Gio n BDHSG Ton 8 Nm hc : 2011-2012 Thanh M, ngy 15/7/2011Chuyn 1TNH CHT CHIA HT CA S NGUYNI. Mc tiuSau khi hc xong chuyn hc sinh c kh nng:1.Bit vn dng tnh cht chiaht ca s nguynd chng minh quan h chia ht, tm s d v tm iu kin chia ht.2. Hiu cc bc phn tch bi ton, tm hng chng minh3. C k nng vn dng cc kin thc c trang b gii ton.II. Cc ti liu h tr:- Bi tp nng cao v mt s chuyn ton 8- Ton nng cao v cc chuyn i s 8- Bi dng ton 8- Nng cao v pht trin ton 8- III. Ni dung1. Kin thc cn nh1. Chng minh quan h chia ht Gi A(n) l mt biu thc ph thuc vo n (nN hoc n Z)a/ chng minh A(n) chia ht cho m ta phn tch A(n) thnh tch trong c mt tha s l m + Nu m l hp s ta phn tch m thnh tch cc tha s I mt nguyn t cng nhau ri chng minh A(n) chia ht cho tt c cc s + Trong k s lin tip bao gi cng tn ti mt s l bi ca k b/. Khi chng minh A(n) chia ht cho n ta c th xt mi trng hp v s d khi chia m cho n * V d1: C/minh rng A=n3(n2- 7)2 36n chia ht cho 5040 vi mi s t nhin n Gii:Ta c 5040 = 24. 32.5.7 A= n3(n2- 7)2 36n = n.[ n2(n2-7)2 36 ] = n. [n.(n2-7 ) -6].[n.(n2-7 ) +6] = n.(n3-7n 6).(n3-7n +6) Ta li c n3-7n 6 = n3 + n2 n2 n 6n -6 = n2.(n+1)- n (n+1) -6(n+1) =(n+1)(n2-n-6)= (n+1 )(n+2) (n-3) Tng t : n3-7n+6 = (n-1) (n-2)(n+3) dDo A= (n-3)(n-2) (n-1) n (n+1) (n+2) (n+3) Ta thy : A l tch ca 7 s nguyn lin tip m trong 7 s nguyn lin tip: - Tn ti mt bi s ca 5 (nn A M5 ) - Tn ti mt bi ca 7 (nn A M7 ) - Tn ti hai bi ca 3 (nn A M9 )- Tn ti 3 bi ca 2 trong c bi ca 4 (nn A M16)VyAchiahtcho5, 7,9,16i mtnguyntcngnhauAM5.7.9.16= 5040V d 2: Chng minh rng vi mi s nguyn a th : a/ a3 a chia ht cho 3 b/ a5-a chia ht cho 5 Gv: Nguyn Vn TTrng THCS Thanh M1Gio n BDHSG Ton 8 Nm hc : 2011-2012 Gii:a/ a3-a = (a-1)a (a+1) l tch ca cc s nguyn lin tip nn tch chia ht cho 3 b/ A= a5-a = a(a2-1) (a2+1) Cch 1:Ta xt mi trng hp v s d khi chia a cho 5- Nu a= 5 k (kZ) th A M5(1)- Nu a= 5kt 1 th a2-1 = (5k2t 1) 2 -1 = 25k2t10kM5 A M5(2)- Nu a= 5kt 2 th a2+1 = (5kt 2)2+ 1 = 25 k2t 20k +5 A M5 (3) T (1),(2),(3) A M5,n ZCch 2: Phn tch A thnh mt tng ca hai s hng chia ht cho 5 :+ Mt s hng l tch ca 5 s nguyn lin tip+ Mt s hng cha tha s 5 Tac: a5-a=a(a2-1)(a2+1)=a(a2-1)(a2-4+5)=a(a2-1)(a2-4)+ 5a(a2-1) = a(a-1)(a+1) (a+2)(a-2)- 5a (a2-1) M = a(a-1)(a+1) (a+2)(a-2) M5 (tch ca 5 s nguyn lin tip ) 5a (a2-1) M5 Do a5-aM5*Cch 3:Davo cch2:Chng minhhiu a5-av tchca 5s nguyn lin tip chia ht cho 5.Ta c: a5-a (a-2)(a-1)a(a+1)(a+2) = a5-a (a2- 4)a(a2-1) = a5-a - (a3- 4a)(a2-1) = a5-a- a5 + a3 +4a3 - 4a = 5a3 5a M5 a5-a (a-2)(a-1)a(a+1)(a+2) M5M(a-2)(a-1)a(a+1)(a+2) M5 a5-a M5(Tnh cht chia ht ca mt hiu)c/ Khi chng minh tnh chia ht ca cc lu tha ta cn s dng cc hng ng thc:an bn = (a b)( an-1 + an-2b+ an-3b2+ +abn-2+ bn-1) (HT 8)an + bn = (a + b)( an-1 - an-2b+ an-3b2 -- abn-2+ bn-1)(HT 9)- S dng tam gic Paxcan:1111 211 33 11 46 41..Mi dng u bt u bng 1 v kt thc bng 1Mi s trn mt dng (k t dng th 2) u bng s lin trn cng vi s bn tri ca s lin trn.Do : Via, b Z, n N: an bn chia ht cho a b( ab)a2n+1 + b2n+1 chia ht cho a + b( a-b)(a+b)n = Bsa +bn ( BSa:Bi s ca a)Gv: Nguyn Vn TTrng THCS Thanh M2Gio n BDHSG Ton 8 Nm hc : 2011-2012 (a+1)n = Bsa +1(a-1)2n = Bsa +1(a-1)2n+1 = Bsa -1* VD3: CMR vi mi s t nhin n, biu thc 16n 1 chia ht cho 17 khi v ch khi n l s chn.Gii:+ Cch 1: - Nu n chn: n = 2k, kN th:A = 162k 1 = (162)k 1 chia ht cho 162 1( theo nh thc Niu Tn)M 162 1 = 255 M17. Vy AM17- Nu n l th : A = 16n 1 = 16n + 1 2 m n l th 16n + 1M16+1=17 (HT 9) Akhng chia ht cho 17+Cch 2: A = 16n 1 = ( 17 1)n 1 = BS17 +(-1)n 1 (theo cng thc Niu Tn)- Nu n chn th A = BS17 + 1 1 = BS17 chia ht cho 17- Nu n l th A = BS17 1 1 = BS17 2 Khng chia ht cho 17 Vy biu thc 16n 1 chia ht cho 17 khi v ch khi n l s chn, n Nd/ Ngoi ra cn dng phng php phn chng, nguyn l Dirichl chng minh quan h chia ht. VD 4: CMR tn ti mt bi ca 2003 c dng: 2004 2004.2004Gii: Xt 2004 s: a1 = 2004 a2 = 2004 2004 a3 = 2004 2004 2004 .a2004 = 2004 20042004 2004 nhm 2004Theo nguyn l Dirichle, tn ti hai s c cng s d khi chia cho 2003.Gi hai s l am v an ( 1 n 0 nn 3n 5 > 0. Ta li c: 3n 5 < 4n +5(v n 0) nn 12n2 5n 25 l s ngyn t th tha s nh phi bng 1 hay 3n 5 = 1 n = 2Khi , 12n2 5n 25 = 13.1 = 13 l s nguyn t.Vy vi n = 2 th gi tr ca biu thc 12n2 5n 25 l s nguyn t 13b/ 8n2 + 10n +3 = (2n 1)(4n + 3)Bin i tng t ta c n = 0. Khi , 8n2 + 10n +3 l s nguyn t 3c/ A = 334n n +. Do A l s t nhinnn n(n + 3) M4. Hai s n v n + 3 khng th cng chn. Vy hoc n , hoc n + 3 chia ht cho 4- Nu n = 0 th A = 0, khng l s nguyn t- Nu n = 4 th A = 7, l s nguyn t-Nu n = 4k vi kZ, k > 1 th A = k(4k + 3) l tch ca hai tha s ln hn 1 nn A l hp s - Nu n + 3 = 4 th A = 1, khng l s nguyn t- Nu n + 3 = 4k vi kZ, k > 1 th A = k(4k -3) l tch ca hai tha s ln hn 1 nn A l hp s.Vy vi n = 4 th 334n n + l s nguyn t 7Bi 7: vui: Nm sinh ca hai bnMt ngy ca thp k cui cng ca th k XX, mt nh khch n thm trng gp hai hc sinh. Ngi khch hi:- C l hai em bng tui nhau?Bn Mai tr li:- Khng, em hn bn em mt tui. Nhng tng cc ch s ca nm sinh mi chng em u l s chn.- Vy th cc em sinh nm 1979 v 1980, ng khng?Ngi khch suy lun th no?Gii:Ch s tn cng ca nm sinh hai bn phI l 9 v 0 v trong trng hp ngocli thtngccchscanmsinhhai bnchhnkm nhau l 1, khng th cng l s chn.Gv: Nguyn Vn TTrng THCS Thanh M7Gio n BDHSG Ton 8 Nm hc : 2011-2012 Gi nm sinh ca Mai l 19 9 ath 1 +9+a+9 = 19 + a. Mun tng ny l s chn th a{1; 3; 5; 7; 9}. Hin nhin Mai khng th sinh nm 1959 hoc 1999. Vy Mai sinh nm 1979, bn ca Mai sinh nm 1980.Chuyn 2 :TNH CHT CHIA HT TRONG NTi t10-12: Mt s du hiu chia ht V d I.Mt s du hiu chia ht 1. Chia ht cho 2, 5, 4, 25 v 8; 125.

1 1 0 0 0... 2 2 0; 2; 4; 6;8.n na a a a a a M M

1 1 0 0... 5 0;5n na a a a a M1 1 0... 4n na a a aM ( hoc 25) 1 04 a a M ( hoc 25) 1 1 0... 8n na a a aM ( hoc 125) 2 1 08 a a a M ( hoc 125)2. Chia ht cho 3; 9.

1 1 0... 3n na a a aM (hoc 9) 0 1... 3na a a + + + M( hoc 9)Nhn xt: D trong php chia N cho 3 ( hoc 9) cng chnh l d trong php chia tng cc ch s ca N cho 3 ( hoc 9).3. Du hiu chia ht cho 11 : Cho 5 4 3 2 1 0... A a a a a a a ( ) ( )0 2 4 1 3 511 ... ... 11 A a a a a a a + + + + + +1 ]M M4.Du hiu chia ht cho 101

