + + + a a aChng minh rng: 9 2 1... a a a = = =Bi 11: CMR: Nubc a =2 th a ca cb ab a+=+. o li c ng khng? Bi 12: Chng minh rng nu : dbba= thdad bb a=++2 22 2 Bi 13:Cho d cd cb ab a+=+. CMR: dcba=Bi 14. Cho t l thc : 2 22 2a b a bc d c d+=+. Chng minh rng: a cb d= . Gii. Ta c : cdabd cb a=++2 22 2 =( )( )( )( )( )( ) d c b ad c d cb a b acdabd cb ad cd cb ab acdab..2222222 22 2=+ ++ + =++=+ ++ +=; ( )( )( )( ) dcbaad cb ad ac cb cabd cabd cadb dabd bcad accb cab a dd c bd c ab a c= = + = + ==++=++=++=++ 1 Bi 15:Chngminh rng nu: 3322+=+vvuuth3 2v u=Bi 16: CMR: Nubc a =2 th a ca cb ab a+=+. o li c ng khng? Bi 17: CMR nu) ( ) ( ) ( y x c x z b z y a + = + = + trong a, b,c khc nhau v khc 0 th : ) ( ) ( ) ( b a cy xa c bx zc b az y== Bi 18:Cho d cd cb ab a+=+. CMR: dcba=Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 16Bi 19: Cho dcba= . Cc s x, y, z, t tha mn:0 = + yb xav0 = + td zcChng minh rng:td zcyd xctb zayb xa++=++ Bi 20:Chngminh rng nu: 3322+=+vvuuth3 2v u=Bi 21: Cho a, b, c, d l 4 s khc 0 tha mn:bd c ac b = =2 2;v03 3 3= + + d c bChng minh rng: dad c bc b a=+ ++ +3 3 33 3 3 Bi 22: CMR nu) ( ) ( ) ( y x c x z b z y a + = + = +.Trong a, b,c khc nhau v khc 0 th : ) ( ) ( ) ( b a cy xa c bx zc b az y== Bi 23: Cho 1 1212c x b x ac bx axP+ ++ +=. Chng minh rng nu 1 1 1ccbbaa= = th gi tr ca P khng ph thuc vo x. Bi 24: Cho bit : ' '' 'a b b c1; 1a b b c+ = + = . CMR: abc + abc = 0. Bi 25: Cho dcba= . Cc s x, y, z, t tha mn:0 = + yb xav0 = + td zcChng minh rng:td zcyd xctb zayb xa++=++ Bi 26: Cho a, b, c, d l 4 s khc 0 tha mn:bd c ac b = =2 2; v03 3 3= + + d c bChng minh rng: dad c bc b a=+ ++ +3 3 33 3 3 Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 17Bi 27: Cho 1 1212c x b x ac bx axP+ ++ +=. Chng minh rng nu 1 1 1ccbbaa= = th gi tr ca P khng ph thuc vo x. Bi 28: Cho t l thc: 2a 13b 2c 13d3a 7 b 3c 7 d+ += ;Chng minh rng: a cb d= . Bi 29: Cho dy t s : bz cy cx az ay bxa b c = =; CMR: x y za b c= = . Thanh M,ngy 10thng 12nm2010 Chuyn : GI TR TUYT I A> MC TIU Thng qua vic gii ton s pht trin c t duy c lp, sng to ca hc sinh, rn ch vt qua mi kh khn.B> THI LNG Tng s :(6 tit) 1) Kin thc cn nh:(1 tit) 2)Cc dng bi tp v phng php gii(5 tit) 1. L thuyt *nh ngha: Khong cch t im a n im 0 trn trc s l gi tr tuyt i ca mt s a( a l s thc) * Gi tr tuyt i ca s khng m l chnh n, gi tr tuyt i ca s m l s i ca n. TQ: Nua a a = > 0Nua a a = < 0Nu x-a > 0=> | | x-a = x-a Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 18Nu x-a s0=> | | x-a = a-x *Tnh cht Gi tr tuyt i ca mi s u khng m TQ: 0 > avi mi a e R C th: | | a =0 a=0 | | a 0 a 0 * Hai s bng nhau hoc i nhau th c gi tr tuyt i bng nhau, v ngc li hai s c gi tr tuyt i bng nhau th chng l hai s bng nhau hoc i nhau. TQ:
== =b ab ab a* Mi s u ln hn hoc bng i ca gi tr tuyt i ca n v ng thi nh hn hoc bng gi tr tuyt i ca n. TQ:a a a s s v0 ; 0 > = s = a a a a a a* Trong hai s m s no nh hn th c gi tr tuyt i ln hn TQ: Nub a b a > < < 0*Trong hai s dng s no nh hn th c gi tr tuyt i nh hn TQ: Nub a b a < < < 0* Gi tr tuyt i ca mt tch bng tch cc gi tr tuyt i. TQ: b a b a . . =* Gi tr tuyt i ca mt thng bng thng hai gi tr tuyt i. TQ:baba=* Bnh phng ca gi tr tuyt i ca mt s bng bnh phng s . TQ:22a a =* Tng hai gi tr tuyt i ca hai s lun ln hn hoc bng gi tr tuyt i ca hai s, du bng xy ra khi v ch khi hai s cng du. TQ: b a b a + > +v0 .> + = + b a b a b a2. Cc dng ton : I. Tm gi tr ca x tho mn ng thc c cha du gi tr tuyt i: 1. Dng 1: k A(x) =( Trong A(x) l biu thc cha x, k l mt s cho trc ) * Cch gii:- Nu k < 0 th khng c gi tr no ca x tho mn ng thc( V gi tr tuyt i ca mi s u khng m )- Nu k = 0 th ta c0 ) ( 0 ) ( = = x A x A- Nu k > 0 th ta c:
== =k x Ak x Ak x A) () () (Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 19Bi 1.1: Tm x, bit: a)4 5 2 = xb) 4124531= x c) 315121= + x d) 871 243= + xGii a 1) | | x = 4 x= 4 a 2)4 5 2 = x2x-5 = 4 * 2x-5 = 4 2x = 9 x= 4,5 * 2x-5 = - 4 2x =5-4 2x=1 x=0,5 Tm li:x = 4,5;x =0,5 b)4124531= x 54-2x= 13 - 14
Bi 1.2: Tm x, bit: a) 213 2 2 = x b)5 , 4 2 5 3 5 , 7 = x c)15 , 2 75 , 3154 = + xBi 1.3: Tm x, bit: a)5 1 1 3 2 = + x b)3 12= xc)5 , 32152= + + x d) 51231= xBi 1.4: Tm x, bit: a)% 54341= + x b) 4541232= x c) 47435423= + x d) 653521435 , 4 = + xBi 1.5: Tm x, bit: a)231:495 , 6 = + xb) 27514 :23411= + x c) 32143: 5 , 2415= + x d) 6324: 3521= +x 2. Dng 2:B(x) A(x) =( Trong A(x) v B(x) l hai biu thc cha x ) * Cch gii: Vn dng tnh cht:
== =b ab ab ata c:
== =) ( ) () ( ) () ( ) (x B x Ax B x Ax B x ABi 2.1: Tm x, bit: Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 20a)2 4 5 + = x x b)0 2 3 3 2 = + x x c)3 4 3 2 = + x x d)0 6 5 1 7 = + + x xa)2 4 5 + = x x* 5x-4=x+2 5x- x =2+4 4x=6 x= 1,5 * 5x-4=-x-2 5x + x =- 2+ 4 6x= 2 x= 13
Vy x= 1,5; x= 13
Bi 2.2: Tm x, bit: a)1 42123 = + x x b)053852745= + x x c) 41343257 = + x x d)0 5216587= + + x x3. Dng 3:B(x) A(x) = ( Trong A(x) v B(x) l hai biu thc cha x ) * Cch 1:Tathynu B(x)(*) (1)Trthnh
== =) ( ) () ( ) () ( ) (x B x Ax B x Ax B x A (ichiugitrixtmcviiu kin ( * ) * Cch 2: Chia khong xt iu kin b du gi tr tuyt i: Nua a a = > 0Nua a a = < 0Ta gii nh sau: ) ( ) ( x B x A = (1) -Nu A(x)0 >th (1) tr thnh: A(x) = B(x) (i chiu gi tr x tm c vi iu kin ) -Nu A (x) 0 ta c x+ 25 =2x Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 21 *Xt x+ 25 < 0 ta c x+ 25 =- 2x Bi 3.1: Tm x, bit: a)x x 2 321 = b)2 3 1 + = x x c)12 5 = x x d)1 5 7 + = x xBi 3.2: Tm x, bit: a)x x 2 9 = +b)2 3 5 = x x c)x x 2 9 6 = + d)21 3 2 = + x xBi 3.3: Tm x, bit: a)x x 4 2 4 = + b)x x = + 2 1 3 c)x x 3 1 15 = + + d)2 5 2 = + x xBi 3.4: Tm x, bit: a)1 5 2 + = x x b)x x = 1 2 3 c)1 2 7 3 + = x x d)x x = + 1 1 2Bi 3.5: Tm x, bit: a)x x = + 5 5 b)7 7 = + x x c)x x 3 4 4 3 = + d)x x 2 7 2 7 = + 4. Dng 4: ng thc cha nhiu du gi tr tuyt i:* Cch gii: Lp bng xt iu kin b du gi tr tuyt i: m x C x B x A = + + ) ( ) ( ) (Cn c bng trn xt tng khong gii bi ton ( i chiu iu kin tng ng ) V d1 : Tm x bit rng1 3 2 1 x x x + = (1) Nhn xt: Nh trn chng ta bin i c biu thc cha du gi tr tuyt i thnh cc biu thc khng cha du gi tr tuyt i. Vy ta s bin i biu thc v tri ca ng thc trn. T s tm c x Gii Xtx 1 = 0x = 1; x 1 < 0 x < 1; x 1> 0x > 1 x- 3 = 0 x = 3; x 3 < 0 x < 3; x 3 > 0x > 3 Ta c bng xt du cc a thc x- 1 v x- 3 di y:
Xt khong x < 1 ta c: (1) (1 x ) + ( 3 x ) = 2x 1 -2x + 4=2x 1 x = 54 (gi tr ny khng thuc khong ang xt) Xt khong 1s xs 3ta c:(1) (x 1 ) + ( 3 x ) = 2x 1 2= 2x 1 x= 32 ( gi tr ny thuc khong ang xt) x13 x 1- 0++ x 3--0+ Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 22Xt khong x > 3 ta c: (1) (x 1 ) + (x 3) = 2x 1 - 4= -1 ( V l) Kt lun: Vy x = 32. VD2 : Tm x | | x+1 + | | x-1 =0 Nhn xt x+1=0 => x=-1 x-1=0 => x=1 Ta lp bng xt du x -1 1 x+1 - 0 + + x-1-- 0 + Cn c vo bng xt du ta c ba trng hp Nu x1Bi 4.1: Tm x, bit:a)12 3 7 5 2 1 3 4 = + + x x x x b)5 9 3 5 1 2 4 3 = + + + + x x x xc)2 , 151851512 = + + x x d)x x x = + +512213213 2Bi 4.2: Tm x, bit: a)8 3 6 2 = + + x xc)9 3 5 = + + x x d)2 4 3 2 = + + x x xe)6 3 2 1 = + + + + x x x f) 11 4 2 2 = + + x x Bi 4.3: Tm x, bit: a)9 8 2 3 2 = + + x x x b)12 2 2 1 3 = + + x x x xc)4 2 2 3 3 1 = + x x x d) x x x = + 2 1 5 e)1 3 2 = + x x x f)3 1 + = + x x x xBi 4.4: Tm x, bit: a)3 5 2 = + x x b)8 5 3 = + + x xc)4 5 2 1 2 = + x x d)1 2 4 3 3 + = + + x x x5. Dng 5: Xt iu kin b du gi tr tuyt i hng lot: ) D(x C(x) B(x) A(x) = + + (1) iu kin: D(x)0 >ko theo0 ) ( ; 0 ) ( ; 0 ) ( > > > x C x B x ADo vy (1) tr thnh: A(x) + B(x) + C(x) = D(x) Bi 5.1: Tm x, bit: a)x x x x 4 3 2 1 = + + + + + b)1 5 4 3 2 1 = + + + + + + + x x x x xGenerated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 23c)x x x x 421532 = + + + + + d)x x x x x 5 4 , 1 3 , 1 2 , 1 1 , 1 = + + + + + + +Bi 5.2: Tm x, bit: a)x x x x x 101101100...101310121011= + + + + + + + +b)x x x x x 100100 . 991...4 . 313 . 212 . 11= + + + + + + + +c)x x x x x 5099 . 971...7 . 515 . 313 . 11= + + + + + + + +d)x x x x x 101401 . 3971...13 . 919 . 515 . 11= + + + + + + + +6. Dng 6: Dng hn hp: Bi 6.1: Tm x, bit: a) 54211 2 = + xb) 22122 2+ = + x x xc) 2 243x x x = +Bi 6.2: Tm x, bit: a) 51211 2 = x b) 5243121= + x c)x x x = +432 Bi 6.3: Tm x, bit: a)x x x = 432b) 43243221 = |.|