5 4 3 2 1 0... A a a a a a a ( ) ( )1 0 5 4 3 2 7 6101 ... ... 101 A a a a a a a a a + + + +1 ]M MII.V dV d 1:Tm cc ch s x, y :a) 134 4 45 x yMb)1234 72 xyMGii:a) 134 4 45 x yMta phi c 134 4 x y chia ht cho 9 v 5 y = 0 hoc y = 5Vi y = 0 th t 134 40 9 x M ta phi c 1+3+5+x+49 M 4 9 5 x x + M khi ta c s 13554vi x = 5 th t : 134 4 9 x yM ta phi c1+3+5+x+4 +5 9 M9 0; 9 x x x M lc ta c 2 s: 135045; 135945.b) Ta c1234 123400 72.1713 64 72 64 72 xy xy xy xy + + + + M MV64 64 163 xy + nn64 xy +bng 72 hoc 144.+ Vi64 xy + =72 thxy =08, ta c s: 123408.+ Vi64 xy + =14 thxy =80, ta c s 123480V d 2 Tm cc ch s x, y 7 36 5 1375 N x y M Gii:Ta c: 1375 = 11.125.( ) ( )125 6 5 125 27 3625 11 5 6 2 3 7 12 11 1N y yN x x x x + + + + M MM MVy s cn tm l 713625V d 3a) Hi s 19911991 19911991...1991soA 1 42 43 c chia ht cho 101 khng?b)Tm n 101nA MGv: Nguyn Vn TTrng THCS Thanh M8Gio n BDHSG Ton 8 Nm hc : 2011-2012 Gii:a) Ghp 2 ch s lin tip nhau th A1991 c 2 cp s l 91;19Ta c:1991.91-1991.19 = 1991. 72M101 nn 1991101 A Mb) 101 .91 .19 72 101 101nA n n n n M M MTIT13 14:II. MT S NH L V PHP CHIA HT A.Tm tt l thuyt 1. nh l v php chia ht:a)nh l Cho a, b l cc s nguyn tu ,0 b , khi c 2 s nguyn q, r duy nht sao cho : a bq r + vi 0 r b , a l s b chia, b l s chia, q l thng s v r l s d. c bit vi r = 0 tha = b.qKhi ta ni a chia ht cho b hay b l c ca a, k hiua b M. Vy b) Tnh cht a) Nua b Mvb c M tha c MMb) Nua b Mvb a M th a = bc) Nua b M,a c Mv (b,c) = 1 tha bc Md) Nuab c M v (c,b) = 1tha c M2.Tnh cht chia ht ca mt tng, mt hiu, mt tch.-Nu )m bm am b a + - Nu)m bm am b a - Nu)m bm aa .bm - Nu m a a n m(n l s t nhin)3.Mt s tnh cht khc: Trong n s t nhin lin tip c mt s chia ht cho n Tch n s t nhin lin tip chia ht cho n! A a M A b Mv (a;b) = 1 a.b A MB.V d:1. Chng minh rng vi mi s nguyn dng n ta c: ( ) 24 1 122 +n nGii:( )( ) ( ) ( )221 1 1 1 2 4! 24 A n n n n n n11 + + + ] ]MGv: Nguyn Vn TTrng THCS Thanh M9a b M c s nguyn q sao cho a = b.qGio n BDHSG Ton 8 Nm hc : 2011-2012 Bi tp t luyn:2. Chng minh rng a. 48 8 62 3 n n n + +vi n chnb.384 9 102 4 + n nvi n l3. Chng minh rng :72 22 4 6 n n n +vi n nguyn4. CMR vi mi s nguyn a biu thc sau:a)a(a 1) (a +3)(a + 2) chia ht cho 6.b) a(a + 2) (a 7)(a -5) chia ht cho 7.c) (a2 + a + 1)2 1 chia ht cho 24d) n3 + 6n2 + 8n chia ht cho 48 (mi n chn)5. CMR vi mi s t nhin n th biu thc:a) n(n + 1)(n +2) chia ht cho 6b) 2n ( 2n + 2) chia ht cho 8.Tit 15 16:3. ng dthc I.L thuyt ng d : a) nh ngha : Cho s nguyn m > 0. Nu 2 s nguyn a, b cho cng s d khi chia cho m th ta ni a ng d vi b theo mun m . K hiu : (mod ) a b m b) Tnh cht a) (mod ) (mod ) a b m a c b c m t tb)(mod ) (mod ) a b m na nb m M Mc)(mod ) (mod )n na b m a b m d) (mod ) (mod ) a b m ac bc m c) Mt s hng ng thc:m ma b a b Mn na b a b + + M(n l) ( ) ( )na b B a b + +II.V d:1.Chng minh:9 992 2 200 + MGii:2 + 2 = 2 = 512 112(mod 200) (1)2 =2 112 (mod 200) . 112 = 1254412 (mod 200) 112 12 (mod 200) 12 = 61917364224 24(mod 200) .112 24.112(mod 200) 2688(mod 200) 88(mod 200) 288(mod 200)(2) T (1) v (2) 2 + 2 = 200(mod 200) hay 9 992 2 200 + MIII,Bi tp t luyn:S dng hng ng thc v ng d1.( ) 7 2 1965 1963 19611966 1964 1962 + + +Gv: Nguyn Vn TTrng THCS Thanh M10Gio n BDHSG Ton 8 Nm hc : 2011-2012 2.( ) 19 14 241917 1917 +3.( ) 200 2 299 9 +4.( ) 183 1 13123456789 5.( ) 1980 1982 1981 19791981 1979 + 6.( ) 120 3 ... 3 3 3100 3 2 + + + +7.( ) 7 5555 22222222 5555 +--------------------------------Tit 17 18:QUY NP TON HCI.PHNG PHP CHNG MINH B1: Kim tra mnh ng vi n = 1?B2: Gi s Mnh ng vi n = k 1. Chng minh mnh ng vi n = k + 1II.V D:1. Chng minh rng vi mi s nguyn dng n th: 2 2 17 8 57n n + ++ MGii: -Vi n = 1:A1 = 7+ 8 = 855 + 57- Gi sAk +57 ngha l 2 2 17 8 57n n + ++ M Ak+1 =7 + 8=7. 7 + 64.8 = 7(7 + 8 ) + 57.8 .V 7 + 8 ( gi thit qui np) v 57.8M57 Ak+1M57Vy theo nguyn l qui npA = 7 + 8 M57.*Ch : Trong trng hp tng qut vi n l s nguyn v n n0. Th ta kim tra mnh ng khi n = n0?III.BI TP:Chng minh : Vi n l s t nhin th:1.( ) 23 2 2 51 4 1 2+ + ++ +n n n2. 11 + 12 M1333.( ) 59 8 5 . 26 51 2 2+ ++ +n n n 4.( ) 5 3 21 3 1 2+ ++n n 5.( ) 18 14 24 22 2 + ++nn-----------------------------------Tit 19-20LUYN TP1. 1025 2 1 c ab A 2. ( )21 5 + c abca B3. ab E sao cho( )32b a ab + 4. A = ( )2b a ab + HD:( )2b a ab + ( )( )29 9 1 + + a b a b a (a + b) 9v (a + b) = 9k k = 1 a + b = 9 9a = 9.8 = 72 a = 8 v b = 1 5. B = ( )2cd ab abcd + Gv: Nguyn Vn TTrng THCS Thanh M11Gio n BDHSG Ton 8 Nm hc : 2011-2012 HD: tab x ; cd y 99x = (x + y)(x + y - 1) 992 Xt 2 kh nng :

1 khng phi l s chnh phngn6 n4 + 2n3 +2n2 = n2.( n4 n2 + 2n +2 ) = n2.[ n2(n-1)(n+1) + 2(n+1) ] = n2[ (n+1)(n3 n2 + 2) ] = n2(n+1).[ (n3+1) (n2-1) ]= n2( n+1 )2.( n22n+2)Gv: Nguyn Vn TTrng THCS Thanh M1522 22Gio n BDHSG Ton 8 Nm hc : 2011-2012 Vi nN, n >1 th n2-2n+2 = (n - 1)2+ 1 > ( n 1 )2 v n2 2n + 2 = n2 2(n - 1) < n2 Vy ( n 1)2 < n2 2n + 2 < n2 n2 2n + 2 khng phi l mt s chnh phng.

Bi 9: Cho 5 s chnh phng bt k c ch s hng chc khc nhau cn ch s 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 Cch 1: Ta bit mt 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 M = a2 c ch s hng n v l 6 th ch s tn cng ca a l 4 hoc 6 a2 a2 4 Theo du hiu chia ht cho 4 th hai ch s tn cng ca M ch c th l 16, 36, 56, 76, 96 Ta c: 1 + 3 + 5 + 7 + 9 = 25 = 52 l s chnh phng.Bi 10: Chng minh rng tng bnh phng ca hai s l bt k khng phi l mt s chnh phng.a v b l nn a = 2k+1, b = 2m+1 (Vi k, m N) a2 + b2= (2k+1)2 + (2m+1)2 = 4k2 + 4k + 1 + 4m2 + 4m + 1 = 4(k2 + k + m2 + m) + 2=4t + 2(Vi t N)Khng c s chnh phng no c dng 4t + 2(t N) do a2 + b2khng th l s chnh phng.Bi 11: 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.V p l tch ca n s nguyn t u tin nn p2 v p khng chia ht cho 4 (1)a. Gi s p+1 l s chnh phng . t p+1 = m2 (m N)V p chn nn p+1 l m2 l m l.t m = 2k+1(k N). Ta c m2 = 4k2 + 4k + 1 p+1 = 4k2 + 4k + 1 p = 4k2 + 4k = 4k(k+1) 4 mu thun vi (1) p+1 l s chnh phngb. p = 2.3.5 l s chia ht cho 3 p-1 c dng 3k+2.Khng c s chnh phng no c dng 3k+2 p-1 khng l s chnh phng .Vy nu p l tch n s nguyn t u tin th p-1 v p+1 khng l s chnh phngBi 12: Gi s N = 1.3.5.72007.Chng minh rng trong 3 s nguyn lin tip 2N-1, 2N v 2N+1 khng c s no l s chnh phng.Gv: Nguyn Vn TTrng THCS Thanh M16Gio n BDHSG Ton 8 Nm hc : 2011-2012 a. 2N-1 = 2.1.3.5.72007 1C 2N 3 2N-1 khng chia ht cho 3 v 2N-1 = 3k+2(k N) 2N-1 khng l s chnh phng.b. 2N = 2.1.3.5.72007V N l N khng chia ht cho 2 v 2N 2 nhng 2N khng chia ht cho 4.2N chn nn 2N khng chia cho 4 d 1 2N khng l s chnh phng.c. 2N+1 = 2.1.3.5.72007 + 12N+1 l nn 2N+1 khng chia ht cho 4 2N khng chia ht cho 4 nn 2N+1 khng chia cho 4 d 1 2N+1 khng l s chnh phng.Bi 13: Cho a = 111 ;b = 10005

2008 ch s 12007 ch s 0Chng minh1 + ab l s t nhin.Cch 1: Ta c a = 111 = 91 102008;b = 10005 = 1000 + 5 = 102008 + 5 2008 ch s 1 2007 ch s 02008 ch s 0

ab+1 =9) 5 10 )( 1 10 (2008 2008+ + 1 = 99 5 10 . 4 ) 10 (2008 2 2008+ + =

,_

+32 102008 1 + ab =

,_

+32 102008 = 32 102008+Ta thy 102008 + 2 = 10002 3 nn32 102008+ N hay 1 + ab l s t nhin.2007 ch s 0Cch 2: b = 10005 = 1000 1 + 6 = 999 + 6 = 9a +6

2007 ch s 02008 ch s 02008 ch s 9 ab+1 = a(9a +6) + 1 = 9a2 + 6a + 1 = (3a+1)2 1 + ab = 2) 1 3 ( + a= 3a + 1 NB. DNG 2 : TM GI TR CA BIN BIU THC L S CHNH PHNGBi1: Tm s t nhin n sao cho cc s sau l s chnh phng:a.n2 + 2n + 12 b.n ( n+3 ) c. 13n + 3 d. n2 + n + 1589Giia. V n2 + 2n + 12 l s chnh phng nn t n2 + 2n + 12 = k2(k N)

(n2 + 2n + 1) + 11 = k2

k2 (n+1)2 = 11 (k+n+1)(k-n-1) = 11Nhn xt thy k+n+1 > k-n-1 v chng l nhng s nguyn dng, nn ta c th vit (k+n+1)(k-n-1) = 11.1 k+n+1 = 11 k = 6k n - 1 = 1n = 4b. tn(n+3) = a2(n N) n2 + 3n = a2 4n2 + 12n = 4a2 (4n2 + 12n + 9) 9 = 4a2Gv: Nguyn Vn TTrng THCS Thanh M1722Gio n BDHSG Ton 8 Nm hc : 2011-2012 (2n + 3)2- 4a2 = 9 (2n + 3 + 2a)(2n + 3 2a) = 9Nhn xt thy 2n + 3 + 2a > 2n + 3 2a v chng l nhng s nguyn dng, nn ta c th vit (2n + 3 + 2a)(2n + 3 2a) = 9.1 2n + 3 + 2a = 9 n = 12n + 3 2a = 1 a = 2c. t 13n + 3 = y2 ( y N) 13(n 1) = y2 16 13(n 1) = (y + 4)(y 4) (y + 4)(y 4) 13 m 13 l s nguyn t nn y + 4 13 hoc y 4 13 y = 13k t 4(Vi k N) 13(n 1) = (13k t 4)2 16 = 13k.(13k t 8) n = 13k2 t 8k + 1Vy n = 13k2 t 8k + 1 (Vi k N) th 13n + 3 l s chnh phng.d. t n2 + n + 1589 = m2 (m N) (4n2 + 1)2 + 6355 = 4m2