\|+ x x xc) 43243221 = x x xBi 6.4: Tm x, bit: a)1 4 1 3 2 = + x x x b)2 1 1 = x c)2 5 1 3 = + x7. Dng 7: 0 B A = + Vndngtnhchtkhngmcagitrtuytidnnphngphpbtng thc. * Nhn xt: Tng ca cc s khng m l mt s khng m v tng bng 0 khi v ch khi cc s hng ca tng ng thi bng 0. * Cch gii chung:0 = + B A B1: nh gi:000> + )`>>B ABA B2: Khng nh:0 = + B A==00BA Bi 7.1: Tm x, y tho mn:a)0 5 3 4 3 = + + y x b)0259= + + y y x c)0 5 4 2 3 = + + y xGenerated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 24Bi 7.2: Tm x, y tho mn: a)0 372435 = + y x b)0132317115 , 1432132= + + + y x c) 0 2008 2007 = + y x* Ch 1: Bi ton c th cho di dng0 s + B Anhng kt qu khng thay i * Cch gii:0 s + B A(1) 000> + )`>>B ABA (2) T (1) v (2) 0 = + B A==00BA Bi 7.3: Tm x, y tho mn: a)0 8 6 1 5 s + + y x b)0 3 4 2 s + + y y x c)0 1 2 2 s + + + y y xBi 7.4: Tm x, y tho mn: a)0 5 11 8 12 s + + y x b)0 1 4 2 3 s + + y y x c)0 10 7 s + + xy y x* Ch 2: Do tnh cht khng m ca gi tr tuyt i tng t nh tnh cht khng m ca lu tha bc chn nn c th kt hp hai kin thc ta cng c cc bi tng t. Bi 7.5: Tm x, y tho mn ng thc: a)0 3 2 = + + y y x b)0 4 32008 2007= + + y y xc)( ) 0 1 20072006= + + y y xd)( ) 0 3 2007 52008= + y y xBi 7.6: Tm x, y tho mn : a)( ) ( ) 0 3 12 2= + + y x b)( ) 0 7 2 5 5 25 4= + y xc)( ) 0214 2 32004= + + y y xd)0212 1 32000= |.|
\| + + y y xBi 7.7: Tm x, y tho mn: a)0 2008 2007 s + y xb) 03210 375s + + y y x c) 025654200820072143212006s + + |.|
\| y x d)0 4 2008 2 20072007 2008s + y y x8. Dng 8: B A B A + = +* Cch gii: S dng tnh cht:b a b a + > +T ta c:0 .> + = + b a b a b aBi 8.1: Tm x, bit: Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 25a)8 3 5 = + + x x b)3 5 2 = + x x c)6 1 3 5 3 = + + x xd)11 5 2 3 2 = + + x x e)2 3 3 2 1 = + + x x x f)2 4 2 5 3 = + + x x xBi 8.2: Tm x, bit: a)2 6 4 = + x x b)4 5 1 = + + + x x c)13 2 3 7 3 = + + x xd)x x x 3 4 2 3 1 5 + = + + e)3 1 1 3 2 = + + + x x x f)4 7 2 = + x x1 - Lp bng xt du b du gi tri tuyt i Bi 1: Tm x, bit: a)8 3 6 2 = + + x xTa lp bng xt du x -3 3 x+3 - 0 + + 2x-6-- 0 + Cn c vo bng xt du ta c ba trng hp * Nu x3) 2- B du gi tr tuyt i theo nguyn tc t ngoi vo trong Bi 1: Tm x, bit: a) 54211 2 = + x
* | | 2x-1 + 12 = 45
| | 2x-1 = 45 -12
| | 2x-1 = 310
Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 26 2x-1= 310 2x = 310 +1 x= 1320
2x-1= -3102x = - 310 +1 x= 720
* | | 2x-1 + 12 =-45
| | 2x-1=-45 - 12 (khng tha mn) 3 - S dng phng php bt ng thc: Bi 1: Tm x, y tho mn ng thc: a)0 3 2 = + + y y x x-y-2 =0x=-1
y+3=0 y= -3 Bi 2: Tm x, y tho mn : a)( ) ( ) 0 3 12 2= + + y x Bi 3: Tm x, y tho mn: a) 0 2008 2007 s + y x Bi 4: Tm x tho mn: a)8 3 5 = + + x xII Tmcp gitr(x;y) nguynthomnngthcchadugitrtuyt i: 1. Dng 1:m B A = +vi0 > m* Cch gii:* Nu m = 0 th ta c0 = + B A==00BA * Nu m > 0 ta gii nh sau: m B A = +(1) Do0 > Ann t (1) ta c:m B s s 0t tm gi tr caBvAtng ng . Bi 1.1: Tm cp s nguyn ( x, y) tho mn: a)0 2008 2007 = + x x b)0 3 2 = + + y y x c)( ) 0 1 22= + + y y xGenerated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 27Bi 1.2: Tm cp s nguyn ( x, y) tho mn: a)0 4 35= + + y y x b)( ) 0 3 54= + y y x c)0 2 3 1 3 = + + + y y xBi 1.3: Tm cp s nguyn (x, y ) tho mn: a)3 2 4 = + + y x b)4 1 1 2 = + + y xc)5 5 3 = + +y x d)7 3 2 5 = + + y xBi 1.4: Tm cp s nguyn ( x, y ) tho mn: a) 5 4 5 3 = + + y x b)12 1 2 4 6 = + + y x c)10 3 3 2 = + +y xd)21 3 4 3 = + +y xBi 1.5: Tm cc cp s nguyn ( x, y ) tho mn: a)3 2 32 = x y b)1 52 = x yc)4 3 22+ = x y d) 2 12 32 = x y2. Dng 2:m B A < +vi m > 0. * Cch gii: nh gim B A < + (1) 000> + )`>>B ABA (2) T (1) v (2)m B A < + s 0t gii bi tonk B A = +nh dng 1 vim k < s 0Bi 2.1: Tm cc cp s nguyn ( x, y ) tho mn: a)3 s +y x b)4 2 5 s + + y x c)3 4 1 2 s + + y xd)4 5 3 s + +y xBi 2.2: Tm cc cp s nguyn ( x, y )tho mn: a)7 2 1 5 s + + y xb)5 3 5 2 4 s + + + y x c)3 1 2 5 3 s + + y x d)7 1 2 4 1 2 3 s + + y x3. Dng 3: S dng bt ng thc:b a b a + > +xt khong gi tr ca n s. Bi 3.1: Tm cc s nguyn x tho mn: a)3 4 1 = + x x b)5 3 2 = + + x x c)7 6 1 = + + x x d)8 3 2 5 2 = + + x xBi 3.2: Tm cc cp s nguyn ( x, y) tho mn ng thi cc iu kin sau. a) x + y = 4 v6 2 = + + y x b) x +y = 4 v5 1 2 = + + x y xc) x y = 3 v3 = +y x d) x 2y = 5 v6 1 2 = + y xBi 3.3: Tm cc cp s nguyn ( x, y ) tho mn ng thi: a) x + y = 5 v4 2 1 = + + y x b) x y = 3 v4 1 6 = + y xc) x y = 2 v4 1 2 1 2 = + + + y x d) 2x + y = 3 v8 2 3 2 = + + + y x4. Dng 4: Kt hp tnh cht khng m ca gi tr tuyt i v du ca mt tch: * Cch gii :) ( ) ( ). ( y A x B x A =nh gi:m x n x B x A y A s s > > 0 ) ( ). ( 0 ) (tm c gi tr ca x. Bi 4.1: Tm cc s nguyn x tho mn:a)( )( ) 0 3 2 < + x x b)( )( ) 0 5 2 1 2 < x x c)( )( ) 0 2 2 3 > + x x d)( )( ) 0 2 5 1 3 > + x xGenerated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 28Bi 4.2: Tm cc cp s nguyn ( x, y ) tho mn: a)( )( ) 1 1 2 + = + y x x b)( )( ) y x x = + 1 3 c)( )( ) 2 1 2 5 2 + + = y x xBi 4.3:Tm cc cp s nguyn ( x, y ) tho mn: a)( )( ) 1 2 3 1 + = + y x x b)( )( ) 1 1 5 2 = + y x x c)( )( ) 0 2 5 3 = + y x x5. Dng 5: S dng phng php i lp hai v ca ng thc: * Cch gii: Tm x, y tho mn ng thc: A = Bnh gi:m A > (1) nh gi:m B s (2) T (1) v (2) ta c: == =m Bm AB ABi 5.1: Tm cc cp s nguyn ( x, y ) tho mn: a)( )22 3 1 2 + = + + y x x b) 3 1121 5+ += + yx xc) ( ) 2 6 2105 32+ = + +xy d) 3 363 1+ += + yx xBi 5.2: Tm cc cp s nguyn ( x, y ) tho mn: a) ( ) 2 5 281 2 3 22+ = + +yx x b) 2 2161 3+ + = + +y yx xc) ( ) 2 3125 3 1 32+ += + +yx x d) 2 4105 1 2+ = + yy xBi 5.3: Tm cc cp s nguyn ( x, y ) tho mn:a)( )3 1147 22 + = + +y yy x b)( )5 2 3204 22+ += + +yxc) 2 200863 2007 2+ = + yx d) 6 5 3305 2+ += + + +yy xIII Rt gn biu thc cha du gi tr tuyt i: -Cch gii chung:Xt iu kin b du gi tr tuyt i ri thu gn: Bi 1: Rt gn biu thc sau vi1 , 4 5 , 3 s s xa)x x A + = 1 , 4 5 , 3 b)1 , 4 5 , 3 + + = x x BBi 2: Rt gn biu thc sau khi x < - 1,3: a)5 , 2 3 , 1 + = x x A b)5 , 2 3 , 1 + = x x BBi 3: Rt gn biu thc: a)7 , 1 5 , 2 + = x x A b) 5251 + = x x B c)3 1 + + = x x CBi 4: Rt gn biu thc khi 7153< 0 ==============&=&=&============== IV Tnh gi tr biu thc: Bi 1: Tnh gi tr ca biu thc: a) M = a + 2ab b vi75 , 0 ; 5 , 1 = = b a b) N = ba 22 vi75 , 0 ; 5 , 1 = = b aBi 2: Tnh gi tr ca biu thc: a)y xy x A + = 2 2vi 43; 5 , 2= = y x b)b ab a B = 3 3vi25 , 0 ;31= = b ac) baC335 =vi25 , 0 ;31= = b ad)1 2 32+ = x x D vi 21= xBi 3: Tnh gi tr ca cc biu thc: a)4 2 3 62 3+ + = x x x Avi 32 = x b)y x B 3 2 =vi3 ;21 = = y xc)x x C = 1 3 2 2vi x = 4d) 1 31 7 52 + =xx xD vi 21= xV Tm gi tr ln nht nh nht ca mt biu thc cha du gi tr tuyt i: 1. Dng 1: S dng tnh cht khng m ca gi tr tuyt i: * Cch gii ch yu l t tnh cht khng m ca gi tr tuyt i vn dng tnh cht ca bt ng thc nh gi gi tr ca biu thc: Bi 1.1: Tm gi tr ln nht ca ccbiu thc: a)5 , 3 5 , 0 = x A b)2 4 , 1 = x B c) 5 42 3+=xxC d) 1 33 2+=xxDe)5 , 1 2 5 , 5 = x E f)14 3 2 , 10 = x F g)12 3 2 5 4 + = y x Gh) 8 , 5 5 , 28 , 5+ =xHi)8 , 5 5 , 2 = x I k)2 4 10 = x Kl)1 2 5 = x L m) 3 21+ =xM n) 4 5 3122+ ++ =xNBi 1.2: Tm gi tr nh nht ca biu thc: a)x A + = 4 , 3 7 , 1 b)5 , 3 8 , 2 + =x B c)x C + = 3 , 4 7 , 3d)2 , 14 4 , 8 3 + = x D e)5 , 17 5 , 7 5 3 4 + + + = y x E f)8 , 5 5 , 2 + = x FGenerated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 30g)8 , 2 9 , 4 + = x G h) 7352+ =x H i)x I + = 9 , 1 5 , 1k)4 1 3 