(2m + 2n +1)(2m 2n -1) = 6355Nhn xt thy 2m + 2n +1> 2m 2n -1 > 0 v chng l nhng s l, nn ta c th vit (2m + 2n +1)(2m 2n -1) = 6355.1 = 1271.5 = 205.31 = 155.41Suy ra n c th c cc gi tr sau: 1588; 316; 43; 28.Bi 2:Tm a cc s sau l nhng s chnh phng:a. a2 + a + 43 b. a2 + 81c. a2 + 31a + 1984Kt qu: a.2; 42; 13b.0; 12; 40c.12; 33; 48; 97; 176; 332; 565; 1728Bi 3:Tm s t nhin n 1 sao cho tng 1! + 2! + 3! + + n! l mt s chnh phng .Vi n = 1 th 1! = 1 = 12l s chnh phng .Vi n = 2 th 1! + 2! = 3 khng l s chnh phng Vi n = 3 th 1! + 2! + 3! = 1+1.2+1.2.3 = 9 = 32 l s chnh phng Vi n 4 ta c 1! + 2! + 3! + 4! = 1+1.2+1.2.3+1.2.3.4 = 33 cn 5!;6!; ; n! u tn cng bi 0 do 1! + 2! + 3! + + n! c tn cng bi ch s 3 nn n khng phi l s chnh phng .Vy c 2 s t nhin n tha mn bi l n = 1; n = 3.Bi 4: Tm n N cc s sau l s chnh phng: a. n2 + 2004 ( Kt qu: 500; 164)Gv: Nguyn Vn TTrng THCS Thanh M18Gio n BDHSG Ton 8 Nm hc : 2011-2012 b. (23 n)(n 3)( Kt qu: 3; 5; 7; 13; 19; 21; 23)c. n2 + 4n + 97 d. 2n + 15Bi 5:C hay khng s t nhin n 2006 + n2 l s chnh phng. Gi s 2006 + n2 l s chnh phng th 2006 + n2 = m2 (m N)T suy ra m2 n2 = 2006 (m + n)(m - n) = 2006 Nh vy trong 2 s m v n phi c t nht 1 s chn (1)Mt khcm + n + m n = 2m 2 s m + n v m n cng tnh chn l (2)T (1) v (2) m + n v m n l 2 s chn

(m + n)(m - n) 4Nhng 2006 khng chia ht cho 4

iu gi s sai. Vy khng tn ti s t nhin n 2006 + n2 l s chnh phng.Bi 6: Bit x Nv x>2. Tm x sao cho x(x-1).x(x-1) = (x-2)xx(x-1) ng thc cho c vit li nh sau: x(x-1) =(x-2)xx(x-1)Do v tri l mt s chnh phng nn v phi cng l mt s chnh phng .Mt s chnh phng ch c th tn cng bi 1 trong cc ch s 0; 1; 4; 5; 6; 9 nn x ch c th tn cng bi 1 trong cc ch s 1; 2; 5; 6; 7; 0 (1)Do x l ch s nn x 9, kt hp vi iu kin bi ta c x N v 2 < x 9 (2)T (1) v (2) x ch c th nhn 1 trong cc gi tr5; 6; 7.Bng php th ta thy ch c x = 7 tha mn bi, khi 762 = 5776Bi 7: Tm s t nhin n c 2 ch s bit rng 2n+1 v 3n+1 u l cc s chnh phng.Ta c 10 n 99 nn 21 2n+1 199. Tm s chnh phng l trong khong trn ta c 25; 49; 81; 121; 169 tng ng vi s n bng 12; 24; 40; 60; 84.S 3n+1 bng 37; 73; 121; 181; 253. Ch c 121 l s chnh phng.Vy n = 40Bi 8: Chng minh rng nu n l s t nhin sao cho n+1 v 2n+1 u l cc s chnh phng th n l bi s ca 24.V n+1 v 2n+1 l cc s chnh phng nn t n+1 = k2 , 2n+1 = m2(k, m N)Ta c m l s l m = 2a+1 m2 = 4a (a+1) + 1

n = 212 m = 2) 1 ( 4 + a a = 2a(a+1) n chn n+1 l k l t k = 2b+1 (Vi b N) k2= 4b(b+1) +1

n = 4b(b+1) n 8(1)Gv: Nguyn Vn TTrng THCS Thanh M192Gio n BDHSG Ton 8 Nm hc : 2011-2012 Ta c k2 + m2 = 3n + 22 (mod3)Mt khc k2 chia cho 3 d 0 hoc 1, m2 chia cho 3 d 0 hoc 1. Nn k2 + m2 2 (mod3) th k21 (mod3)m2 1 (mod3) m2 k2 3 hay (2n+1) (n+1) 3 n 3(2)M (8; 3) = 1 (3)T (1), (2), (3) n 24.Bi 9: Tm tt c cc s t nhin n sao cho s 28 + 211 + 2n l s chnh phng .Gi s 28 + 211 + 2n = a2 (a N) th 2n = a2 482 = (a+48)(a-48) 2p.2q = (a+48)(a-48) Vi p, q N ; p+q = nv p > q a+48 = 2p

2p 2q = 96 2q (2p-q -1) = 25.3 a- 48 = 2q

q = 5 v p-q = 2 p = 7 n = 5+7 = 12Th li ta c:28 + 211 + 2n = 802 C.DNG 3: TM S CHNH PHNG Bi 1: Cho A l s chnh phng gm 4 ch s. Nu ta thm vo mi ch s ca A mtn v th ta c s chnh phng B. Hy tm cc s A v B.Gi A = abcd = k2. Nu thm vo mi ch s ca A mtn v th ta c s B = (a+1)(b+1)(c+1)(d+1) = m2vi k, m N v 32 < k < m < 100a, b, c, d N ; 1 a 9 ; 0 b, c, d 9 Ta c A = abcd = k2

B = abcd + 1111 = m2 m2 k2 = 1111 (m-k)(m+k) = 1111(*)Nhn xt thy tch (m-k)(m+k) > 0 nn m-k v m+k l 2 s nguyn dng.Vm-k < m+k < 200 nn (*) c th vit(m-k)(m+k) = 11.101Do m k == 11 m = 56A = 2025m + k = 101n = 45 B = 3136 Bi 2: Tm 1 s chnh phng gm 4 ch s bit rng s gm 2 ch s u ln hn s gm 2 ch s sau 1 n v.t abcd = k2 ta c ab cd = 1v k N, 32 k < 100 Suy ra 101cd = k2 100 = (k-10)(k+10) k +10 101 hoc k-10 101M (k-10; 101) = 1 k +10 101Gv: Nguyn Vn TTrng THCS Thanh M20Gio n BDHSG Ton 8 Nm hc : 2011-2012 V 32 k < 100 nn 42 k+10 < 110 k+10 = 101 k = 91 abcd = 912 = 8281Bi 3: Tm s chnh phng c 4 ch s bit rng 2 ch s u ging nhau, 2 ch s cui ging nhau.Gi s chnh phng phi tm l aabb = n2vi a, b N,1 a 9; 0 b 9Ta c n2 = aabb = 11.a0b = 11.(100a+b) = 11.(99a+a+b) (1)Nhn xt thy aabb 11 a + b 11M 1 a 9 ;0 b 9 nn 1 a+b 18 a+b = 11Thay a+b = 11 vo (1) c n2 = 112(9a+1) do 9a+1 l s chnh phng .Bng php th vi a = 1; 2; ; 9 ta thy ch c a = 7 tha mn b = 4S cn tm l 7744Bi 4: Tm mt s c 4 ch s va l s chnh phng va l mt lp phng.Gi s chnh phng l abcd . V abcd va l s chnh phng va l mt lp phng nn t abcd = x2 = y3 Vi x, y NV y3 = x2 nn y cng l mt s chnh phng .Ta c 1000 abcd 9999 10 y 21 v y chnh phng y = 16 abcd = 4096 Bi 5: Tm mt s chnh phng gm 4 ch s sao cho ch s cui l s nguyn t, cn bc hai ca s c tng cc ch s l mt s chnh phng.Gi s phi tm l abcdvi a, b, c, d nguyn v 1 a 9 ; 0 b,c,d 9abcd chnh phng d{ 0,1,4,5,6,9}d nguyn t d = 5t abcd = k2 < 10000 32 k < 100k l mt s c hai ch s m k2 c tn cng bng 5 k tn cng bng 5Tng cc ch s ca k l mt s chnh phng k = 45 abcd = 2025Vy s phi tm l 2025Bi 6: Tm s t nhin c hai ch s bit rng hiu cc bnh phng ca s v vits bi hai ch s ca s nhng theo th t ngc li l mt s chnh phngGi s t nhin c hai ch s phi tm lab ( a,b N, 1 a,b 9 )S vit theo th t ngc li ba Ta cab-ba

= ( 10a + b ) 2 ( 10b + a )2 = 99 ( a2 b2 ) 11 a2 - b2 11Hay ( a-b )(a+b ) 11 V 0 < a - b 8 , 2 a+b 18 nn a+b 11 a + b = 11Gv: Nguyn Vn TTrng THCS Thanh M212 22 2Gio n BDHSG Ton 8 Nm hc : 2011-2012 Khi ab -ba = 32 . 112 . (a - b)ab -ba l s chnh phng th a - b phi l s chnh phng do a-b = 1 hoc a - b = 4 Nu a-b = 1 kt hp vi a+b = 11 a = 6, b = 5, ab = 65 Khi 652 562 = 1089 = 332 Nu a - b = 4 kt hp vi a+b = 11 a = 7,5 ( loi )Vy s phi tm l 65Bi 7:Cho mt s chnh phng c 4 ch s. Nu thm 3 vo mi ch s ta cng c mt s chnh phng. Tm s chnh phng ban u ( Kt qu: 1156 )Bi 8:Tm s c 2 ch s m bnh phng ca s y bng lp phng ca tng cc ch s ca n. Gi s phi tm lab vi a,b N v 1 a 9 , 0 b 9Theo gi thit ta c : ab = ( a + b )3

(10a+b)2 = ( a + b )3 ab l mt lp phng v a+b l mt s chnh phngt ab= t3 ( t N ) , a + b = l 2 ( l N )V 10 ab 99 ab= 27 hoc ab = 64 Nu ab = 27 a + b = 9 l s chnh phng Nuab = 64 a + b = 10 khng l s chnh phng loiVy s cn tm l ab = 27Bi 9: Tm 3 s l lin tip m tng bnh phng l mt s c 4 ch s ging nhau.Gi 3 s l lin tip l 2n-1, 2n+1, 2n+3( n N)Ta cA= ( 2n-1 )2 + ( 2n+1)2 + ( 2n+3 )2= 12n2 + 12n + 11Theo bi ta t 12n2 + 12n + 11 = aaaa= 1111.avi a l v 1 a 9 12n( n + 1 ) = 11(101a 1 )