2 = x K l)1 2 3 2 + = x L m)1 4 1 5 = x MBi 1.3: Tm gi tr ln nht ca biu thc: a) 3 7 3 4155+ ++ =xA b) 7 21 15 82131+ +=xB c) 8 5 4 5 32054+ + + ++ =y xCd) 6 1 2 3 2 2246+ + + + =x y xD e) ( ) 14 5 5 321322+ + + ++ =x y xEBi 1.4: Tm gi tr ln nht ca biu thc: a) 4 5 711 5 7 2+ ++ +=xxA b) 6 7 2 213 7 2+ + + +=yyB c) 8 1 632 1 15+ ++ +=xxCBi 1.5: Tm gi tr nh nht ca biu thc: a) 24 7 5 485+ ++ =xA b) 35 8 6 51456+ =yB c) 35 1 2 3 3281215+ + + =x y xCBi 1.6: Tm gi tr nh nht ca biu thc: a) 5 6 4 333 6 4 21+ ++ +=xxA b) 14 5 214 5 6+ ++ +=yyB c) 12 7 368 7 15+ + + =xxC2. Dng 2: Xt iu kin b du gi tr tuyt i xc nh khong gi tr ca biu thc: Bi 2.1: Tm gi tr nh nht ca biu thc: a)x x A + + = 2 5 b)6 2 1 2 + + = x x B c)x x C 3 8 5 3 + + =d)5 4 3 4 + + = x x D e)x x E 5 3 6 5 + + = f)x x F 2 5 7 2 + + =Bi 2.2: Tm gi tr nh nht ca biu thc: a)5 2 3 2 + + = x x A b)x x B 3 4 1 3 + = c) 1 4 5 4 + + = x x CBi 2.3: Tm gi tr ln nht ca biu thc:a)4 5 + + = x x A b)4 2 3 2 + + + = x x B c)x x C 3 7 1 3 + =Bi 2.4: Tm gi tr ln nht ca biu thc: a)6 2 5 2 + + = x x A b)x x B 3 8 4 3 + = c)7 5 5 5 + + = x x CBi 2.5: Tm gi tr nh nht ca biu thc: a)5 1 + + = x x A b)5 6 2 + + = x x B c)1 2 4 2 + + = x x C3. Dng 3: S dng bt ng thcb a b a + > +Bi 3.1: Tm gi tr nh nht ca biu thc: a)3 2 + + = x x A b)5 2 4 2 + + = x x B c)1 3 2 3 + + = x x CGenerated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 31Bi 3.2: Tm gi tr nh nht ca biu thc: a)4 1 5 + + + + = x x A b)8 2 3 7 3 + + + = x x B c)12 5 4 3 4 + + + = x x CBi 3.3: Tm gi tr nh nht ca biu thc: a)7 5 2 3 + + + = x x x A b)5 1 4 3 1 + + + + = x x x Bc)3 5 2 4 2 + + + = x x x C d)3 1 1 6 5 3 + + + + + = x x x DBi 3.4: Cho x + y = 5 tm gi tr nh nht ca biu thc: 2 1 + + = y x ABi 3.5: Cho x y = 3, tm gi tr ca biu thc: 1 6 + + = y x BBi 3.6: Cho x y = 2 tm gi tr nh nht ca biu thc: 1 2 1 2 + + + = y x CBi 3.7: Cho 2x+y = 3 tm gi tr nh nht ca biu thc: 2 2 3 2 + + + + = y x D DY S T NHIN VIT THEO QUY LUT, DY CC PHN S VIT THEO QUY LUT( tip) Bi 1 : Tnh tng: 2 + 4 6 8 + 10 + 12 14 16 + 18 + 20 22 24 - 2008 Hng dn: Bi 2:Cho. 100 99 ... 4 3 2 1 + + + = Aa) Tnh A. Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 32b) A c chia ht cho 2, cho 3, cho 5 khng ? c) A c bao nhiu c t nhin. Bao nhiu c nguyn ? Hng dn: Bi 3:Cho... 31 25 19 13 7 1 + + + = Aa) Bit A = 181. Hi A c bao nhiu s hng ? b) Bit A c n s hng. Tnh gi tr ca A theo n ? Hng dn: Bi 4: Cho.... 31 25 19 13 7 1 + + + = Aa) Bit A c 40 s hng. Tnh gi tr ca A. b) Tm s hng th 2004 ca A. Hng dn: Bi 5: Tm gi tr ca x trong dy tnh sau: 655 ) 47 ( ) 42 ( ... ) 12 ( ) 7 ( ) 2 ( = + + + + + + + + + + x x x x xHng dn: Bi 6: a) Tm x bit :x + (x+1) + (x+2) + (x+3) + + (x+2009) = 2009.2010 b)TnhM = 1.2+2.3+3.4+ + 2009. 2010 Hng dn: Bi 7:Tnh tng:100001 . 99999 10001 . 9999 1001 . 999 101 . 99 11 . 9 + + + + = SHng dn: Bi 8:Cho 100 3 23 ... 3 3 3 + + + + = ATm s t nhin n bit rng 2A + 3 = 3n Hng dn: Bi 9:Cho 2004 3 23 .... 3 3 3 + + + + = Aa) Tnh tng A. b) Chng minh rng130 A . c) A c phi l s chnh phng khng ? V sao ? Hng dn: Bi 10:a) Cho 2004 2003 3 23 3 ... 3 3 3 1 + + + = AChng minh rng: 4A -1 l lu tha ca 3. b) Chng minh rng A l mt lu tha ca 2 vi
2004 2003 5 4 32 2 ... 2 2 2 4 + + + + + + = A Hng dn: Bi 11:Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 33a) Cho 60 3 22 ... 2 2 2 + + + + = AChng minh rng A chia ht cho 3, 7 v 15. b)Chng minh rng tng 2 + 22 + 23 + + 22003 + 22004chia ht cho 42 Hng dn: Bi 12: ChoA = 2 + 22 + 23 + ............+299 + 2100 Chng t A chia ht cho 31 Hng dn: Bi 13: Cho S = 5 + 52 + 53 + . . . . + 596
a, Chng minh: S 126 b, Tm ch s tn cng ca S Hng dn: Bi 14: Cho30 . 29 ...... 3 . 2 . 1 = A 60 . 59 ........ 33 . 32 . 31 = Ba) Chng minh: B chia ht cho 302b) Chng minh: B - A chia ht cho 61. Hng dn: Bi 15: Cho 2002 2001 4 3 22 2 ... 2 2 2 3 + + + + + + = A v 20032 = BSo snh A v B. Hng dn: Bi 16: Cho M = 2 3 99 1003 3 3 ... 3 3 + + + + + . a. M c chia ht cho 4, cho 12 khng ? v sao? b.Tm s t nhin n bit rng 2M+3 =3n. Hng dn: Bi 17: Cho biu thc:M = 1 +3 + 32+ 33 ++ 3118+ 3119 a) Thu gn biu thc M. b) Biu thc M c chia ht cho 5, cho 13 khng? V sao? Hng dn: Bi 18:Tm s t nhin n bit:20042003) 1 (2...1016131=++ + + +n n Hng dn: Bi 19: a) Tnh:2 2 2 2.....1.3 3.5 5.7 99.101+ + + +Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 34b) Cho *) 3 (310 . 737 . 434 . 13N nn nS e++ + + + = Chng minh: S < 1 Hng dn: Bi 20: So snh: 2 2 2 2...60.63 63.66 117.120 2003A = + + + + v 5 5 5 5...40.44 44.48 76.80 2003B = + + + +Hng dn: Bi 21: a) Tnh 340123811541881401101+ + + + + = A b) Tnh: 2005 . 20042....1511016131+ + + + + = Mc)Tnh tng: 100 . 99 . 981...4 . 3 . 213 . 2 . 11+ + + = SHng dn: Bi 22: So snh:100 3 221...2121211 + + + + + = A vB = 2. Hng dn: Bi 23: So snh: 2 2 2 2...60.63 63.66 117.120 2006A = + + + +v 5 5 5 5...40.44 44.48 76.80 2006B = + + + +Hng dn: Bi 24. Tnh a. A = 2 2 2 2 2.15 35 63 99 143+ + + +b. B = 3+3 3 3 3...1 2 1 2 3 1 2 3 4 1 2 ... 100+ + + ++ + + + ++ + + +. Hng dn: Bi 25: Tnh gi tr cc biu thc: a)A = 1 . 9913 . 971...95 . 5197 . 3199 . 11991971...51311+ + + + ++ + + + + b)B = 991...3972981991001...413121+ + + ++ + + + Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 35Hng dn: Bi 26: Chng minh rng: 100 - 10099...4332211001...31211 + + + + = |.|
\|+ + + +Hng dn: Bi 27: Tnh BA bit: A = 2001...413121+ + + + v B = 11992198...197319821991+ + + + +Hng dn: Bi 28: Tm tch ca 98 s u tin ca dy: ;....3511 ;2411 ;1511 ;811 ;311Hng dn: Bi 29: Tnh tng 100 s hng u tin ca dy sau: ;...3361;1761;661;61 Hng dn: Bi 30: Tnh BA bit: A = 20 . 19118 . 171...6 . 514 . 312 . 11+ + + + +B = 201191...131121111+ + + + +Hng dn: Bi 31: Tm x, bit: 110 . 1001....12 . 2111 . 11110 . 101...102 . 21101 . 11+ + + = |.|
\|+ + + xHng dn: Bi 32: Tnh :a) 2 31 ...nS a a a a = + + + + +, vi ( 2, a n N > e ) b) 2 4 6 211 ...nS a a a a = + + + + + , vi ( 2, a n N > e ) c) 3 5 2 12...nS a a a a+= + + + +, vi (*2, a n N > e ) Hng dn: Bi 33: Cho 2 3 99 1001 4 4 4 ... 4 ,4 A B = + + + + + = . Chng minh rng: 3BA< . Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 36Hng dn: Bi 34: Tnh gi tr ca biu thc: 50 200 )9 99 999 ... 999...9)9 99 999 ... 999...9a Ab B= + + + += + + + +ch sch s Hng dn: Chuyn 1:gii ton cha du gi tr tuyt i. 1-Kin thc c bn: > =00 x xx xxGenerated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 37x x x x x = > > ; ; 0 y x y xy x y x > + s + 2- Cc dng ton c bn: * Dng ton 1: Tnhx bit 1) 511 = x 2) 133:532 = x3)02125 = + x4) x149 . 471....5 . 313 . 11= + + +5) 2 100 . 971.....7 . 414 . 11 x= + + +6) 1015 2101 . 974....9 . 545 . 14 += + + +x7) 51210011 ....411311211 = + |.|