101a 1 3 2a 1 3 V 1 a 9 nn 1 2a-1 17 v 2a-1 l nn 2a 1 { 3; 9; 15 } a { 2; 5; 8 } V a l a = 5 n = 21 3 s cn tm l 41; 43; 45Bi 10: Tm s c 2 ch s sao cho tch ca s vi tng cc ch s ca n bng tng lp phng cc ch s ca s .ab (a + b ) = a3 + b3Gv: Nguyn Vn TTrng THCS Thanh M222 22Gio n BDHSG Ton 8 Nm hc : 2011-2012 10a + b = a2 ab + b2 = ( a + b )2 3ab 3a( 3 + b ) = ( a + b ) ( a + b 1 )a + b v a + b 1 nguyn t cng nhau do a + b = 3a hoca + b 1 = 3aa+ b 1 = 3 + ba + b = 3 + b a = 4 , b = 8hoca = 3 , b = 7Vy ab= 48hoc ab= 37. Chuyn 3Cc phng php phn tch a thc thnh nhn tI.CC PHNG PHP C BN1. Phng php t nhn t chung Tm nhn t chung l nhng n, a thc c mt trong tt c cc hng t. Phn tch mi hng t thnh tch ca nhn t chung v mt nhn t khc. Vit nhn t chung ra ngoi du ngoc, vit cc nhn t cn li ca mi hng t vo trong du ngoc (k c du ca chng).V d 1. Phn tch cac a thc sau thnh nhn t.28a2b2 - 21ab2 + 14a2b = 7ab(4ab - 3b + 2a)2x(y z) + 5y(z y ) = 2(y - z) 5y(y - z) = (y z)(2 - 5y)xm + xm + 3 = xm (x3 + 1) = xm( x+ 1)(x2 x + 1)2. Phng php dng hng ng thc- Dng cc hng ng thc ng nh phn tch a thc thnh nhn t.- Cn chu y n vic vn dung hng ng thc.V d 2. Phn tch cac a thc sau thnh nhn t.9x2 4 = (3x)2 22 = ( 3x 2)(3x + 2)8 27a3b6 = 23 (3ab2)3 = (2 3ab2)( 4 + 6ab2+ 9a2b4)25x4 10x2y + y2 = (5x2 y)2Gv: Nguyn Vn TTrng THCS Thanh M23Gio n BDHSG Ton 8 Nm hc : 2011-2012 3. Phng php nhm nhiu hng t Kt hp cc hng t thch hp thnh tng nhm. p dng lin tip cc phng php t nhn t chung hoc dng hng ng thc.V d 3. Phn tch cac a thc sau thnh nhn t2x3 3x2 + 2x 3 = ( 2x3 + 2x) (3x2 + 3) = 2x(x2 + 1) 3( x2 + 1) = ( x2 + 1)( 2x 3)x2 2xy + y2 16 = (x y)2 - 42 = ( x y 4)( x y + 4)4. Phi hp nhiu phng php- Chn cc phng php theo th t u tin.- t nhn t chung.- Dng hng ng thc.- Nhm nhiu hng t.V d 4. Phn tch cac a thc sau thnh nhn t3xy2 12xy + 12x = 3x(y2 4y + 4) = 3x(y 2)23x3y 6x2y 3xy3 6axy2 3a2xy + 3xy == 3xy(x2 2y y2 2ay a2 + 1)= 3xy[( x2 2x + 1) (y2 + 2ay + a2)]= 3xy[(x 1)2 (y + a)2]= 3xy[(x 1) (y + a)][(x 1) + (y + a)]= 3xy( x 1 y a)(x 1 + y + a)II.PHNG PHA P TCH MT HNG T THNH NHIU HNG T1. i vi a thc bc hai (f(x) = ax2 + bx + c)a)Cch 1 (tch hng t bc nht bx):Bc 1: Tm tch ac, ri phn tch ac ra tch ca hai tha s nguyn bng mi cch.a.c = a1.c1 = a2.c2 = a3.c3 = = ai.ci = Bc 2: Chn hai tha s c tng bng b, chng hn chn tch a.c = ai.ci vi b = ai + ciBc 3: Tch bx = aix + cix. T nhm hai s hng thch hp phn tch tip.Gv: Nguyn Vn TTrng THCS Thanh M24Gio n BDHSG Ton 8 Nm hc : 2011-2012 V d 5. Phn tch a thc f(x) = 3x2 + 8x + 4 thnh nhn t.Hng dn- Phn tch ac = 12 = 3.4 = (3).(4) = 2.6 = (2).(6) = 1.12 = (1).(12)- Tch ca hai tha s c tng bng b = 8 l tch a.c = 2.6 (a.c = ai.ci).- Tch 8x = 2x + 6x (bx = aix + cix)Li gii3x2 + 8x + 4 = 3x2 + 2x + 6x + 4 = (3x2 + 2x) + (6x + 4)= x(3x + 2) + 2(3x + 2)= (x + 2)(3x +2)b)Cch 2 (tch hng t bc hai ax2)- Lm xut hin hiu hai bnh phng :f(x) = (4x2 + 8x + 4) x2 = (2x + 2)2 x2 = (2x + 2 x)(2x + 2 + x) = (x + 2)(3x + 2)- Tch thnh 4 s hng ri nhm :f(x) = 4x2 x2 + 8x + 4 = (4x2 + 8x) ( x2 4) = 4x(x + 2) (x 2)(x + 2)= (x + 2)(3x + 2) f(x) = (12x2 + 8x) (9x2 4) = = (x + 2)(3x + 2)c) Cch 3 (tch hng t t do c)- Tch thnh 4 s hng ri nhm thnh hai nhm: f(x) = 3x2 + 8x + 16 12 = (3x2 12) + (8x + 16) = = (x + 2)(3x + 2)d)Cch 4 (tch 2 s hng, 3 s hng)f(x) = (3x2 + 12x + 12) (4x + 8) = 3(x + 2)2 4(x + 2) = (x + 2)(3x 2)f(x) = (x2 + 4x + 4) + (2x2 + 4x) = = (x + 2)(3x + 2)e) Cch 5 (nhm nghim): Xem phn III.Ch : Nu f(x) = ax2 + bx + c c dng A2 2AB + c th ta tch nh sau : f(x) = A2 2AB + B2 B2 + c = (A B)2 (B2 c)V d 6. Phn tch a thc f(x) = 4x2 - 4x - 3 thnh nhn t.Hng dnGv: Nguyn Vn TTrng THCS Thanh M25Gio n BDHSG Ton 8 Nm hc : 2011-2012 Ta thy 4x2 - 4x = (2x)2 - 2.2x. T ta cn thm v bt 12 = 1 xut hin hng ng thc.Li giif(x) = (4x2 4x + 1) 4 = (2x 1)2 22 = (2x 3)(2x + 1)V d 7. Phn tch a thc f(x) = 9x2 + 12x 5 thnh nhn t.Li giiCch 1 : f(x) = 9x2 3x + 15x 5 = (9x2 3x) + (15x 5) = 3x(3x 1) + 5(3x 1) = (3x 1)(3x + 5)Cch 2 : f(x) = (9x2 + 12x + 4) 9 = (3x + 2)2 32 = (3x 1)(3x + 5)2. i vi a thc bc t 3 tr ln (Xem mc III. Phng php nhm nghim)3. i vi a thc nhiu binV d 11. Phn tch cac a thc sau thnh nhn ta) 2x2 - 5xy + 2y2 ;b)x2(y - z) + y2(z - x) + z2(x - y).Hng dna) Phn tich a thc nay tng t nh phn tich a thc f(x) = ax2 + bx + c.Ta tach hang t th 2 :2x2 - 5xy + 2y2 = (2x2 - 4xy) - (xy - 2y2) = 2x(x - 2y) - y(x - 2y)= (x - 2y)(2x - y)a) Nhn xet z - x = -(y - z) - (x - y). Vi vy ta tach hang t th hai cua a thc :x2(y - z) + y2(z - x) + z2(x - y) = x2(y - z) - y2(y - z) - y2(x - y) + z2(x - y) == (y - z)(x2 - y2) - (x - y)(y2 - z2) = (y - z)(x - y)(x + y) - (x - y)(y - z)(y + z)= (x - y)(y - z)(x - z)Chu y :1) cu b) ta co th tach y - z = - (x - y) - (z - x) (hoc z - x= - (y - z) - (x - y))2) a thc cu b) la mt trong nhng a thc co dang a thc c bit. Khi ta thay x = y (y = z hocz = x) vao a thc thi gia tricua a thc bng 0. Vi vy, ngoa i cach phn tich bng cach tach nh trn, ta con cach phn tich bng cach xet gia triring (Xem phn VII).Gv: Nguyn Vn TTrng THCS Thanh M26Gio n BDHSG Ton 8 Nm hc : 2011-2012 III.PHNG PHAP NHM NGHIMTrc ht, ta ch n mt nh l quan trng sau :nh l : Nu f(x) c nghim x = a th f(a) = 0. Khi , f(x) c mt nhn t l x av f(x) c th vit di dng f(x) = (x a).q(x)Lc tch cc s hng ca f(x) thnh cc nhm, mi nhm u cha nhn t lx a. Cng cn lu rng, nghim nguyn ca a thc, nu c, phi l mt c ca h s t do.V d 8. Phn tch a thc f(x) = x3 + x2 + 4 thnh nhn t.Li giiLn lt kim tra vi x = 1, 2,4, ta thy f(2) = (2)3 + (2)2 + 4 = 0. a thc f(x) c mt nghim x = 2, do n cha mt nhn t l x + 2. T , ta tch nh sauCch 1 : f(x) = x3 + 2x2 x2 + 4 = (x3 + 2x2) (x2 4) = x2(x + 2) (x 2)(x + 2) = (x + 2)(x2 x + 2).Cch 2 : f(x) = (x3 + 8) + (x2 4) = (x + 2)(x2 2x + 4) + (x 2)(x + 2) = (x + 2)(x2 x + 2).Cch 3 : f(x) = (x3 + 4x2 + 4x) (3x2 + 6x) + (2x + 4) = x(x + 2)2 3x(x + 2) + 2(x + 2) = (x + 2)(x2 x + 2).Cch 4 : f(x) = (x3 x2 + 2x) + (2x2 2x + 4) = x(x2 x + 2) + 2(x2 x + 2) = (x + 2)(x2 x + 2).T nh l trn, ta c cc h qu sau :H qu 1. Nu f(x) ctng cc h s bng 0 th f(x) c mt nghim l x = 1. T f(x)c mt nhn t l x 1.Chng hn, a thc x3 5x2 + 8x 4 c 1 + (5) + 8 + (4) = 0 nn x = 1 l mt nghim ca a thc. a thc c mt nhn t l x 1. Ta phn tch nh sau :f(x) = (x3 x2) (4x2 4x) + (4x 4) = x2(x 1) 4x(x 1) + 4(x 1) = (x 1)( x 2)2H qu 2. Nu f(x) c tng cc h s ca cc lu tha bc chn bng tng cc h s ca cc lu tha bc l th f(x) c mt nghim x = 1. T f(x) c mt nhn t lx + 1.Chng hn, a thc x3 5x2 + 3x + 9 c 1 + 3 = 5 + 9 nn x = 1 l mt nghim ca a thc. a thc c mt nhn t l x + 1. Ta phn tch nh sau :Gv: Nguyn Vn TTrng THCS Thanh M27Gio n BDHSG Ton 8 Nm hc : 2011-2012 f(x) = (x3 + x2) (6x2 + 6x) + (9x + 9) = x2(x + 1) 6x(x + 1) + 9(x + 1)= (x + 1)( x 3)2 H qu 3. Nu f(x) c nghim nguyn x = a v f(1) v f(1) khc 0 thv u l s nguyn.V d 9. Phn tch a thc f(x) = 4x3 - 13x2 + 9x - 18 thnh nhn t.Hng dnCc c ca 18 l 1, 2, 3, 6, 9, 18.f(1) = 18, f(1) = 44, nn 1 khng phi l nghim ca f(x).D thykhng l s nguyn nn 3, 6, 9, 18 khng l nghim ca f(x). Ch cn 2 v 3. Kim tra ta thy 3 l nghim ca f(x). Do , ta tch cc hng t nh sau : = (x 3)(4x2 x + 6)H qu 4. Nu( l cc s nguyn) c nghim hu t , trong p, q Z v (p , q)=1, th p l c a0, q l c dng ca an .V d 10. Phn tch a thc f(x) = 3x3 - 7x2 + 17x - 5 thnh nhn t.Hng dnCc c ca 5 l 1, 5. Th trc tip ta thy cc s ny khng l nghim ca f(x). Nh vy f(x) khng c nghim nghuyn. Xt cc s, ta thy l nghim ca a thc, do a thc c mt nhn t l 3x 1. Ta phn tch nh sau :f(x) = (3x3 x2) (6x2 2x) + (15x 5) = (3x 1)(x2 2x + 5).IV.PHNG PHAP THM V BT CNG MT HNG T1. Thm v bt cng mt hng t lm xut hin hiu hai bnh ph ngV d 12. Phn tch a thc x4 + x2 + 1 thnh nhn tLi giiCch 1 : x4 + x2 + 1 = (x4 + 2x2 + 1) x2 = (x2 + 1)2 x2 = (x2 x + 1)(x2 + x + 1).Cch 2 : x4 + x2 + 1 = (x4 x3 + x2) + (x3 + 1) = x2(x2 x + 1) + (x + 1)(x2 x + 1)= (x2 x + 1)(x2 + x + 1).Gv: Nguyn Vn TTrng THCS Thanh M28Gio n BDHSG Ton 8 Nm hc : 2011-2012 Cch 3 : x4 + x2 + 1 = (x4 + x3 + x2) (x3 1) = x2(x2 + x + 1) + (x 1)(x2 + x + 1)= (x2 x + 1)(x2 + x + 1).V d 13. Phn tch a thc x4 + 16 thnh nhn tLi giiCch 1 : x4 + 4 = (x4 + 4x2 + 4) 4x2 = (x2 + 2)2 (2x)2 = (x2 2x + 2)(x2 + 2x + 2)Cch 2 : x4 + 4 = (x4 + 2x3 + 2x2) (2x3 + 4x2 + 4x) + (2x2 + 4x + 4) = (x2 2x + 2)(x2 + 2x + 2)2. Thm v bt cng mt hng t lm xut hin nhn t chungV d 14. Phn tich a thc x5 + x - 1 thanh nhn tLi giaiCach 1.x5 + x - 1 = x5 - x4 + x3 + x4 - x3 + x2 - x2 + x - 1 = x3(x2 - x + 1) - x2(x2 - x + 1) - (x2 - x + 1) = (x2 - x + 1)(x3 - x2 - 1).Cach 2. Thm vab t x2 :x5 + x - 1 = x5 + x2 - x2 + x - 1 = x2(x3 + 1) - (x2 - x + 1) = (x2 - x + 1)[x2(x + 1) - 1] = (x2 - x + 1)(x3 - x2 - 1).V d 15. Phn tich a thc x7 + x + 1 thanh nhn tLi giaix7 + x2 + 1 = x7 x + x2 + x + 1 = x(x6 1) + (x2 + x + 1) = x(x3 1)(x3 + 1) + (x2+ x + 1)= x(x3 + 1)(x - 1)(x2 + x + 1) + ( x2 + x + 1) = (x2 + x + 1)(x5 - x4 x2- x + 1)Lu y : Cac a thc dangx3m + 1 + x3n + 2 + 1 nh x7 + x2 + 1, x4 + x5 + 1 u cha nhn tla x2 + x + 1.V.PHNG PHA P I BINt n ph a v dng tam thc bc hairi s dng cac phng php c bn.V d 16. Phn tich a thc sau thanh nhn t :Gv: Nguyn Vn TTrng THCS Thanh M29Gio n BDHSG Ton 8 Nm hc : 2011-2012 x(x + 4)(x + 6)(x + 10) + 128Li giaix(x + 4)(x + 6)(x + 10) + 128 = (x2 + 10x)(x2 + 10x + 24) + 128t x2 + 10x + 12 = y, a thc a cho co dang :(y - 12)(y + 12) + 128 = y2 - 16 = (y + 4)(y - 4) = (x2 + 10x + 16)(x2 + 10x + 8) = (x + 2)(x + 8)(x2 + 10x + 8)Nhn xt: Nh phng php i bin ta a a thc bc 4 i vi x thnh a thc bc 2 i vi y.V d 17. Phn tich a thc sau thanh nhn t :A = x4 + 6x3 + 7x2 - 6x + 1.Li giaiCach 1. Gia s x 0. Ta vit a thc di dang :.t thi . Do o :A = x2(y2 + 2 + 6y + 7) = x2(y + 3)2 = (xy + 3x)2= = (x2 + 3x - 1)2.Dang phn tich nay cung ung vi x = 0.Cach 2. A = x4 + 6x3 - 2x2 + 9x2 - 6x + 1 = x4 + (6x3 -2x2) + (9x2 - 6x + 1) = x4 + 2x2(3x - 1) + (3x - 1)2 = (x2 + 3x - 1)2.VI.PHNG PHAP H S BT INHV d 18. Phn tich a thc sau thanh nhn t :x4 - 6x3 + 12x2 - 14x - 3Li giaiTh vi x= 1; 3 khng l nghim ca a thc, a thc khng conghim nguyn cng khng conghim hu t. Nh vy a thc trn phn tich c thnh nhn tthiphi c dng(x2 + ax + b)(x2 + cx + d) = x4 +(a + c)x3 + (ac+b+d)x2 + (ad+bc)x + bdGv: Nguyn Vn TTrng THCS Thanh M30Gio n BDHSG Ton 8 Nm hc : 2011-2012 = x4 - 6x3 + 12x2 - 14x + 3.ng nht cac h sta c :Xet bd= 3 vi b, d Z, b { 1, 3}. Vi b = 3 thid = 1, hiu kin trn trthanh 2c = -14 - (-6) = -8. Do o c = -4, a = -2.Vy x4 - 6x3 + 12x2 - 14x + 3 = (x2 - 2x + 3)(x2 - 4x + 1).VII.PHNG PHA P XET GIA TRIRINGTrong phng phap nay, trc ht ta xac inh dang cac nhn tcha bin cua a thc, ri gan cho cac bin cac gia tri cu th xac inh cac nhn t con lai.V d 19. Phn tich a thc sau thanh nhn t :P = x2(y z) + y2(z x) + z(x y).Li giaiThay x bi y th P = y2(y z) + y2( z y) = 0. Nh vy P cha tha s (x y).Ta thy nu thay x bi y, thay y bi z, thay z bi x th p khng i (a thc P c th hon v vng quanh). Do nu P cha tha s (x y) th cng cha tha s (y z), (z x). Vy P c dng k(x y)(y z)(z x).Ta thy k phi l hng s v P c bc 3 i vi tp hp cc bin x, y, z, cn tch (x y)(y z)(z x) cng c bc 3 i vi tp hp cc bin x, y, z.V ng thcx2(y z) + y2(z x) + z2(x y) = k(x y)(y z)(z x) ng vi mi x, y, z nn ta gn cho cc bin x ,y, z cc gi tr ring, chng hn x = 2, y = 1, z = 0 ta c:4.1 + 1.(2) + 0 = k.1.1.(2)suy ra k =1Vy P = (x y)(y z)(z x) = (x y)(y z)(x z)VIII.PHNG PHAP A VMT S A THC C BIT1. a va thc : a3 + b3 + c3 - 3abcVi du 20. Phn tich a thc sau thanh nhn t :a) a3 + b3 + c3 - 3abc.Gv: Nguyn Vn TTrng THCS Thanh M31Gio n BDHSG Ton 8 Nm hc : 2011-2012 b)(x - y)3 + (y - z)3 + (z - x)3.Li giaia) a3 + b3 + c3 - 3abc = (a + b)3 - 3a2b - 3ab2 + c3 - 3abc= [(a + b)3 + c3] - 3ab(a + b + c)= (a + b + c)[(a + b)2 - (a + b)c + c2] - 3ab(a + b + c)= (a + b + c)(a2 + b2 + c2 - ab - bc -ca)b)tx - y = a, y - z = b, z - x = c thi a + b + c. Theo cu a) ta co :a3 + b3 + c3 - 3abc = 0 a3 + b3 + c3 = 3abc.Vy (x - y)3 + (y - z)3 + (z - x)3 = 3(x - y)(y - z)(z - x)2. a va thc : (a + b + c)3 - a3 - b3 - c3Vi du 21. Phn tich a thc sau thanh nhn t :a) (a + b + c)3 - a3 - b3 - c3.b)8(x + y + z)3 - (x + y)3 - (y + z)3 - (z + x)3.Li giaia) (a + b + c)3 - a3 - b3 - c3 = [(a + b) + c]3 - a3 - b3 - c3= (a + b)3 + c3 + 3c(a + b)(a + b + c) - a3 - b3 - c3= (a + b)3 + 3c(a + b)(a + b + c) - (a + b)(a2 - ab + b2)= (a + b)[(a + b)2 + 3c(a + b + c) - (a2 - ab + b2)]= 3(a + b)(ab + bc + ca + c2) = 3(a + b)[b(a + c) + c(a + c)]= 3(a + b)(b + c)(c + a).b)t x + y = a, y + z = b, z + x = c thi a + b + c = 2(a + b + c).a thc a cho co dang : (a + b + c)3 - a3 - b3 - c3 Theo kt qua cu a) ta co:(a + b + c)3 - a3 - b3 - c3 = 3(a + b)(b + c)(c + a)Hay 8(x + y + z)3 - (x + y)3 - (y + z)3 - (z + x)3= 3(x + 2y + z)(y + 2z + x)(z + 2x + y)Gv: Nguyn Vn TTrng THCS Thanh M32Gio n BDHSG Ton 8 Nm hc : 2011-2012 II. Bi tp: Bi tp 1: Phn tch a thc thnh nhn t.1. 16x3y + 0,25yz321. (a + b + c)2 + (a + b c)2 4c22. x 4 4x3 + 4x222. 4a2b2 (a2 + b2 c2)23. 2ab2 a2b b323. a 4 + b4 + c4 2a2b2 2b2c2 2a2c24. a 3 + a2b ab2 b324. a(b3 c3) + b(c3 a3) + c(a3 b3)5. x 3 + x2 4x - 4 25. a 6 a4 + 2a3 + 2a26. x 3 x2 x + 1 26. (a + b)3 (a b)37. x 4 + x3 + x2 - 1 27. X 3 3x2 + 3x 1 y38. x 2y2 + 1 x2 y228. X m + 4 + xm + 3 x - 110. x 4 x2 + 2x - 1 29. (x + y)3 x3 y311. 3a 3b + a2 2ab + b230. (x + y + z)3 x3 y3 z312. a 2 + 2ab + b2 2a 2b + 1 31. (b c)3 + (c a)3 + (a b)313. a 2 b2 4a + 4b 32. x3 + y3+ z3 3xyz14. a 3 b3 3a + 3b 33. (x + y)5 x5 y515. x 3 + 3x2 3x - 1 34. (x2 + y2)3 + (z2 x2)3 (y2 + z2)316. x 3 3x2 3x + 117. x 3 4x2 + 4x - 118. 4a2b2 (a2 + b2 1)2 19. (xy + 4)2 (2x + 2y)2 20. (a2 + b2 + ab)2 a2b2 b2c2 c2a2 Bi tp 2: Phn tch a thc thnh nhn t.1. x2 6x + 8 23. x3 5x2y 14xy22. x2 7xy + 10y224. x4 7x2 + 13. a2 5a - 14 25. 4x4 12x2 + 14. 2m2 + 10m + 8 26. x2 + 8x + 75. 4p2 36p + 56 27. x2 13x + 366. x3 5x2 14x 28. x2 + 3x 187. a4 + a2 + 1 29. x2 5x 24Gv: Nguyn Vn TTrng THCS Thanh M33Gio n BDHSG Ton 8 Nm hc : 2011-2012 8. a4 + a2 2 30. 3x2 16x + 59. x4 + 4x2 + 5 31. 8x2 + 30x + 710. x3 10x - 12 32. 2x2 5x 1211. x3 7x - 6 33. 6x2 7x 2012. x2 7x + 12 34. x2 7x + 1013. x2 5x 14 35. x2 10x + 1614. 4 x2 3x 1 36. 3x2 14x + 1115. 3 x2 7x + 4 37. 5x2 + 8x 1316. 2 x2 7x + 3 38. x2 + 19x + 6017. 6x3 17x2 + 14x 3 39. x4 + 4x2 - 518. 4x3 25x2 53x 24 40. x3 19x + 3019. x4 34x2 + 225 41. x3 + 9x2 + 26x + 2420. 4x4 37x2 + 9 42. 4x2 17xy + 13y221. x4 + 3x3 + x2 12x - 20 43. - 7x2 + 5xy + 12y222. 2x4 + 5x3 + 13x2 + 25x + 15 44. x3 + 4x2 31x - 70Bi tp 3: Phn tch a thc thnh nhn t.1. x4 + x2 + 1 17. x5 -x4 - 12. x4 3x2 + 9 18. x12 3x6 + 13. x4 + 3x2 + 4 19. x8 - 3x4 + 14. 2x4 x2 1 20. a5 + a4 + a3 + a2 + a + 15. x4y4 + 4 21. m3 6m2 + 11m - 66. x4y4 + 64 22. x4 + 6x3 + 7x2 6x + 17. 4 x4y4 + 1 23. x3 + 4x2 29x + 248. 32x4 + 1 24. x10 + x8 + x6 + x4 + x2 + 19. x4 + 4y425. x7 + x5 + x4 + x3 + x2 + 110. x7 + x2 + 1 26. x5 x4 x3 x2 x - 211. x8 + x + 1 27. x8 + x6 + x4 + x2 + 112. x8 + x7 + 1 28. x9 x7 x6 x5 + x4 + x3 + x2 + 1Gv: Nguyn Vn TTrng THCS Thanh M34Gio n BDHSG Ton 8 Nm hc : 2011-2012 13. x8 + 3x4 + 1 29. a(b3 c3) + b(c3 a3) + c(a3 b3)14. x10 + x5 + 115. x5 + x + 116. x5 + x4 + 1Bi tp 4: Phn tch a thc thnh nhn t.1.x2 + 2xy 8y2 + 2xz + 14yz 3z22.3x2 22xy 4x + 8y + 7y2 + 13.12x2 + 5x 12y2 + 12y 10xy 34.2x2 7xy + 3y2 + 5xz 5yz + 2z25.x2 + 3xy + 2y2 + 3xz + 5yz + 2z26.x2 8xy + 15y2 + 2x 4y 37.x4 13x2 + 368.x4 + 3x2 2x + 39.x4 + 2x3 + 3x2 + 2x + 1Bi tp 5: Phn tch a thc thnh nhn t:1.(a b)3 + (b c)3 + (c a)32.(a x)y3 (a y)x3 (x y)a33.x(y2 z2) + y(z2 x2) + z(x2 y2)4.(x + y + z)3 x3 y3 z35.3x5 10x4 8x3 3x2 + 10x + 86.5x4 + 24x3 15x2 118x + 247.15x3 + 29x2 8x 128.x4 6x3 + 7x2 + 6x 89.x3 + 9x2 + 26x + 24Bi tp 6: Phn tch a thc thnh nhn t.1.a(b + c)(b2 c2) + b(a + c)(a2 c2) + c(a + b)(a2 b2)2.ab(a b) + bc(b c) + ca(c a)3.a(b2 c2) b(a2 c2) + c(a2 b2)Gv: Nguyn Vn TTrng THCS Thanh M35Gio n BDHSG Ton 8 Nm hc : 2011-2012 4.(x y)5 + (y z)5 + (z x)55.(x + y)7 x7 y76.ab(a + b) + bc(b + c) + ca(c + a) + abc7.(x + y + z)5 x5 y5 z58. a(b2 + c2) + b(c2 + a2) + c(a2 + b2) + 2abc9. a3(b c) + b3(c a) + c3(a b)10. abc (ab + bc + ac) + (a + b + c) 1Bi tp 7: Phn tch a thc thnh nhn t.1.(x2 + x)2 + 4x2 + 4x 122.(x2 + 4x + 8)2 + 3x(x2 + 4x + 8) + 2x23.(x2 + x + 1)(x2 + x + 2) 124.(x + 1)(x + 2)(x + 3)(x + 4) 245.(x2 + 2x)2 + 9x2 + 18x + 206.x2 4xy + 4y2 2x + 4y 357.(x + 2)(x + 4)(x + 6)(x + 8) + 168.(x2 + x)2 + 4(x2 + x) 129.4(x2 + 15x + 50)(x2 + 18x + 72) 3x2Chuyn 1 :PHN TCH A THC THNH NHN TTit 1 3 : Cc v d v phng php giiV d 1: Phn tch a thc thnh nhn t a. ( ) ( ) 1 12 2+ + a x x ab. n nx x x + +31 .Gii: a. Dng phng php t nhn t chung ( ) ( ) 1 12 2+ + a x x a=x x a a ax +2 2( ) ( ) ( )( ) 1 ax a x a x a x axb. Dng phng php t nhn t chung ri s dng hng ng thcn nx x x + +31 .( ) ( ) 1 13 + x x xn( )( ) ( ) ( ) ( ) [ ]( )( ) 1 11 1 1 1 1 11 22 2+ + + + + + + + + + + n n nn nx x x xx x x x x x x x xV d 2: Phn tch a thc thnh nhn t :a. x8 + 3x4 + 4.b. x6 - x4 - 2x3 + 2x2 .Gii: a.Dng phng php tch hng t ri s dng hng ng thcGv: Nguyn Vn TTrng THCS Thanh M36Gio n BDHSG Ton 8 Nm hc : 2011-2012 x8 + 3x4 + 4 = (x8 + 4x4 + 4)- x4= (x4 + 2)2 - (x2)2 = (x4 - x2 + 2)(x4 + x2 + 2)b.Dng phng php t nhn t chung ,tch hng t ,nhm thch hp s dng hng ng thcx6 - x4 - 2x3 + 2x2 = x2(x4 - x2 - 2x +2)( ) ( ) [ ]( ) ( ) [ ] ( ) ( ) [ ]( ) [ ] 2 2 11 1 1 1 11 2 1 22 2 22 2 2 222 22 2 4 2+ + + + + + + + x x x xx x x x x xx x x x xV d 3: Phn tch a thc thnh nhn t :a.abc bc c b ac c a ab b a 4 2 4 4 22 2 2 2 2 2 + + +b. 2007 2006 20072 4+ + + x x xGii: a.Dng phng php tch hng t ri nhm thch hp:abc bc c b ac c a ab b a 4 2 4 4 22 2 2 2 2 2 + + +( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) [ ]( )( )( ) c a c b b ac b c c b a b a bc c ac ab b ab a bc b a c b a ac b a ababc bc c b ac abc c a ab b aabc bc c b ac c a ab b a + + + + + + + + + + + + + + +2 22 2 2 2 2 22 2 2 2 2 22 2 4 2 4 24 2 4 4 2222 2 2 2 2 22 2 2 2 2 2b.Dng phng php t nhn t chung ri s dng hng ng thc2007 206 20072 4+ + + x x x( )( )( ) ( )( )( ) 2007 11 2007 1 12007 2007 20072 22 22 4+ + + + + + + + + + + x x x xx x x x x xx x x xV d 4: Phn tch a thc thnh nhn t : a. abc c b a 33 3 3 + +b.( )3 3 3 3c b a c b a + + .Gii: S dng cc hng ng thc ( )( ) ab b a b a b a + + +2 2 3 3( )( ) [ ] ab b a b a 32 + + ( ) ( ) b a ab b a + + 33.Do : + + abc c b a 33 3 3( ) [ ] ( ) abc b a ab c b a 3 33 3 + + + ( ) ( ) ( ) [ ] ( )( )( ) ca bc ab c b a c b ac b a ab c c b a b a c b a + + + + + + + + + + + 2 2 22 23b.( ) ( ) [ ] ( )3 3 3 3 3 3 3c b a c b a c b a c b a + + + + +( )( ) ( ) [ ] ( ) ( )( ) ( ) ( )( )( ) b a c a c b ca bc ab a c bc bc b c b a c b a a c b a c b+ + + + + + + + + + + + + + + + 3 3 3 3 322 2 2 2V d 5: Cho a + b + c = 0. Chng minh rng :a3 + b3 + c3 = 3abc.Gii: V a + b + c = 0 ( ) ( )abc c b a abc c b ac b a ab b a c b a3 0 333 3 3 3 3 33 3 3 3 3 + + + + + + + + V d 6: Cho 4a2 + b2= 5ab, v 2a > b > 0. Tnh 2 24 b aabPGii: Bin i 4a2 + b2= 5ab 4a2 + b2- 5ab = 0Gv: Nguyn Vn TTrng THCS Thanh M37Gio n BDHSG Ton 8 Nm hc : 2011-2012 ( 4a - b)(a - b) = 0 a = b.Do 313 4222 2 aab aabPV d 7:Cho a,b,c v x,y,z khc nhau v khc 0. Chng minh rng nu: 1 ; 0 + + + +czbyaxzcybxa th1 ;222222 + +czbyaxGii: 0 0 0 + + + + + + cxy bxz ayzxyzcxy bxz ayzzcybxa11 . 212222222222222 + + + ++ + +