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\| x8)1512 100 . 99 .... 4 . 3 3 . 2 2 . 1 = + + + + x 9) 511 ) 2 )( 49 ... 2 1 (2 2 2 = + + + x* Dng 2: Tm x bit 1) 533 = x2)0825= x3)02355 = x4) 31151. 2 = x5)25 , 1 5 , 2 75 , 1 = x 6)13 5 2 = x 7) 32732313 = x8) 10117 3512 = x 9)9 ) 5 2 (2= x 10)42= x 11) 41) 7 3 (2= x* Dng 3: Tm x, y, zbit 1)0 = + + z y x 2)0 7 2 5 3 = + y x3)0313252211 = + + z y x 4)0 )31( )21( ) 1 (2 2 2= + + z y x5)0 4 3 3 2 2 1 = + + y y x 6)0 ) 1 )( 1 ( 1 = + + x x x*Dng 4: Tnh gi tr ca cc biu thc sau. 1)5 22+ = x x A vi31 = x2) 2 22 ) 3 ( 5 2 y xy x x xy B + + =vi x=y=2 3)1 2 2412+ + = x x x Cvi 21= x4)3 6 32+ = x x D vi1 = x5)xy y x E 7 5 2 + = vi0 2 = +y x6)xy y x G 6 3 22 2+ =vi0 2 1 = + y x* Dng 5: Rt gn cc biu thc sau 1)13 3 9 2 5 + + = x x x Mvi5 , 6 > x2) N= 3 2 1 + + + + x x xvi1 2 s x3) P= 15 5 7 3 5 2 + x x xvi3 > x*)Dng 6: Tm gi tr ln nht , gi tr nh nht. Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 381, Tm gi tr nh nht ca: 25 , 0 5 , 0 2 5 , 4 = x C2, Tmgi tr ln nht ca :75 , 0 5 , 4 3 + + = x D3, Tm gi tr nh nht ca :2004 2005 + = x x E3- Cc bi ton t hc : Bi 1: Tnh gi tr biu thc: A= 2x+2xy-y vi | x| = 2,5 v y = -3/4 Bi 2: Tm x , y bit:a) 2.| 2x-3|= 1/ 2 b) 7,5 -3 |5-2x|=-4,5 c) | 3x-4|+ |3y+5| = 0 Bi 3: Tm gi tr nh nht: a)| 3x- 8,4| -14,2 b)|4x-3|+|5y+7,5| +17,5 Ba 4: Tm gi tr ln nht: F= 4- |5x-2|- | 3y+12|CHUYN : CC BI TON V S THP PHN- S THC- CN BC HAI. Bi ton 1: Vit cc s thp phn sau di dng phn s ti gin 0,(1);0,(01); 0,(001); 1,(28);0,(12);1,3(4);0,00(24); 1,2(31);3,21(13) Bi ton 2: Tnh a) 10,(3)+0,(4)-8,(6) b)| | ) 21 ( , 4 : ) 6 ( 3 , 2 ) 1 ( , 12 c)) 2 ( 4 , 0313 ) 3 ( , 0 +Bi ton 3: Tnh tng cc ch s trong chu k khi biu din s 99116di dng s thp phn v hn tun hon. Bi ton 4: Tnh tng ca t v mu ca phn s ti gin biu din s thp phn 0,(12) Bi ton 5: Tnh gi tr ca biu thc sau v lm trn kt qu n hng n v a) 75 , 625 , 2 ). 19 , 8 81 , 11 ( += A b) 31 , 2 125 , 0 . 44 ). 25 , 6 : 5 6 , 4 (++= BBi ton 6: Rt gn biu thc ) 3 ( 8 , 0 ) 6 ( , 1 5 , 2) 6 ( 1 , 0 ) 3 ( , 0 5 , 0 + += MBi ton 7: Chng minh rng:0,(27)+0,(72)=1 Bi ton 8: Tm x bit a)) 2 ( , 0 .) 6 ( 1 , 1 ) 3 ( , 0) 3 ( , 0 ) 6 ( 1 , 0=++xb) 8550) 3 ( 0 , 0133) 384615 ( , 0 ) 3 ( , 0=+ + x c)| | 10 ) 62 ( , 0 ) 37 ( , 0 = + x d) 0,(12):1,(6)=x:0,(4) Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 39 e) x:0,(3)=0,(12) Bi ton 9: Cho phn s) ( ;6 ) 2 )( 1 (5 2 32 3N mm m mm m mA e+ + ++ + +=a) Chng minh rng A l phn s ti gin. b) Phn s A c biu din thp phn l hu hn hay v hn tun hon? v sao? CHUYN : CC BI TON V S THP PHN- S THC- CN BC HAI. Bi ton 10: So snh cc s sau a) 254100 5 , 0 v5 :169911||.|
\| b)9 25 +v9 25 +c) CMR: vi a, b dng thb a b a + + Bi ton 11: Tm x bit a) x l cn bc hai ca cc s: 16; 25; 0,81; a2 ;( )23 2 b)( ) x x 2 3 3 22 = c)( ) ( ) 0 1 2 12 2= + x xBi ton 12: Tm x bit a)0 2 = x x b)x x =c)( )16912= xBi ton 13: Cho 11+=xxA . CMR vi 916= xv 925= xth A c gi tr l mt s nguyn Bi ton 14: Tm cc s nguyn x cc biu thc sau c gi tr l mt s nguyn a) xA7=b) 13=xBc) C=32 x Bi ton 15: Cho 31+=xxATm s nguyn x A c gi tr l s nguyn Bi ton 16: thc hin php tnh ( ) ( )( ) ( ))`(((
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812 2: 2 :75:712 : 7 : 25 , 5 4 , 2 : 2 22222 2 Bi 17: Tnh gi tr biu thc sau theo cch hp l. ( )343472742647 71491491122 |.|
\|+ + = AGenerated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 40Bi ton 18: Tnh bng cch hp l. ( )( )37452042521 251965122 = M Bi ton 19: Tm cc s x, y, z tho mn ng thc ( ) ( ) 0 2 22 2= + + + + + z y x y xBi ton 20: thc hin php tnh ( )( )4451704:2 377683112 :449.328 225 :311822(((
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\|+ = MCHUYN : NHN, CHIA S HU T - P DNG. ********** Bi ton 1: Tnha) 3 11 12.31 0, 75.84 23 23 b) 1 1 1 1 12 3 : 4 3 73 2 6 7 2| | | |+ + + ||\ . \ .c) 5 5 4 54 : 5 :9 7 9 7| | | | + ||\ . \ . d) 1 5 5 1 313 2 10 .230 464 27 6 25 43 10 1 21 : 12 147 3 3 7| | + |\ .| | | |+ ||\ . \ .e) 25 9 125 274 25 : :16 16 64 8 | |+ |\ .g) 2 1 343 2 4| | + |\ .
Bi ton 2: Tnh a) 1 1 1....1.2 2.3 99.100A = + + +b) 1 1 11 1 ..... 12 3 1Bn| || | | |= | ||+\ .\ . \ . vin N ec) 1 1 166. 124.( 37) 63.( 124)2 3 11C| |= + + + |\ .d) 7 33 3333 333333 333333334 12 2020 303030 42424242D| |= + + + |\ .
Bi ton 3: Tnh 1 1 11 (1 2) (1 2 3) .... (1 2 3 .... 16)2 3 16A = + + + + + + + + ++ + Bi ton 4: Tm x bit a) 3(2 3) 1 04x x| | + = |\ . b) 2 5 33 7 10x + =c) 21 1 213 3 3x + =d) 3 3 22 17 8 5x + =e) 1(5 1) 2 03x x| | = |\ .g) 3 1 3:7 7 14x + =Bi ton 5: Cho 1 1 11 1 ..... 12 3 10A| || | | |= | ||\ .\ . \ .. So snh A vi 19Bi ton 6: Cho 1 1 11 1 ..... 14 9 100B| || | | |= | ||\ .\ . \ .. So snh B vi 1121Bi ton 7: Tnh 2 3 193 33 7 11 1931 9. : .193 386 17 34 1931 3862 25 2 (( | | | | + + + ||((\ . \ . Bi ton 8: Cho 1,11 0,19 13.2 1 1: 22, 06 0, 54 2 4A+ | |= + |+\ .
7 1 235 2 0, 5 : 28 4 26B| |= |\ . a) Rt gn A, B b) Tmx Z e A 3/ GTT ca mt tng khng ln hn tng cc GTT /x + y / / / / / y x + s Hiu khng nh hn hiu cc GTT / x-y/ >/x/ - /y/ 4/ GTT:Vi a > 0 th:/x /= a x= a / x /> a
a xa x / x/0 x > 0 th / x -5 / =x-5 vi x 5< 0 x< 5 th / x 5 / = - x + 5 Xt c 2 trng hp ng vi hai khang gi tr ca bin x a/ Nux>5 th A = 3 (2x 1 ) ( x 5 ) =5x + 2 Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 42 b/ Nux < 5th A = 3 ( 2x 1 ) ( -x + 5 ) =7x 8 3. Dng: Tnh gi tr ca bin trong ng thc c cha du GTT:
Bi 3 : Tm x . Bit2 / 3x 1 / + 1=5 Gii :Ta c/ 3x-1 /= 2Nn3x 1=+2 v -2 Xt c hai trng hp : a/3x 1=2 =>x=1 b/3x -1=2=>x= -31 Bi4 : Vi gi tr no ca a,b ta c ng thc: /a ( b 2 ) / = a ( 2 b )? Gii : Ta bin i/a (b 2 )/ = / a ( 2 b )/ (1) v /A/ = /-A/ / A /=A A > 0 Do (1) xy ra 4 trng hp : a/ a=0thb ty b/ b=2th a ty c/ a>0 thb0, b > 0th (1)a + b = a b b = - b (khng xy ra ) b/a>0, bs 0 th (1)a = b = a + b ng thc ny lun lun ng.Vy : a>0, bs 0 tha mn bi ton .
c/ a < 0 , b> 0 th (1)a + b = -a b a = - b . Vy a < 0 v b = -a tha mn bi ton . d/ a < 0 , bs 0 th (1)a + b = -a + b a = -a ( khng xy ra ) Kt lun : Cc gi tr a,b phi tm l a>0, bs 0 hoc a < 0 , b> 0 4. DngTm GTNN , GTLN ca biu thc cha du GT tuyt i : Bi 6: a/Tm GTNNca A =2 / 3x 1 /- 4 Vi mi x ta c / 3x 1 />0 =>2 / 3x 1 / >0 Do 2 / 3x-1 /-4 > -4 Vy GTNN ca A = -4 ti3x 1 = 0 x = 1/3 b/ Tm GTNN ca B= 1,5 + /2 - x / HD:B t GTNN bng 1,5 ti=2
c/Tm GTNN ca C = /x-3/ Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 43 HD:Ta c x 0 0 / 3 / 0 = => > => > GTNN x
Bi 7:a/ Tm GTLN caB = 10-4 / x-2/ Vi mi x tac / x 2 />0=>- /4 / x-2 / s 10 Do 10--4 / x-2 /s 10 VyGTLN caB = 10 tix = 2 b/ Tm GGLNcaB =-/ x+2 / HD: C= - /x+2/ 2 0 0 = = => s khix GTLN c/ Tm GTLN caC=1 - /2x-3/ HD:D = 1-/2x-3/ 2 / 3 0 1 = => s khix GTLNlla Bi 8:Tm GTNN caC =3 / /6 x vi x l s nguyn- Xt / x /> 3=>C > 0 -Xt/ x / / x / = 0;1hoc 2 => c = -2 ;-3 hoc -6 Vy GTNNcaC = -6x = 2 ; -2 . Bi 9TmGTLN ca C =x -/ x / - Xtx >0=>C=x-x=0 (1) - Xtx C=x (- x )=2x< 0 (2) T (1) v (2) ta thy C s 0 Vy GTLNcaC=0x >0
Bi 10 : Tm gitr biu thc :a/A =6 x 4 / / 2 32 3+ + x x vi x = -2/3 (s 20/9)b/B=2/x/ - 4/y/vi x = v y = - 3(s-8) Bi 11:Rt gn biu thc : a/ 3 (x-1 ) 2 / x + 3 / (s :x 9vi x3 >;5x+ 3 vi x< 3) b/2 / x 3/-/ 4x-1 /(s: = 2x+5 vi x < ; Bng -6x+7 vi s x < 3v bng-2x -5vi x>3. Bi 12 :Tm GTNN ca cc biu thc : a /A=2 / 3x 2 /- 1 =>GTNN ca A = -1x = 2/3 b/ B= 5 / 1 4x /- 1=>GTNNca B = -1 x = 1/4 Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 44c/ C= x2 + 3 / y 2 /- 1 => GTNNca C = -1 x = 0 ; y = 2 d/ D= x+/ x /( xt x > 0 ;c < 0)=> GTNNca D =0xs0 Bi 13: Tm GTLN ca cc biu thc : e/E=5 -/2x-1 /=>GTLNca E = 5 x = 1/2 f/ F=3 / 2 /1+ x =>GTLNca F =1/3 x =2 g/G=/ /2xx +vi x l s nguyn HD : Xt 3 TH :* x1 2 s s C* x= 1 C = 1*xx xxG2121 + =+= > Ta thy G ln nht khi x2 nh nht . M x2 ln nht x nh nht tc x = 1 khi G = 3 =>GTLN ca G = 3 x= 3 BI 14: Tm x sao cho : a/ /x - 2 /< 4 HD:Ta bit/x/< a-a < x < a Nn /x-2/ 0 x > 1 x-3 > 0 x > 3 Ta c bng xt ducc a thc x-1 ; x-3 nh sau : x1 3 x - 1- 0 +/+ x- 3- / -0+ ngthc (1) (-x+1)+(-x+3)=6(x-1)+(3-x)= 6 (x-1)+(x-3) = 6 -2x=2 0x = 42x = 10 x=-1 (khng c gi tr x = 5 (gi tr ny thucno tho mn (1) ( gi tri ny thuc khong ang xt)khong ang xt) Vy x = -1 v x = 5 th tho mn (1) b/ 7 5 2 = + + x x x-25 x+2 - 0+/+ x-5 - / - 0 + * Xt khong x 5 Ta oc2x=10x = 5 ( loi) Kt lun:-2 5 s s x c/ 4 2 3 = + x x x x-34 x+3 - 0+/+ x- 4 - / - 0 + *Xt khongx < 3 ta c-2x = 7 x= -3,5( thuc khong ang xt) *Xt khong -3 4 s s xta c 0x= 1=> khng c gi tr no ca x tho mn. * Xt khong x>4Ta c-2x = -7 x = 3,5 khng thuc khong ang xt . Kt lun :vy x = -3,55 V d 17:Tm x , Bit: 1 3 1 + < + x x x (2) Tng t: -Xt khong x< 1 Ta c(2) =>(1-x)+*3-x)(x-1)+(3-x) Ta c cc gi tr13 => ta c (x-1)+(x-3) + > yNn: /2x-338/+/3y+ 0 /457= == = += 4573340457303382yxyx Dng 5. a x f a a x f < < < ) ( ) ( ( a l hng s dong) BI 22: Tm x. a/Bit5 1 3 < xHD :a/ 5 1 3 < x -5< 3x - 1 xHD: ......