,_

+ + + +czbyaxabccxy bxz ayzczbyaxczbyaxczbyaxTit 4 -9 Bi tp vn dng - T luyn1. Phn tch a thc thnh nhn t :a.122 x xb.15 82+ + x xc.16 62 x xd.32 3+ + x x x2. Phn tch a thc thnh nhn t :( ) ( ) 15 2222 x x x x .3. Phn tch a thc thnh nhn t 1.(a - x)y3 - (a - y)x3 + (x - y)a3.2.bc(b + c) + ca(c + a) + ba(a + b) + 2abc.3.x2 y + xy2 + x2 z+ xz2+ y2 z + yz2 + 2xyz.4. Tm x,y tha mn: x2 + 4y2 + z2 = 2x + 12y - 4z - 14.5. Cho a +| b + c + d = 0. Chng minh rng a3 + b3 + c3 + d 3= 3(c + d)( ab + cd).6. Chng minh rng nu x + y + z = 0 th : 2(x5 + y5 + z5) = 5xyz(x2 + y2 + z2).7. Chng minh rng vi x,y nguyn th :A = y4 + (x + y) (x + 2y) (x + 3y) (x + 4y) l s chnh phng.8. Bit a - b = 7. Tnh gi tr ca biu thc sau: ( ) ( ) ( ) 1 3 1 12 2+ + + b a ab ab b b a a9. Cho x,y,z l 3 s tha mn ng thi:' + + + + + +1113 3 32 2 2z y xz y xz y x. Hy tnh gi tr biu thc P =( ) ( ) ( )1997 9 171 1 1 + + z y x .10.a.Tnh 2 2 2 2 2 2 2101 100 99 ... 4 3 2 1 + + + + .b.Cho a + b + c = 9 v a2 + b2 + c2 = 53.Gv: Nguyn Vn TTrng THCS Thanh M38Gio n BDHSG Ton 8 Nm hc : 2011-2012 Tnh ab + bc + ca.11. Cho 3 s x,y,z tha mn iu kin x + y + z = 0 v xy + yz + zx = 0. Hy tnh gi tr ca Biu thc :S = (x-1)2005 + (y - 1)2006 + (z+1)200712.Cho 3 s a,b,c tha iu kin : c b a c b a + + + +1 1 1 1. Tnh Q = (a25 + b25)(b3 + c3)(c2008 - a2008).==========o0o==========HNG DN:1. Phn tch a thc thnh nhn t :a. ( )( ) 3 4 122+ x x x xb. ( )( ) 5 3 15 82+ + + + x x x xc. ( )( ) 8 2 16 62 + x x x xd. ( )( ) 3 2 1 32 2 3+ + + + x x x x x x2. Phn tch a thc thnh nhn t :( ) ( ) ( )( ) 3 5 15 22 2 222+ x x x x x x x x .3. Phn tch a thc thnh nhn t 1.(a - x)y3 - (a - y)x3 + (x-y)a3( )( )( )( ) a y x a y a x y x + + 2.bc(b + c) + ca(c + a) + ba(a + b) + 2abc( )( )( ) a c c b b a + + + 3.x2 y + xy2 + x2 z+ xz2+ y2 z + yz2 + 2xyz( )( )( ) x z z y y x + + +4. x2 + 4y2 + z2 = 2x + 12y - 4z - 14( ) ( ) ( )2 2 22 | 3 2 1 + + z y x5. T a + b + c + d = 0 ( ) ( )3 3d c b a + + Bin i tip ta c :a3 + b3 + c3 + d 3= 3(c + d)( ab + cd).6. Nu x + y + z = 0 th : ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( )2 2 22 2 2 5 5 52 2 2 5 5 52 2 2 2 2 2 3 3 33 3 32* ; 6 2 2333z y x xyz zx yz xy xyzz y x xyz zx yz xy xyz z y xz y x xyz zx yz xy xyz z y xz y x xyz z y x z y xxyz z y x+ + + + + + + + + + + + + + + + + + + + + + + +Nhng:( ) ( )2 2 2 22 0 z y x zx yz xy xyz z y x + + + + + + (**)Thay (**) vo (*) ta c:2(x5 + y5 + z5) = 5xyz(x2 + y2 + z2).7. Vi x,y nguyn th :A = y4 + (x + y) (x + 2y) (x + 3y) (x + 4y)( )22 25 5 y xy x + + 8. Bin i ( ) ( ) ( ) ( ) ( ) 1 1 3 1 12 2 2+ + + + b a b a b a ab ab b b a a9. T ' + + + +113 3 3z y xz y x( ) ( )( )( ) x z z y y x z y x z y x + + + + + 33 3 3 3Gv: Nguyn Vn TTrng THCS Thanh M39Gio n BDHSG Ton 8 Nm hc : 2011-2012

+ + +000x zz yy x2 P10.a. S dng hng ng thc a2 - b2 ; S -=5151b. S dng hng ng thc (a + b + c)2; P = 1411. T gi thit suy ra: x2 + y2 + z2 = 0 suy ra : x = y = z = 0;S = 0

12. T: c b a c b a + + + +1 1 1 1. : (a + b)(b + c)(c + a) = 0 Tnh cQ = 0==========o0o========== Chuyn 4:BT NG THC V GI TR LN NHT, GI TR Nh NHTA.CC PHNG PHP CHNG MINH BT NG THC1.Ph ng php i t ng ng *) chng minh:B A Ta bin i n nB A B A B A ...1 1(y l bt ng thc ng)Hoc t bt ng thc ngn nB A, ta bin i B A B A B A B An n n n 1 1 1 1...V d 1.1 Chng minh rng vi mi s thc a, b, c ta lun c: a) 2 2 2) ( ) ( 2 b a b a + +(1)b)ca bc ab c b a + + + +2 2 2(1)Gi ia) Ta c: (1) 0 ) ( 0 2 0 ) ( ) ( 22 2 2 2 2 2 + + + b a ab b a b a b a(2)Do bt ng thc (2) ng nn bt ng thc (1) c chng minh.b) Ta c: (2) 0 ) 2 ( ) 2 ( ) 2 ( 0 ) ( 2 ) ( 22 2 2 2 2 2 2 2 2 + + + + + + + + + a ca c c bc b b ab a ca bc ab c b a 0 ) ( ) ( ) (2 2 2 + + a c c b b a (2)Bt ng thc (2) lun ng suy ra iu phi chng minh.V d 1.2 Chng minh rng:a) ) )( ( ) ( 23 3 4 4b a b a b a + + +(1)b) ) )( ( ) ( 33 3 3 4 4 4c b a c b a c b a + + + + + +(1)Gi i.a) Ta c: (1) 0 ) ( ) ( 0 ) ( 2 23 4 3 4 4 3 3 4 4 4 + + + + + ab b b a a b ab b a a b a 0 ) )( ( 0 ) ( ) (3 3 3 3 b a b a b a b b a a

0432) ( 0 ) ( ) (222 2 2 211]1

+ ,_

+ + b ba b a b ab a b a(2)Do bt ng thc (2) ng suy ra iu phi chng minh.b) Ta c:(1)) ( 3 3 34 3 3 3 3 4 3 3 4 4 4 4c bc ac c b ab b c a b a a c b a + + + + + + + + + +

Gv: Nguyn Vn TTrng THCS Thanh M40Gio n BDHSG Ton 8 Nm hc : 2011-2012 0 ) ( ) ( ) (3 3 4 4 3 3 4 4 3 3 4 4 + + + + + ac c a a c bc c b c b ab b a b a 0 ) ( ) ( ) ( ) ( ) ( ) (2 2 2 2 2 2 2 2 2 + + + + + + + + a ca c a c c bc b c b b ab a b a (2)Do bt ng thc (2) ng suy ra iu phi chng minh.V d 1.3 . Chng minh rng:a) 2 2 2 2 2) ( ) )( ( by ax y x b a + + +(1) b) 2 2 2 2 2 2) ( ) ( d b c a d c b a + + + + + + (1)Gi ia)Ta c:(1) 2 2 2 2 2 2 2 2 2 2 2 22 y b abxy x a y b x b y a x a + + + + +

0 ) ( 0 22 2 2 2 2 + bx ay x b abxy y a (2)Bt ng thc (2) lun ng suy ra iu phi chng minh.b) Ta c:(1) 2 2 2 2 2 2 2 2 2 2) ( ) ( ) )( ( 2 d b c a d c b a d c b a + + + + + + + + +

bd ac d c b a + + + ) )( (2 2 2 2(2)+) Nu 0 < +bd ac th (2) ng.+) Nu 0 +bd ac th (2) 2 2 2 2 2) ( ) )( ( bd ac d c b a + + + 0 ) ( 22 2 2 2 2 2 2 2 2 2 2 2 2 + + + + bd ad d b abcd c a d b c b d a c a(*)Bt ng thc (*) lun ng suy ra iu phi chng minh.V d 1.4 . Cho a, b, c > 0. Chng minh rng:c b aaccbba+ + + +2 2 2(1)Gi iTa c:(1)02 2 2 2 3 3 + + ab c ac b bc a b c a b c a

0 ) 2 ( ) 2 ( ) 2 (2 2 2 2 2 2 + + + + + c bc b bc c ab a ac b ab a ab

0 ) ( ) ( ) (2 2 2 + + c b bc c a ac b a ab (*)Bt ng thc (*) lun ng suy ra iu phi chng minh.V d 1.5 . Cho a, b, c > 0.CMR: abc c b a c b a c b a c b a 3 ) ( ) ( ) (2 2 2 + + + + + (1)Gi iGi s0 , > c b a.Khi ta c: (1) 0 ) ( ) ( ) ( 32 2 2 + + + c b a c b a c b a c b a abc0 ) ( ) ( ) ( 32 2 2 + + + + + ab bc ac c c ac ba bc b b bc ac ab a a abc0 ) )( ( ) )( ( ) )( ( + + b c a c c a b c b b c a b a a0 ) )( ( ) )( (2 2 + + c b c a c bc b ac a b a0 ) )( ( ) ( ) (2 + + c b c a c c b a b a(*)Bt ng thc (*) lun ng (V 0 , > c b a). Suy ra iu phi chng minh.2. Phng php bin i ng nht.Gv: Nguyn Vn TTrng THCS Thanh M41Gio n BDHSG Ton 8 Nm hc : 2011-2012 chng minh BT:B A . Ta bin i biu thcB Athnh tng cc biu thc c gi tr khng m.V d 2.1 . Chng minh rng:a)ad ac ab d c b a + + + + +2 2 2 2(1) b)bc ac ab c b a 8 4 4 4 42 2 2+ + +(1)Gi ia) Ta c: 4 4 4 422222222 2 2 2ad adac acab abaad ac ab d c b a +

,_

+ +

,_

+ +

,_

+ + + +04 2 2 222 2 2 + ,_

+ ,_

+ ,_

adacabaad ac ab d c b a + + + + + 2 2 2 2 (pcm)b) Ta c:) 8 4 ( 4 ) 4 4 ( 8 4 4 4 42 2 2 2 2 2bc ac c b ab a bc ac ab c b a + + + + + +

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

0 ) 2 2 (2 + c b abc ac ab c b a 8 4 4 4 42 2 2+ + + (pcm)V d 2.2 .Chng minh rng:a) 3 3 3) ( ) ( 4 b a b a + +vi 0 , > b a.b) 3 3 3 3 3 3) ( ) ( ) ( ) ( 8 a c c b b a c b a + + + + + + +vi 0 , , > c b a.c)abc c b a c b a 24 ) (3 3 3 3+ + + + + vi 0 , , c b a.Gi ia) Ta c:[ ] 0 ) )( ( 3 ) ( ) ( 4 ) ( ) ( ) ( 42 2 2 2 3 3 3 + + + + + b a b a b a b ab a b a b a b a (pcm)b) Ta c: 0 ) )( ( 3 ) )( ( 4 ) )( ( 3 ... ) ( ) ( ) ( ) ( 82 2 2 3 3 3 3 3 3 + + + + + + + + + + c b c b c a c a b a b a a c c b b a c b ac) Ta c: 0 ) ( 3 ) ( 3 ) ( 3 .. 24 ) (2 2 2 3 3 3 3 + + + + c b a b a c c a b abc c b a c b aV d 2.3 .Vi a, b, c > 0. Chng minh rng:a) b a b a + +4 1 1b) c b a c b a + + + +9 1 1 1c) 23+++++ c aca cbc baGi ia) 0) () () (2) (4 ) ( ) ( 4 1 12 2 2+++ + + + ++ +b a abb ab a abb ab ab a abab c b a b a bb a b a b)) () ( ) ( ) ( 9 1 1 1c b a abcc b a ab c b a ac c b a bcc b a c b a + ++ + + + + + + ++ + + +

0) ( ) ( ) (2 2 2++aca cbcc babb ac)

,_

++ ,_

++ ,_

+ +++++ 21212123b aca cbc bac aca cbc ba

) ( 2) ( ) () ( 2) ( ) () ( 2) ( ) (b ab c a ca cc b a bc bc a b a+ + ++ + ++ + Gv: Nguyn Vn TTrng THCS Thanh M42Gio n BDHSG Ton 8 Nm hc : 2011-2012

,_

++ + ,_

++ + ,_

++ c b b aa cb a c ac ba c c bb a1 1) (21 1 1) (21 1 1) (21

0) )( () () )( () () )( () (212 2 21]1

+ +++ +++ +c b b aa cb a c ac bc b c ab aV d 2.4 a) Cho 0 ; b a. Chng minh rng:ab b a 2 + (Bt ng thc C Si)b) Cho 0 ; ; c b a. Chng minh rng: 33 abc c b a + + (Bt ng thc C Si)c) Cho c b a v z y x . Chng minh rng: ) ( 3 ) )( ( cz by ax z y x c b a + + + + + +(BT Tr-B-Sp)Gi ia) Ta c: 0 ) ( 22 + b a ab b ab) Ta c: [ ] 0 ) ( ) ( ) ( ) (2132 3 3 2 3 3 2 3 3 3 3 3 3 + + + + + + c a b c b a c b a abc c b ac) Ta c: 0 ) )( ( ) )( ( ) )( ( ) ( 3 ) )( ( + + + + + + + + a c z x c b x z b a x y cz by ax z y x c b aV d 2.5 Cho a, b, c > 0. Chng minh:a) c b acabbcaabc+ + + + b) ) ( 32 2 2c b acabbcaabc+ + + +Gi ia) Ta c:0) (.2) (.2) (.2) (2 2 2++ + + + +bcc b acaa c babb a cc b acabbcaabcb) Ta c: 0) ( ) ( ) (21) ( 322 222 222 22 2 2211]1