< > 2153231 1 1531 1 15xxxx
b/ Tm x . Bit 25 4 5 2 > + x.......
s>
s > 81321 5 221 5 2xxxx Bi 24. C bao nhiu cp s nguyn (x;y) tho mn iu kin sau : a/4 = +y x HD:Nu x =0 th y =4 ta c2 cp s l (0;4)v(0;-4) Nu x= 1th y =3 ta c 4 cp s l ((1;3);(-1;-3); (1;-3);(-1;3) Nu x= 2th y =2 ta c 4 cp s l :................... Nu x= 3th y =1 ta c 4 cp s l :................... Nu x= 4th y =0 ta c 2 cp s l .................... Vy ta c tt c16 cp s tho mn ng thc cho. b/4 < +y xHD.Tng t c tt c 7 + 10 +6+2= 25 cp s tho mn BT cho BI 25: Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 51a/ Tm x tho mn : ( x + 2/3 ) ( 1/4 - x )>0HD: a/ Cch 1 (x + 2/3 )( 1/4 - x) > 0) (4 / 13 / 20 4 / 10 3 / 24 / 13 / 20 4 / 10 3 / 2a ngthoixayr khongthedoxxxxxxxx>
> > + -2/3 +< = = 3 B = 1800 , => B = 600, =800;C= 400 &1B= 1200, 1A=1000; 1C= 1400 BI5 : V thm v dng nh l gc ngoi . Chng minh : A B =B A +aA O b B
$2. CC TRNG HP BNG NHAU CA HAI TAM GIC
Tam gicTam gic vung TH 1. C-C-C Cnh huyn + Cnh gc vung TH 2. C-G-C Hai cnh gc vungGenerated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 53ICABD TH 3. G-C-G Cnh GV+ G.nhn k ; C.Huyn +G.nhn $ 3. TAM GICVMT S TAM GIC C BIT: Tam gicA. CnA.UA VUNGAvung cn nh ngha A,B,C khng thng hng A ABC: AB = AC A ABC : AB=BC=AC : ABC AA= 900 090: = A A ABC AB=AC Quan h cc gc + C B + =1800 A C 1 > B C 1 > C B =B=2180 A =180B20 = = = C B A 600 C B + = 900 C B += 450 Quan h cc cnh 1 cnh< Tngv > Hiu 2cnh cn li AB=AC AB=BC=AC BC2 2 2AC AB + = BC > AB BC> AC AB=AC= cBC= c 2 BI 6 :Chotam gicABCc = 80 ,B = 60 . Hai tia phn gic ca gc B v C ct nhau ti I . V tia phn gic ngoi ti nhB ct tia CItiD . Chng minh gc BDC= gc C? HD: Tnh gc C = 40 . Tnh gc BDC = 1800 (90 +30)= 40 =>gcC =gc
BI 7 :Cho tam gicABC c gcA = 2B vB = 3C. a/Tnh gc A ;B ; C ? b/ Gi E giao im ca ng thng AB vi tia phn gic ca gc ngoi tinhC. Tnh gcAEC?
B HD : a/Qui v gc C =>gc A+B+C =10C => gc C = 180 =>B = 54 ; = 108 . b/K tiaAC x k b vi gc ACB=> gc AC x = 162 A=> AC E = 81 v 2= C B +=54+18 =72 =>gcE =180(81+72)= 27 . Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 54 C E BI 8 : Cho tam gic ABC c cc gc A;B;C t l vi 3;2;1 .Hi tam gic ABC l tam gic g ? HD : Ta c gc A:B:C=3:2:1 => gc A =90 => Tamgic ABC vung ti A . BI 9 :Cho tam gic ABC c chu vi bng 21 cm . di 3 canh l 3 s l lin tip vAB < BC < CA . Tim di 3 cnh ca tam gic A. Bit. PQR ABC A = A A HD : Gi di3 cnh AB = 2n + 1 ,BC= 2n +3 vCA = 2n +5 . Ta c AB+BC+AC= 6n = 12 => n= 2=>AB= PQ= 5 ;BC=QR=7,CA=RP=9 cm B C BI 10: Cho gc xy . Trn tia O x ly A , B v trn Oy ly C,D sao choOA=OC ; AB = CD . Chng minh rng: a/ CDB ABD b CDA ABC A = A A = A / &? D C HD :) ( & ) ( cgc ABD CDB cgc CDA ABC A = A A = A A B BI 11 :Cho tam gic ABC.Bit AB = 3 cm , BC = 5 cm v CA = 4 cm .Gi ng thng qua A v song song vi BC l a .ng qua B song song vi CA l b v ng thng qua C v song song vi AB l c . Gi M,N,P theo th t giao im cc ng thng b v c ; a v c; a v b . Tm di cc cnh tam gic MNP?
A HD : Chng minh. ); ( MCB BAP ABC gcg CNA ABC A = A = A A = A =>Cc cnh ca tam gic MNP di gp i cc cnh tng ng ca tam gic ABC => MN=2AB =6cm ;Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 55NP = 2BC = 10 cm vNP =2CA = 8cm . BC M BI 12 : Gi M trung im cnh BC ca tam gic ABC , k BH AMv CK AM . Chng minh: a/BH //CK Ab/M trung im ca HK c/HC // BK? HH D :a/BH // CKv cng vung gc viAM . B M Cb/MK MH CKM BHM = => A = A c/BK HC gocKBC gocHCB KBM HCM // => = => A = A BI 13 :Cho tam gicLMN c 3 gc u nhn . Ngi ta v pha ngoi tam gic y ba tam gic u LMA ; MNB vNLC . Chng minh rng :LB = MC = NA ? HD :CM NA cgc MLC ALNBL NA cgc LMB AMN= => A = A= => A = A) () ( => LB = MC = NA . L A M N B BI 14:Cho tamgic ABC c = 90 ; B = 60 . Phn gic gc B;gc C ct nhau tai I vAI ct BCtiM . a/ Chng minh gcBIC l gc t? b/ Tnh gcBIC ? A HD:a/ GcI 1 > gc A1Gc ngoi tam gic BIM Gc I2 > gc A2gc ngoitam gic CIM gc I > gcA = 90 = > gc BIC l gc t .
C b/...=>gc BIC = 180 45 = 135 . MB Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 56BI 15 :Cho tam gicABC cgc B gc C= 20 . Tia phn gic gc A ct BC tiD. Tnh s c gcADC ? gc ADB ? AHD :=> Ta c=> + = + =2 2 ; 1 1 A C D A B DC B D D 2 1 == 200
M2 1 D D + = 180 => 1D =1000,2D = 800
B D C BI 16 :Cho tam gic ABCc ba gc nhn . V on thng AD vung gc v bng AB ( D khc pha C i vi AB )V onthng AE vung gc v bng AC ( E khc pha B i vi AC ) . Chng minh rng :a/DC=BE? b/DC BE? E HD : a/) (gcg ABE ADC A = A =>DE = BE D c./ Gi H l giao im AB vi CD v K l giao Aim DC vi BE. 0& 90 ADH KBHgocDAH BKH A A = = B C BI 17 :Cho tam gic ABC c gc B = 2C . Tia phn gic gc B ct AC D . Trn tia i BD ly im E sao choBE=AC .Trn tia i CBly im K sao cho CK = AB . Chng minh rng:AE=AK ? HD :Chng minh gc ABE = gc ACK A =>) (cgc KCA ABE A = A =>AE=AK . D B CK E BI 18 :Cho tam gicABC vi K l trung im AB v E trung imAC . Trn tia i tia KC ly im M sao choKM = KC . Trn tia i EB ly imNsao choEN = EB. Chng minhA l trung im caMN?
HD: BC AM gocKBC gocKAM cgc BKC AKM // ) ( => = => A = ABC AN BC AN CEB AEN // & = => A = AM ANM AM//BC;AN//BC=>M;A;N thng hng (1) AM=BC;AN=BC=>AM=AN (2) Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 57 T (1) &(2)=> A l trung im caMN . K E B C BI 19 :Cho tam gic ABC. V v pha ngoi tam gic ABC cc tam gic vung ti A lAADB ; ACE Ac AB = AD ; AC = AE . K AH vung gc BC; DM vung gc AHv EN vung gcAH . Chng minh rng a/ DM = AH N E b/MN i qua trung imDE. DM A HD : a/...=>AH DM BAH ADM = => A = A
b/...=> tng t cu a => EN=AH =>DM=EN Chng minh DM//EN v gi O giao im MN v B HC DE =>OE OD gcg ENO DMO = => A = A ) (. BI 20 : Cho tam gic ABC. gi D trung im AB v E trung im AC. V im F sao cho El trung im ca DF . Chng minh rng : Aa/DB=CF b/ D FC BDCA = A DE F c/DE // BC &DE =BC21 HD:a/ ...=>CF BD CF AD (cgc) F = => = => A = A CE AED BC b/ ...=>) (cgc FCD DBC A = A c/ ...=>BC DE DF DE DF BC FCD BDC2121= => = => = => A = A . BI 21 :Cho tam gic ABC . Trn cnh AB ly im D ; E sao cho AD = BE. Qua D v E v cc ng song song BC chng ct AC theo th t M vN . Chng minhDM + EN = BC? AHD: Qua N vng thng //AB ct BC ti K.Tac EN//BK EB//NK nn chng minh c NK=EB;EN=BK AD= NK ( v cng bng EB ). Chng minhKC DM cgc NKC ADM = => A = A ) (...=>.... EN BF C Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 58 BI 22 :Cho tam gic ABC c = 600. Cc tia phn gic gc B,gc C ct nhau ti I v ct AC ; AB theo th t D ; E . Chng minh :ID = IE? A HD : ...=>01 16021202 = =+= +C BC B
04 1060 120: = = => = => A I I C I B BIC EI DIK phn gic 02 160 = = => I I C I B
IE ID IK ID gcg CIK CDIIK IE gcg BIK BIE= => = => A = A= => A = A) () ( BK C BI 23 : Cho hai on thng AB v CD ct nhau ti E . Cc tia phn gicE B D E C A&
ct nhau K . Chng minh :2 C D B C A BC K B+= ? K D HD: GiK l giao im CK&BE. H l giao im BK&DE AH Xt 1 1 & C A B K AGC KGB + = + => A A(1) G Xt2 2 & B D C K DHB KHC + = + => A A(2) ET (1) &(2) =>2 D A K + = => 2 D AK+= C B Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 59BI 24 :Chotam gic ABCvi M trung im BC . Trn na nt phng khng cha C b AB v A x vung gcAB v lyD sao cho AD = AB . Trn na mt phng khng cha B b AC v Ay vung gc AC v ly AE = AC . Chng minh : a/AM =21ED b/AM DE H EHD :a/ chng t DE =2AM to ra on thng gp i AM bng cch trn tia i MA ly MK = MA v i chng minhDE=AK-Xt) ( ); ( : & AC BK AE gt AB AD DAE ABK = = = A A V ) 180 ( 180 02 10= + = + A A vi A E A D(1) )( 180 01A vibu A C B A K B A C B B B = + + = + => + = +(2) Vy :2 DEAM DE AK DAE ABK E A D K B A = => = => A = A => = B M Cb/ Gi H l giao imAM&DE ; Ta c
0 0 09090 90 = => = + => = + H D A H A D D H A D K A B Bi25Min trong gc nhn xy v Oz sao cho xz=21 yz .Qua im A thucOy v AH vung gc O x ct Oz B.Trn tia Bz ly D sao choBD = OA . Chng minh tam gc AOD cn?
HD : chng minh AO = AD ta v DE = OB ATa thy: E B 90 & 900 0A B E A H B O E B A B E A = => = = = o o=>) (cgc ADE AOB A = A=> AO=AD =>AAOD cn ED B O H h A Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 60BI 26 :Cho gc xz= 1200.Oy l tia phn gicxz ; Ot l tia phn gic ca gc xy . M l im mim trong gc yOz. V MA vung gc O x,V MB vung gc Oy,V MC vung gc Ot . Chng minh0C = MA MB?
H:GiE , Il giao im ca MC vi Oy;O x. y=>AEOI u => OC=EK . zM V EHOI EK MA ;d dng chng minh c BMH=MB;EK=OCMA-MB = MA MH = HA = EK = OC H E t C OI x A K BI 27 :Cho tam giccnABC c = 100 . Tia phn gic gc B ct AC D. Chng minhBC= BD+AD . AHD : Ta c 0 02 140 20 + = + = C B Dtrn cnh BC ly cc im K , E sao cho ) 1 ( ) ( 80& 600 0DK DA gcg BDK BDA E D B K D B = => A = A => = =Chng minh tam gac DKE cn ti D =>DK = DE (2) V chng minh tamgic DEC cn ti E=>DE=EC (3) T (1),(2).(3) =>AD=EC=> BC = BE+EC=BD+AD BKEC BI 28 :Cho tam gic ABC nhn. V ng cao BD ,CE . Trn tia i BD ly im I. Trn tia i CE ly im K sao choBI = AC , CK = AB .Chng minhAAIK vung cn? HD : Ch/minh AK AI cgc KCA ABI = A = A ). ( AGc AIK=90 (v gc E = gc K (cmt) Suy ra : tam gic AIK vung cn
B C
Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 61BI 29:Cho gc xy = 90 Ly im A trn O x v im B trn Oy . Ri ly im E trn tia i O x v im F trn tia Oy sao choOE =OB v O F = OA . a/ Chng minhAB = E F v AB E F b/ Gi M,N l trung im AB, E F Chng minh tamgic OMN vung cn ? HD : a/ F E AB & F E AB cgv) E(2 = => A = A O OAB b/ => = = => A = A 90do OM & ) ( gocM ON OM cgc ONE OMB y OMN vung cn B F N M E O Ax
BI30: Cho tam gicu ABC, Trn 2 cnh AB,AC ln lt ly 2 im M v N sao choAM = CN . Gi O l giao im CM v BN . Chn ninh rng : Aa/CM=B N b/ S o gc BOC khng i khi M v N di ng trnAB,AC tho mn iu kin AM = CN. M HD: a/ 1 1 & ) ( B C BN CM cgc CBN ACM = = => A = A b/0 0 02 10120 60 180 ) ( 180 C= = + = A C B O BOCcoB N O
B C
Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 62 Chuyn :
TAM GIC VUNG TAM GIC CN ..... A. TAM GIC VUNG : 1/ nh ngha tam gic vung ? tam gic vung cn ? 2/ Tnh cht : - Tam gic ABC : =90 090 = + C B - nh l PyTago: 2 2 2 090: AC AB BC A ABC + = = A - B ba s Py ta go: (3;4;5); (5;12;15);(6;8;10);(8;15;17);(20;21;29).... - Cc h thc trong tam gic vung:
. . ; .. . ; 90:2 20BC CH AC BC BH ABAC AB BC AH BC AH A ABC= == => = A; - BC AM MC AB A ABC21; 90: = = = A SAMB = AMCS- Tam gic vung c mt gc nhnbng60 (30 ) l na tam gic u( cnh bng cnh huyn ). - Cc trng hp hai tam gic vung bng nhau: 2 cgv-Chuyn
Ton nng cao: BI 1:Cho tam gic ABC vung ti A v gc C = 45 . V phn gic AD.Trn tia i AD ly AE = BC.Trn tia i CA ly CF = AB . Chng minh :a/BE=CFb/ BE=BF . Hng dn: a/ Chng minh : BE =B0135= F CA Ch/minh : CF BE cgc FCB BAE = A = A ) (D b/ 0 090 90: = + => = A F F B A A ABFAC F M:BF BE F B hayEB F B A cmt B F => == + => =009090 ) (
BI 2:Cho tam gic ABC c BC = 2 AB . M trung im BC; D trung im BM . Chng minh :AC = 2 AD
A Hng dn: Trn tia i AD ly DE = DA => D M E D B A ME AB cgc EMD ADB ; ) ( = = => A = A=> AB=ME= ) 1 (21MC ME BC = =>(1) Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 63
BD M CMt khc:) ( ; 2 1gocngoai M A B B A M C M M A M E + = + =M:M B A M cmt B M ); (2 1= =Vy : C M A E M A = (2) vAM chung(3) E T (1),(2) v(3) suy ra 2AD AC AC AE AMC MCME = => = => A = A BI 3: Cho tam gicABC vung ti A v gc B= 60 . V tia C xBC v ly CE = CA ( CE v CA cng pha vi BC). Ko di CB v ly F sao cho BF = BA . Chng minh : a/ACE Au b/ E,A,F thng hng ? Hng dn: a/Ta c CA = CE (gt)=>CEAcan A Chng minh tipgc ACE = 60 Suy ra :CAE A u E b/Ta c : BA = BF (gt) => BFAcan ASuy ra : gc BA F = 30 ; A Vy:0 0 0 0180 60 90 30 = + + = + + E A C C A B A B F Ta suy raba im F;A;E thng hng .EAF F B C BI 4: Cho tam gicABC vung ti A . Tia phn gic gc B v C ct nhau ti O .Qua O k ng song song BC,ct AB ti D v ct AC ti E . Chng minh : a/Gc BOC khng i . b/DE=DB+EC A HD :a/0 0 02 20135 45 180 ) ( 180= = + = C B C O B b/DO DB DBOcan = => AOEO EC E = => A can OC D E Vy DB+EC=DO+OE=DE BC BI 5 :Cho tam gic ABC: Gc B = 2 gc C. K AH vung gc BC (H thuc BC) . Trn tia i BA ly BE = BH . ng thng EH ct AD ti F.Chn minh :FH = FA = FC . AHng dn: Ta c BH= BE =>ABEH cn => 1 H E =MC H H B H B H H 2 2& 2 2 1 2 1+ => = = = => = FVy tam gic FHC cn =>HF = HC (1) Mt khc : = 9020 090 F&H H A C = BVy tam gic FAH cn => FA = FH (2) Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 64HCT (1) v (2) => HF = FA = FCE Bi 6:Cho tam gic ABC :gc A = 90 . mim ngoi tam gic v cc tam gic vung cn ABD, AC F ( AB = BDvAC = CF). a/ Chng minh :D,A,F thng hng ? b/ TA v F k cc ng D D' ', FFvung gc xung BC . Chng minh :BC FF D D = +' ' HD:a/ = 45+90+45 = 90 =>A,D,F thng hng A b/ K AHBC => BC HC BH FF DHC FF AHC CFFBH BAH DBD= + = + =>= => A = A= => A = A' '' '' 'DDD B C Bi 7:Cho 0120: = A C A B ABCK AD phn gic gc A .T A hDE AB ; DF AC . a/ Tam gic DE F tam gic g? b/ QuaCv ng thng // AD ct AB tiM, tam gic ACM l tam gic g ? A HD:a/ Chng minh DE = DF v gc EDF = 60 =>Au Fb/Tam gic ACM u . E BDC BI 8: Tam gic ABC cAB > AC .T trung im M ca BC k ng thng vung gcvi tia phn gic gc A v ct tia phn gic ti H ct AB,AC ln lt ti E v F . Chng minh rng: a/ BE=CFb/ AE =2;2AC ABBEAC AB =+ c/ gc BME = 2 B B C A HD:a/Chng minh gc F = gc E K CD // AB =>BE=CD(1) AMA CDF cn => CF=CD (2)=> BE=CF b/Ta cAE = AB - BE M AE=A F= AC+CF=>2AE=AB+AC AE=2 AC AB + Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 65E Tng t : 2BE=AB-AC =>BE = 2 AC AB M C c/ Ta c : 2B- B CAE MB B- B CA E M2BB - EE MB & F- B CA F= => == = E C F B.TAM GIC CN B SUNG KIN THC: 1.Trong mt tam gic vung c mt gc nhn bng 30 th cnh i din vi gc y bng na cnh huyn.2. Mt tam gic vung c gc nhn bng 30 (hay bng 60 ) th tam gic vung bng na tam gic u.Cnh i din gc vung l cnhtam gic u v cnh i din gc nhn 60 l chiu cao tam gic u. 3.Trong mt tam gic vung c mt cnh gc vung bng na cnh huyn th gc i din cnh vi cnh gc vung y bng 30 . 4.Trong tam gic cn: -Hai trung tuyn ng vi 2cnh bn bng nhau. -Hai phn gic ng vi 2 cnh bn bng nhau. -Hai ng cao ng vi 2 cnh bn bng nhau. TON CHO HS GII: BI 9:Cho tam gic nhn ABC c gc = 60 . ng cao BD. Gi M,N ln lt l trung im AB ; AC. a/Xc nh dng ca tam gicBMD? Tam gicAMD? b/ Trn tiaAB ly im E sao cho AE=AN . Chng minh CE vung gcAB ? HD: A D M EN B C Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 66Xt tam gic vung ABDc DM l trung tuyn ng vi cnh huynAB nn: MD=MA=MB=AB:2 => Tam gic ABD v tam gic AMD cn. M =60 => tam gic AMD u. b/Xt tam gic AEN c AE=AN=>tam gic AEN cn+=60 =>tam gic AEN u=>EN=NA=CN=AC:2. Vy tam gic EAC c trung tuyn EN=AC:2=>tam gic EAC vung ti E => CE vung gc AB BI 10: Cho tam gicABC vung ti A. Trn cnh BC ly 2 im M,N sao choBM=BA;CN=CA. Tnh gc MN ? HD:B N =1=M 1 A C Tam gic BAM cn ti B=> 21801BM=Tam gic CAN cn ti C=>21801CN=Vy :0145 135 180 ) ( 180= = + = N M N A M BI 11: Cho tam gicABC ng cao AHv trung tuyn AM chia gc A thnh 3 gc bngnhau. a/ Chng minh tam gic ABC vung? b/Tam gicBM l tam gic u ? HD: A I B H M C a/ V MI vung gc AC . Chng minh Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 67
0 060& 302121) . . ( = = => = = = => + A = A C A H C MC BM MH BH n g h C MAH MAIVy BC= (60.3):2=90 => Tam gic ABC vung ti A. b/Ta c gc C=30 ;gc B=60 ;AM=BM=1/2BC=>tam gic ABM cn c mt gc bng 60 => tam gic ABM u. BI 12:Cho tam gic ABC c gc B= 75 ,gc C bng 60 . Ko diBC mt on CD sao choCD=1/2BC .Tnh gcADB? HD: A H 1 2 2 1 1
BCD - K BH vung gc AC. Xt tam gica vung BHC vung ti H v gc C=60 => gc ) 1 ( 302121300101HD HB HDBcan B C A D can CDH CD CH BC CH B = => A => = = = A => = => = => =-Xt tam gic HAB vung tiH c gc B2=75-30=45 =>tam gic HAB vung cn=>HA=HB(2).T (1) v (2)=>HD=HA=>Tam gic HAD cn. Ta suy ra 0 01 245 15 301521= + = => = = B D A H D NH L:PY-TA-GO KIN THC B SUNG: 1.Trong tam gic vung cn c cnh bn bng a th cnh huyn bnga 22.Khong cchgii 2 im trong mt phng to :
21 221 221 221 222 2 1 1) ( ) ( ) ( ) ( ) ; ( ); ; ( y y x x AB y y x x AB y x B y x A + = => + = =>
BI 13:Cho tam gic ABC cAB=24; BC=40 v AC=32. Trn cnh AC ly M sao choAM =7. Chng minh rng : Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 68 a/Tam gic ABC vung? b/gc AMB = 2gc C.
HD:A 7 M 24 32 B 40C a/ Tam gic ABC c:BC 1600 40 . 402= = AB 1600 32 . 32 24 . 242 2= + = +AC VyiA ABCvuongta BC AC AB A => = = + 16002 2 2 b/ Chng minh ram gic MBC cn: BM=25 7 3225 7 242 2= = = AM AC Suy ra : gc MBC=gc C. M gc AMB=gc MBC+gc C ( gc ngoi) Vygc AMB = 2. gc C BI 14:Cho tam gic ABC cAB=25;AC = 26 . ng cao AH =24 . Tnh BC? A A 2524 24 2625 26 BH C H BC (H1) (H.2) - Tnh cHB= 7 ;HC=10 -Nu gc B nh=>H nm gia BC=>BC=BH+HC=10+7=17(h1) -Nu gc B t => H nm ngoi BC=>BC=HC--HB=10-7=3(h2) BI 15: di hai cnh gc vung ca tam gic vung t l 8 v 15. Cnh huyn 51 cm. Tnh di 2 cnh gc vung ? Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 69HD:Gi s tam gic ABC vung ti A. =>AB=8k v AC=15k Ta c 3 2601 289 51 ) 15 ( ) 8 (2 2 2 2 2 2= => = => = + = + k k k AC ABVy AB= 8.3= 24 mvAC=15.3=45 m BI 16: Cho tam gic ABC vung ti A. ng cao AH,trn ly im D. Trn tia i HA ly E sao cho HE=AD. ng vung gc AH tiD ct AC ti F . Chng minhEB vung gcE F? HD: A DF BH C E V AD=HE=>AH=DE p dng nh l Py ta go vao tam gic vung ABF;ABH;ADF;BHE;DE F ta c: 2 2 2 2 2 2 2 2 2 2 22 2 2 2 2 2 2) ( ) () ( ) ( FF E BE DF DE HE BH DF HE DE HB BFDF AD AH BH A AB F B+ = + + + = + + + =+ + + = + = Vy theo nh l o Py ta go=> tam gic BE F vung ti E=> EB vung gc E F BI 17: Mt cy trecao 9 m. B gy ngang thn. Ngn cy chm t v ch gc3m .Hi im gy cch gc bao nhiu mt ? HD :B = C x? = AD Gi AB chiu cao cy tre . im gy C . Ngn cham t cch gc 3 m l im Cth CB=CD . Tam gic vung ACD c :met x x xCD AD AC4 ) 9 ( 32 2 22 2 2= => = += + Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 70 BI 18: Trong mt phng tocho cc im A(5;4); B(2;3) v C(6;1). Tnh cc gc tam gic ABC? HD x 4A(5;4) 3B(2;3) 1 C(6;1) x O2 5 6 Ta c : =2AB ) 1 ( 10 ) 3 4 ( ) 2 5 (2 2= + ) 1 4 ( ) 6 5 (2 2 + = AC) 2 ( 102= 20 ) 3 1 ( ) 2 6 (2 2 2== + = BC T (1) v (2) => tam gic ABC cn v AB ABCvuong BC AC A => = = + 202 2 2 Vy gc A =90 . gc B = gc C= 45 CC TRNG HP BNG NHAU CTAM GIC VUNG BI 19: Cho tam gic ABC. Trung tuyn AM cng l phn gic . a/ Chng minh tam gic ABC cn. b/Cho AB=37; AM =35 . Tnh BC ? HD: (H.1)AA FD H K (H.1) B MC (H.2)B EC a/ V thmMH vung gc AB & MK vung gc AC. Chng minh cantaiA ABC C B cgv ch KMC HMBMKA MH gn ch KAM HAMA => = => + A = A= => + A = A ) () ( b/Tam gic ABC cn =>AH vung gcBC =>BM= 24 122 2= => = BC AM ABGenerated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 71 BI 20: Cho tam gic c ba ng cao bng nhau. a/ Chng minh tam gic u ? b/Cho bit mi ng cao c di 23 a. Tnh di mi cnh tam gic ? HD.(H.2) Tam gic ABC c ba ng cao bng nhau l: AD=BE=C F. a/ Ta chng minh. ;........ ) ( deu ABC A C C B cgv ch ECB FBC A => = => = => + A = A b/ Gi di mi cnh l x.Xt tam giac ADC vung ti D ca x CD AD AC = => + =2 2 2 : BI 21: Cho tam gic ABC cn ti v =80 . Gi O l im nm trong tam goc sao cho gcOBC=30 ;gc OCB=10 . Chng minh tam gic COA cn.? M M AA O O BC BC (H.1)( H.2) HD ( Xem H.1) Tam gic ABC cn gc =80 => gocB=Gc C= 50 Vthm tam gic u BCM9 M,A thuc na mt phng b BC) gc MCA=60-50=10 0 030 2 : 60 ) ( = = = => A = A C M A B M A CCC AMC AMB. ) ( can COA CA CO gcg AMC OBC A => = => A = A BI 22:Cho tam gic ABC cn ti A v gc = 100 .Goi O l im nm trn tia phn gic gc C sao cho gcCBO=30 . Tnh gcCAO ? HD: (Xem hnh 2) V tam gic u BCM9M,A cng na mt phng b BC). Chng minh tng t bi 19=>020 2 : 40= = => A O C A COAcantaiC Suy ra:080 2 : 20) - (180 O AC = = Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 72Bi 23:Chotam gic cn ABC (AB=AC. K ng vung gc AB ti B v vung gc AC ti C. hai ng ny ct nhau ti D. a/Chng minh AD l phn gic gc A ? b/So snhAD & CD? HD:(H1)A A( Hnh 2) 12DE B C B MN C D (xem h.1) a/ Chng minh tam gic ABD=tam gic ACD(Ch+cgv)=>2 1 A A = Suy raAD phn gic gc b/ Suy ra AD=CD ( 2 cnh tng ng) BI24:ChotamgiccnABC9AB=AC)DlmtimthucABvElmt imthucACsaochoAD=AE.TDvEhngvunggcviBC.Chng minh BM=CN? HD: ( xem hnh 2)Chng minh BD=EC&gc B = gc C Suy ra tam gic BDM=tam gic ECN(Ch+gn)=>BM=CN BI 25: Cho gc xytrn O x ly im A. Trn O y ly im B. Gi M trung im AB.TA,BhngthngAE;BFcngvunggcvitiaOM.Chngminh AE=BF? HD: Chng minh tam gic MAE=tam gic MBF x (Ch+gn)=>AE=BF
A EM F O y BI 26:Cho tam gic ABC cc tia phn gic gc B,gc C ct nhau ti O. K OE,O F,OG th t vung gc vi AC,AB,BC. Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 73 a/Chng minhOE = O F=O b/Tia AO ct BC ti D . Chngminhgc BOD=gc gc COG HD: A 1 2 E F O 2 2 1 1 B GDC a/ Chng minh:OG F O OE (2) & (1) uTOG(2) OE gn) COE(ch COG(1) OG F O gn) BOG(ch = = =>= => + A = A= => + A = ABO b/C C C B B B A A AO AOE21 &21 ;21AF2 1 2 1 2 1= = = = = = => A = A Suy ra02 190 2 : 180 = = + + C B A (1) Mt khc tam gic vung BOG(gc G=90 )=> ) 2 ( 90 01= + G O B BT (1) v(2) =>) 3 ( 2 1G O B C A = + T (3) v (4)=> G O C D O B D O G D O C D O G G O B D O C G O B , = + = = = BIU THC I S GI TR CA BIU THC I S Bi 1:Tnh gi tr ca biu thc : Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 74A = x2 + 4xy 3y3 vi; 5 = x 1 = yBi 2: Cho x y = 9, tnh gi tr ca biu thc : B = x yyy xx+++39 439 4 ( x= - 3y ; y= - 3x)Bi 3: Tnh gi tr ca cc biu thc sau : a) A = 16 168 8 4 4 2 2 22) 2 )( 2 )( 2 )( 2 (y xy x y x y x y x x++ + + vi x = 4 v y = 8 b) B = 2m2 3m + 5 vim= 1 c) C = 2a2 3ab + b2 vi1 = a vb= 2Bi 4:Xc nh cc gi tr ca bin biu thc sau c ngha : a) 412+xxb) 112+xx c) y xyc by ax3 + + Bi 5: Tnh gi tr ca biu thc : N=1 23 62 +xx x vix = 21 Bi 6 : Tm cc gi tr ca bin : a)A= (x + 1)(y2 6) c gi tr bng 0 b) B = x2 12x + 7 c gi tr bng 7Bi 7 : Tnh gi tr ca biu thc sau : A = 2 22 23 103 5y xy x+ vi 5 3y x=Bi 8: Cho x, y, z= 0 v x y z = 0 .Tnh gi tr ca biu thc B =|.|
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\| zyyxxz1 1 1Bi 9:a) Tm GTNN ca biu thc C = ( x+ 2)2 + ( y -)51 2 10 b) Tm GTLN ca biu thc sau : D = ( ) 5 3 242+ x Bi 10: Cho biu thc E= 25xx.Tm cc gi tr nguyn ca x : a) E c gi tr nguynb) E c gi tr nh nhtBi 11: Tm cc GTNN ca cc biu thc sau : a) (x 3)2+ 2 b) (2x + 1)4 1c) (x2 16)2 +3 y - 2Bi 12: Tm GTNN ca biu thc :A = 10 2 + x xBi 13: Tm cc gi tr nguyn ca x ,biu thc sau nhn gi tr nguyn : A = 1 515 10++xx Bi 14: Cho f(x) = ax + b trong a, b eZChng minh rng khng th ng thi c f(17) = 71 v f(12) = 35Bi 15Cho f(x) = ax2 + bx + c .Chng minh rng khng c nhng s nguyn a, b, c no lm cho f(x) = 1 khi x = 1998 v f(x) = 2 khi x = 2000 Bi 16: Chng minh rng biu thc P = x8 x5 + x2 x + 1 lun nhn gi tr dng vi mi gi tr ca x. Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 75Bi 17: Tm gi tr ln nht v gi tr nh nht ca biu thc : B =3 1 + x xvi x 117sBi 18: Chng minh cc ng thc sau : a) x2 y2 = (x+ y) (x- y) b) x3 + y3 = (x+ y) ( x2 xy + y2)c) a(a b) b(b- a) = a2 b2
d) a( b- c) b(a + c) + c( a b) = - 2bce) a( 1- b) + a( a2 1) = a (a2- b)f) a(b x) + x(a + b) = b( a + x)Bi 20: Rt gcn biu thc i s sau : a) A = ( 15x + 2y) -( ) ( ) | | y x x + + 5 3 2 b) B = - (12x + 3y) + (5x 2y) - ( ) | | 5 2 13 + y xBi 21: t tha s chung vit cc tng sau y thnh tch : a) ab + bd ac cd b) ax + by ay bx c) x2 xy xy + y2
d) x2+ 5x + 6 Bi 22: Chng t rng : a) Biu thc x2 + x + 3 lun lun c gi tr dng vi mi gi tr ca x . b) Biu thc 2x2 + 3x 8 khng nhn gi tr dng vi mi gi tr ca x. Bi 23*:Tm x, y l cc s hu t bit rng: a)11= +xx b)52= +xxc)x y x = + 3 3 3 d)(x-2) 5 252+ n+ y- 2= 0 (neN) Bi 24: Tm x, y l cc s nguyn bit: a) 12+=xxy b*) 13 2+=xxy N THC,N THC NG DNG Bi tp c bn Bi 1: Cng v tr cc n thc : a)3a2 b+ (- a2b) + 2a2b (- 6a2b) b)(-7y2) + (-y2) (- 8y2)c)(-4,2p2) + ( - 0,3p2) + 0,5p2 + 3p2 d) 5an + (- 2a)n + 6an
Bi 2: Thc hin cc php tnh sau : a) 236 3x x x+ +b) 3ab.52ac 2a.abc - 31a2bc c) 232|.|
\|ac .c2 - 52a2.(c.c)2 + 32ac2.ac - 41a2c2
Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 76Bi 3: Cho cc n thc A = x2y v B = xy2 .Chng t rng nu x,y eZ v x + y chia ht cho 13 th A + B chia ht cho 13Bi 4: Cho biu thc : P = 2a2n+1 3a2n + 5a2n+1 7a2n + 3a2n+1+ ( n eN)Vi gi tr no ca a th P > 0Bi 5: Cho biu thc: Q = 5xk+2 + 3xk + 2xk+2 + 4xk + xk+2 + xk ( k eN)Vi gi tr no ca x v k th Q < 0Bi 6: Tm x bit : xn 2xn+1 + 5xn 4xn+1 = 0 ( n eN; n= 0)Bi 7: Bit A = x2yz , B = xy2z; C = xyz2 v x+ x + z = 1Chng t rng A + B + C = xyzBi 8: Tm cc n thc ng dng vinhau trong cc n thc sau: y x y ax x y x y x3 3 5 2 3 3 591; ; 5 ; 4 ; 3 ;71 Bi9:Tnh tng : a) 5 2 5 2 5 2344321z y z y z y + b)3 3 337xy bxy axy + Bi10: Rt gn cc biu thc sau : a) 10n+1- 66.10n b) 2n+ 3 + 2n +2 2n + 1 + 2nc)90.10k 10k+2 + 10k+1 d) 2,5.5n 3 .10 + 5n 6.5n- 1 Nng cao Bi 1: Chobiu thc M = 3a2x2 + 4b2x2- 2a2x2 3b2x2 + 19 ( a= 0; b= 0)Tm GTNN ca MBi 2 : Cho A = 8x5y3 ; B = - 2x6y3 ; C = - 6x7y3 .Chng t rng : Ax2 + Bx + C = 0Bi 3: Chngminh rng vi n eN*
a) 8.2n + 2n+1 c tn cng bng ch s khngb) 3n+3 2.3n + 2n+5 7.2n chia ht cho 25Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 77c)4n+3 + 4n+2 4n+1 4n chia ht cho 300Bi 4: Cho A = ( - 3x5y3)4 v B = ( 2x2z4)5 .Tmx,y,z bit A + B = 0Bi 5: Rt gn: a) M + N P vi M = 2a2 3a + 1 , N = 5a2 + a , P = a2 4b) 2y x -( ) | | { } x y x y y x + 5 3 2vi x =a2 + 2ab + b2 , y = a2 2ab + b2
c) 5x 3 -1 2 xBi 6: Tm x,bit : a) (0,4x 2) (1,5x + 1) ( - 4x 0,8) = 3,6b) (343+ x ) |.|
\| 432x-|.|
\|+161x=|.|
\|+ 431x -|.|
\| 331xBi 7: Tm s t nhinabc ( a > b > c) sao cho :cab bca abc + += 666Bi 8: C s t nhinabc no m tngcab bca abc + +l mt s chnh phng khng ? Bi9 : Tnh tng : a)(- 5x2y + 3xy2 + 7)+ ( - 6x2y + 4xy2 5)b)(2,4x3 -10x2y) + (7x2y 2,4x3+3xy2) c)(15x2y 7xy2-6y2) + (2x2- 12x2y + 7xy2) d)(4x2+x2y -5y3)+( y x xy x2 2 3635 )+(33103yx+ )+ (3 2 2 310 4 15 6 x y x xy y ) Bi 10: Rt gn biu thc saua/ (3x +y -z) (4x -2y + 6z) ( ) ( )( ) ( )3 2 3 23 3 3 2 38 3 , 2 9 , 6 8 1 , 3 7 , 5 /7 5 2 5 6 /y y x xy y xy y x cy x x y x x b+ + + + +
d)K= 2x.(-3x + 5) + 3x(2x 12) + 26xe) |.|
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\| +=545 2557926332 x x xxxMBi 11: Tm x bit: a) x +2x+3x+4x+..+ 100x = -213 b) 61413121 = x xc)3(x-2)+ 2(x-1)=10d)4231 =+ x x Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 78e)12111110109988776 ++=++ x x x x x x f) 1427133812231132 +++=+++ x x x x g)13 2 = x h) 312 8 4 2 3 = + x xi) 323 5 2 31 + = + x xk) 2 + x + 2 x=3 m)(2x-1)2 5 =20n) ( x+2)2 = 3121 p) ( x-1)3 = (x-1) q*) (x-1)x+2 = (x-1)2 r*)(x+3)y+1 = (2x-1)y+1 vi y l mt s t nhin
Ch : S CHNH PHNG MT S DNG BI TP V S CHNH PHNG. I/ MC TIU: 1/ Kin thc:n tp cho hc sinh v s chnh phng v mt s tnh cht c lin quan cng nh mt s phng php gii ton da vo s chnh phng. 2/ K nng: Hc sinh c k nng p dng tnh cht nhn bit s chnh phng v giimt s dng ton c lin quan. 3/ Thi : Gio dc hc sinh tnh chnh xc v vn dng vo thc t. II/ L THUYT:1.nh ngha: S chnh phng l mt s bng bnh phng ca mt s t nhin V d: 225 15 ; 9 32 2= = Cc s 9; 225 l bnh phng ca cc s t nhin : 3; 15 c gi l s chnh phng 2. Mt s tnh cht: a) S chnh phng ch c th tn cng l : 0; 1; 4; 5; 6; 9 khng th tn cng bi2; 3; 7; 8. b) Mt s chnh phng c ch s tn cng l 5 th ch s hng chc l 2. Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 79Tht vy ,gi s25 a M = =. 25 100 100 ) 5 10 (2 2+ + = + a a a V ch s hng chc ca 2100 av 100a l s0 nn ch s hng chc ca sM l 2 c)Mt s chnhphng c ch s hng n v l 6 th ch s hng chc ca n l s l. Tht vy, gi s s chnh phngN=a2c ch s tn cng l 6 th ch shng n v ca s a ch c th l 4 hoc 6.Gi s hai ch s tn cng ca s a lb4(nu lb6 th chng minh tng t ), Khi b42 = (10b+4)2 = 100b2 + 80b + 16. V ch s hng chc ca s 100b2 v 80b l s chn nn ch s hng chc ca N l s l. d) Khi phn tch ra tha s nguyn t ,s chnh phng ch cha cc tha s nguyn t vi s m chn ,khng cha tha s nguyn t vi s m l . Tht vy ,gi s A =m2=ax .by.cz trong a,b,c ,l cc s nguynt khc nhau,cn x,y,zl cc s nguyn t dng th th , A = m2= (ax by cz)2 = a2x.b2y.c2z
T tnh cht ny suy ra -S chnh phng chia ht cho 2 th chia ht cho 4. -S chnh phng chia ht cho 3 th chia ht cho 9. -S chnh phng chia ht cho 5 th chia ht cho 25. -S chnh phng chia ht cho 8 th chia ht cho 16. 3/ Nhn bit mt s chnh phng: 4/ Hng ng thc vn dng: (ab)2 = a22ab + b2 v a2 b2 = (a + b)(a b) 5. Cc v d: V d 1.Chng minh rng : a) Mt s chnh phng khng th vit c di dng 4n+2 hc 4n +3 (neN); b)Mt s chnh phng khng th vit di dng 3n+2(neN). Gii a)Mt s t nhin chn c dng 2k (keN), khi (2k)2 = 4k2l s chia ht cho 4 cn s t nhin l c dng 2k+1 (keN) , Khi (2k+1)2 = 4k2+ 4k +1 l s chia cho 4 d 1.Nh vy mt s chnh phng hoc chia ht cho 4 hoc chia cho 4 d 1 ,do khng th vit c di dng 4n+2 hoc 4n+3(neN) b)Mt s t nhin ch c th vit di dng 3k hoc 3k 1 (ke N) khi bnh phng ca n c dng(3k)2 =9k2 l s chia ht cho 3 ,hoc c dng (3k 1)2= 9k2 6k +1 l s khi chia cho 3 th d 1.Nh vy mt s chnh phng khng th vit di dng 3n+2(neN). V d 2: Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 80Cho 5 s chnh phng bt k c ch s hng chc khc nhaucn chs hng n v u l 6. Chng minh rng tng cc ch s hng chc ca 5 s chnh phng l mt s chnh phng. Gii Cch 1 . Ta bit rng 1 s chnh phng c ch s hng n v l 6 th ch s hng chc ca n l s l .V vy ch s hng chc ca 5 s chnh phng cho l: 1, 3, 5, 7 ,9 khi tng ca chng bng :1+3+5+7+9=25 =52 l s chnh phng. Cch 2. Nu mt s chnh phng c M=a2 c ch s hng n v l 6 th ch s tn cng ca s a l s chn, do a2 nn a2 4. Theo du hiu chia ht cho 4 th 2 ch s tn cng ca s Mch c th l 16,36,56,76,96.T ,ta c : 1+3+5+7+9=25=52l s chnh phng V d3: Tm s t nhin n c 2 ch s, bit rng 2 s 2n+1 v 3n+1 ng thi l 2 s chnh phng Tr li n l s t nhin c 2 ch s nn 10 n < 100, do 21 2n+1 < 201Mt khc 2n+1 l s chnh phng lnn 2n+1 ch c th nhn mt trong cc gi tr :25; 49; 81; 121; 169. T n ch c th nhn mt trong cc gi tr 12, 24, 40, 60,84. Khi s 3n+1 ch c th nhn mt trong cc gi tr :37; 73; 121; 181;253. Trong cc s trn ch c s 121=112 l mt s chnh phng. Vy s t nhin c 2 ch s cn tm l n=40. V d 4: Chng minh rng nu p l tch ca n s nguyn t u tin th p-1 v p+1 khng th l cc s chnh phng Gii V p l tch ca n s nguyn t u tin nn p chia ht cho 2 v p khng chia ht cho 4 (1) a) Gi s p+1 l s chnh phng .t p+1 =m2(meN) V p l s chn nn p+1 l s l , do m2 l s l ,v th m l s l . t m=2k+1 (keN) Ta c m2 = (2k+1)2 = 4k2+ 4k+ 1 , suy ra p+1= 4k2+ 4k+ 1
do p=4k(k+1)l s chia ht cho 4, mu thun vi (1) Vy p+1 khng l s chnh phng b)Ta c p = 2.3.5l s chia ht cho 3. Do p-1 = 3k+2khng l s chnh phngVy nu p l tch ca n s nguyn t u tin th p-1 v p+1 khng l s chnh phng III/ BI TP: BI TPBI GII Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 81Bi1:Tmstnhinnbit rngtrong3mnhsauc2 mnhngvmtmnh sai:1/ n c ch s tn cng l 2 2/n+20lmtschnh phng 3/n69lmtschnh phng Nu mnh(1)ngtht(2)suyran+20 cstncngl 2;Tmnh (3)suyra n 69cchstncngl3.Mtschnh phngkhngcchstncngl2hoc3. Nh vy nu (1) ng th (2) v (3) u sai, tri githit.Vymnh(1)saivmnh(2) v (3) ng. t n + 20 = a2; n 69 = b2 (a, b e N v a > b) => a2 b2 = 89 => (a + b)(a b) = 89.1 Do : a b 89a b 1+ = = suyra a = 45. Vy n = 452 20 = 2005 Bi2:ChoNltngca2s chnhphng.Chngminh rng: a/2Ncngltngca2s chnh phng. b/N2cngltngca2s chnh phng. Gi N = a2 + b2 (a, b e N) a/2N=2a2+2b2=a2+b2+2ab+a2+b2 2ab=(a+b)2+(ab)2ltngca2s chnh phng. b/ N2 = (a2 + b2)2 = a4 + 2a2b2 + b2 = a4 2a2b2 + b2+ 4a2b2 = (a2 b2)2 + (2ab)2 Bi 3:Cho A,B,C,D lccs chnhphng.Chngminh rng:(A + B)(C + D) l tng ca 2 s chnh phng. Theobitonth:A=a2;B=b2;C=c2;D= d2; Nn: (A + B)(C + D) = (a2 + b2)(c2 + d2) == a2c2 + a2d2 + b2c2 + b2d2 = a2c2 + b2d2 + 2abcd 2abcd + a2d2 + b2c2 = (ac + bd)2 + (ad bc) l tng ca 2 s chnh phng. Bi4:Cho3snguynx,y,z sao cho: x = y + z. Chng minh rng:2(xy+xzyz)ltng ca 3 s chnh phng. V x = y + z => x y z = 0 => (x y z)2 = 0 => x2 + y2 + z2 2xy 2xz + 2yz = 0 => 2(xy + xz yz) = x2 + y2 + z2 Bi5:Choa,b,c,dlccs nguyn tho mn: a b = c + d. Chng minh rng: a2 + b2 + c2 + d2lunltngca3schnh phng. T a b = c + d => a b c d = 0=> 2a(a b c d) = 0 Nn ta suy ra: a2 + b2 + c2 + d2 = a2 + b2 + c2 + d2 + 2a(a b c d) = (a b)2 + (a c)2 + (a d)2 Bi6:Cho2schnhphng lin tip. Chng minh rng tng ca2scngvitchca chnglmtschnhphng l. Tac:n2+(n+1)2+n2(n+1)2=n4+2n3+ 3n2 + 2n + 1 = = (n2 + n + 1)2 n2 + n l mt s chn n2 + n + 1 l mt s l.Suy ra (n2 + n + 1)2 l mt s chnh phng l. Bi 7: Cho an = 1 + 2 + 3 + ... + n a/ Tnh an+1 a/ T bi ton ta suy ra: an+1 = 1 + 2 + 3 + ... + (n + 1) Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Cc chuynBi dng HSG Ton lp 7 82b/ Chng minh rng an + an+1 l mt s chnh phng b/an+an+1= (1 n)n2++ (1 n 1)(n 1)2+ + += (n 1)(n n 2)2+ + + == (n + 1)2 C. MT S BI TP TNG T Bi 1.Cho 2 s t nhin A v B trong s A ch gm c 2m ch s 1,s B ch gm m ch s 4. Chng minh rng : A+B +1 l s chnh phng. Bi 2. Tm mt s t nhin c 2 ch s, bit rng hiu cc bnh phng ca s v s vit bi hai ch s ca s nhng theo th t ngc li l mt s chnh phng. Bi3. Tm s chnh phng c 4 ch s ,bit rng ch s hng trm , hng nghn ,hng chc, hng n v l 4 s t nhin lin tip tng dn. Bi 4. T cc ch s 1, 2, 3, 4, 5, 6ngi ta lp tt c cc s c 6 ch s , mi s gm cc ch s khc nhau. Hi trong cc s lp c c s no chia ht cho 11 khng ? C s no l s chnh phng khng? Bi 5 Ngi ta vit lin tip cc s : 1, 2, 3,, 1994 thnh mt hng ngang theo mt th t tu . Hi s to thnh theo cch vit trn c th l s chnh phng khng?
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