,_

+

,_

+

,_

+ + ,_

+ +caa c bbcc b aabb a cc b acabbcaabc + + ,_

+ + ) ( 32 2 22c b acabbcaabc) ( 32 2 2c b acabbcaabc+ + + +V d 2.6 .Chng minh rng: a)ab b a ++++ 1211112 2 (nu 1 > ab)b) ab b a ++++ 1211112 2 (nu 22 2 + b a )c)ab b a + 1211112 2(nu 1 , 1 < < b a)d) ab b a ++++ 11) 1 (1) 1 (12 2 (nu 0 , > b a)Gi iGv: Nguyn Vn TTrng THCS Thanh M43Gio n BDHSG Ton 8 Nm hc : 2011-2012 a) Ta c: 0) 1 )( 1 )( 1 () 1 ( ) (111111111211112 222 2 2 2+ + + ,_

+++ ,_

++++++ ab b aab b aab b ab a ab b ab) Ta c: ) 1 )( 1 )( 1 () 1 ( ) (1211112 222 2ab b aab b aab b a + + + ++++V ' + +0 10 1122 2a ba bb aa b0) 1 )( 1 )( 1 () 1 ( ) (2 22+ + + ab b aab b a (pcm)c) Ta c: 0) 1 )( 1 )( 1 () 1 ( ) (111111111211112 2 22 2 22 2 2 2 ,_

+ ,_

+ ab b ab a b aab b ab a ab b ad) Ta c: 0) 1 ( ) 1 ( ) 1 () 1 ( ) (11) 1 (1) 1 (12 22 22 2+ + + + ++++ ab b aab b a abab b a3. Phng php s dung tnh cht ca bt ng thc C s ca phng php ny l cc tnh cht ca bt ng thc v mt s bt ng thc c bn nh:+) Nu b a v c b th c a +) Nu b a v 0 . > b a th b a1 1+) Nu ' 00n mb a th n b m a . . .+) 0 ) (2 b a+) Nu 0 , > b a th b a b a + +4 1 1V d 3.1 Cho 1 > +b a.Chng minh: 814 4> +b a Gi iTa c: 812) (212) (0 ) (2 2 24 422 2 2>+ + >+ + b ab ab ab a b aV d 3.2 Vi a, b, c > 0. Chng minh rng:a)ca bc abaccbba+ + + +3 3 3b)c b aaccbba+ + + +232323Gi ia) Ta c: 2 22y xy xyx + (vi 0 , > y x)Gv: Nguyn Vn TTrng THCS Thanh M44Gio n BDHSG Ton 8 Nm hc : 2011-2012 ca bc ab a ca c c bc b b ab aaccbba+ + + + + + + + + ) ( ) ( ) (2 2 2 2 2 23 3 3b) Ta c: y xyx2 323 (vi 0 , > y x) c b a a c c b b aaccbba+ + + + + + ) 2 3 ( ) 2 3 ( ) 2 3 (232323V d 3.3 Cho a, b, c > 0. CMR:a)c b acabbcaabc+ + + +b) 22 2 2c b ab aca cbc ba + ++++++Gi ia) Ta d dng chng minh c + cbcaabc2(pcm)b) Ta d dng chng minh c +++ac bc ba42(pcm)V d 3.4Cho 0 , , > z y x tho mn iu kin: 41 1 1 + +z y x. Chng minh rng:1212121+ +++ +++ + z y x z y x z y xGii: Ta c:Vi 0 , > b a th b a b a + +4 1 1. T suy raz y x x z y x x z y x z y x1 1 2 4 4) ( ) (16216+ + ++++ + ++ +(1)z y x z y y x z y y x z y x1 2 1 4 4) ( ) (16216+ + ++++ + ++ +(2)z y x x z z y x z z y z y x2 1 1 4 4) ( ) (16216+ + ++++ + ++ + (3)T (1), (2) v (3) + + + +++ +++ +164 4 4216216216z y x z y x z y x z y x (pcm)V d 3.5.a) Cho a, b, c l di ba cnh ca mt tam gic. Chng minh rng:c b a b a c a c b c b a1 1 1 1 1 1+ + ++ ++ +b) Cho a, b, c > 0 tha mn:ca bc ab abc + + . Chng minh:1633 213 213 21+ +++ +++ + b a c a c b c b aGi ip dng BT b a b a + +4 1 1 (vi ) 0 , > b aTa c:b b a c b c b a224 1 1 ++ +(1)Gv: Nguyn Vn TTrng THCS Thanh M45Gio n BDHSG Ton 8 Nm hc : 2011-2012 c c b a c a c b224 1 1 ++ +(2)a a c b a b a c224 1 1 ++ +(3)T (1), (2) v (3) (pcm)b) Tng t:p dng BT b a b a1 1 4+ + (vi ) 0 , > b aTa c:c b a c c b a c c b a c c b a 2121 1 1) ( 24 4) ( 2 ) (163 216+ + + ++++ + ++ +(1)a c b a a c b a a c b a a c b 2121 1 1) ( 24 4) ( 2 ) (163 216+ + + ++++ + ++ +(2) b a c b b a c b b a c b b a c 2121 1 1) ( 24 4) ( 2 ) (163 216+ + + ++++ + ++ + (3)T (1), (2) v (3)

,_

+ + ,_

+ +++ +++ + c b a b a c a c b c b a1 1 133 213 213 2116 (*)Mt khc:11 1 1 + + + + c b aca bc ab abc(**)T (*) v (**) (pcm)V d 3.5 Cho a, b, c > 0. Chng minh rng:a) 2 < < m y x th m ym xyx++ Gii: a) P = 8 ) 4 )( 2 ( 8 ) 6 ( y y y y. Vy GTLN ca P = 8. t c khi x = 2, y = 4 b) Q = a a S S a y a y a a S y y S ) ( ) )( ( ) ( ) ( + . Vy GTLN ca Q = (S a)a t c khi x = S a, y = a b) a th c b c hai hai bi n Cho a thc: P(x,y) = ax2 + bxy + cy2 + dx + ey + h(1), vi a, b, c 0Ta thng a P(x, y) v dng: P(x, y) = mF2(x, y) + nG2(y) + k(2)P(x, y) = mH2(x, y) + nG2(x) + k(3)Trong G(y), H(x) l hai biu thc bc nht mt n, H(x, y) l biu thc bc nht hai n.Chng hn nu ta bin i (1) v (2) via, (4ac b2) 0.a 4Pah aey adx acy abxy x a y x 4 4 4 4 4 4 ) , (2 2 2+ + + + + 2 2 2 2 2 2 24 ) 2 ( 2 ) 4 ( 2 4 4 4 d ah y bd ae b ac bdy adx abxy d y b x a + + + + + + + + + 222222 24) 2 (442) 4 ( ) 2 (b acbd aeahdb acbd aey b ac d by ax + ,_

+ + + + (Tng t nhn hai v ca (1) vi 4c chuyn v (3))V d 3.1 a) Tm GTNN ca P = x2 + y2 + xy + x + yb) Tm GTLN ca Q = 5x2 2xy 2y2 + 14x + 10y 1Gi ia) 4P = 1 2 3 ) 2 4 4 1 4 ( 4 4 4 4 42 2 2 2 2 + + + + + + + + + + + y y y x xy y x y x xy y x 3434313 ) 1 2 (22 ,_

+ + + + y y xVy GTNN ca P = 34. t c khi 31 y x.b) 5Q 80 ) 6 3 ( ) 7 5 ( 5 50 70 10 10 252 2 2 2 + + + + y y x y x y xy x. Q 16 16 ) 2 (59) 7 5 (512 2 + y y xVy GTLN ca Q = 16. t c khi x = 1, y = 2V d 3.2 Gv: Nguyn Vn TTrng THCS Thanh M53Gio n BDHSG Ton 8 Nm hc : 2011-2012 Tm cp s (x, y) vi y nh nht tha mn iu kin: x2 + 5y2 + 2y 4xy 3 = 0 (*)Gi iTa c (*) 1 3 0 ) 1 )( 3 ( 4 ) 1 ( 4 ) 1 ( ) 2 (2 2 2 + + + + y y y y y y x.Vy GTNN ca y = 3. t c khi x = 6. Vy cp s (x, y) = (6; 3)V d 3.3 Cho x, y lin h vi nhau bi h thc:x2 + 2xy + 7(x + y) + 7y2 + 10 = 0 (**). Hy tm GTLN, GTNN ca biu thc:S = x + y + 1.Gi iTa c (**) 0 40 4 28 28 8 42 2 + + + + + y y x xy x 9 4 ) 7 2 2 (2 2 + + + y y x 0 ) 2 )( 5 ( 9 ) 7 2 2 (2 + + + + + + y x y x y x ' + + + +0 20 5y xy x(v 5 2 + + + + y x y x) 4 S1 Vy GTNN ca S = 4. t c khi x = 5, y = 0. GTLN ca S = 1. t c khi x= 2, y = 0.II. PH NG PHP MI N GI TR V d 1 . Tm GTLN, GTNN ca A = 13 2 422++ +xx x.Gi iBiu thc A nhn gi tr a khi phng trnh:a = 13 2 422++ +xx x c nghim 0 3 . 2 4 ) 1 (2 + a x x a c nghim. (1)Nu a = 1 th phng trnh (1) c nghim x = 42 .Nu a1 th phng trnh (1) c nghim khi 5 1 0 5 4 0 '2 + + a a a .Vy GTNN ca biu thc A l - 1. t c khi2 x . V GTLN ca biu thc A l 5. at c khi 22 x .V d 2 . Tm GTLN, GTNN ca biu thc:B = 71 22 2+ ++ +y xy x.Gi iBiu thc B nhn gi tr b khi phng trnh b = 71 22 2+ ++ +y xy x c nghim.Gv: Nguyn Vn TTrng THCS Thanh M54Gio n BDHSG Ton 8 Nm hc : 2011-2012 0 1 7 22 2 + + b y by x bx(2)c nghim.Trong x ln, y l tham s v b l tham s c iu kin.Nu b = 0 0 1 2 + + y x Nu b 0 th PT (2) c nghim x khi0 ) 1 7 2 ( 4 1 02 + b y by b(3)Coi (3) l bt phng trnh n y. BPT ny xy ra vi mi gi tr ca y khi 0 ) 1 4 28 ( 4 162 2 2 + + + b b b bVy GTNN ca biu thc B = 145. t c khi 57 x v 514 y. GTLN ca biu thc B = 21. t c khi 1 x v 2 y.III. phng php s dng Bt ng thc.1. S dung BT C - Si.V d 1.1 . Tm GTLN, GTNN ca biu thc: A = x x 3 7 5 3 + vi 3735 x.Ta c: A2 = ) 3 7 )( 5 3 ( 2 2 ) 3 7 )( 5 3 ( 2 3 7 5 3 x x x x x x + + + A22 A 2 . Vy gi tr nh nht ca biu thc A l2 . t c khi 35 x v 37 y. (Biu thc c cho di dng tng hai cn thc. Hai biu thc ly cn c tng l hng s)V d 1.2 . Cho x, y > 0 tho mn iu kin x + y 6. Tm GTNN ca biu thc: P y xy x8 62 3 + + + .Gi iTa c: P 19 4 6 98.226.232 6 .23 82623) (23 + + + + + + + + + yyxxyyxxy x.Vy gi tr nh nht ca biu thc P l 19. t c khi x = 2 v y = 4.V d 1.3 . Tm GTLN ca biu thc : M = xyx y y x 3 2 + Vi 2 ; 3 y x.Gi i Ta c :M = yyxx 2 3 +

p dng BT C - Si ta c : Gv: Nguyn Vn TTrng THCS Thanh M55Gio n BDHSG Ton 8 Nm hc : 2011-2012 )) 42263 32) 2 ( 22) 3 ( 3yyxxyyxx M 4263+ Vy Gi tr ln nht ca biu thc M l 4263+ . t c khi x = 6 v y = 4.V d 1.4 . Cho x, y, z > 0 tho mn: 2111111+++++ z y x (1). Tm GTLN ca biu thc P = xyz.Gi iT (1)) 1 )( 1 (21 1 11111111z yyzzzyyz y x + ++++ ,_

+ +

,_

+ +Tng t: ) 1 )( 1 (211x zzxy + ++V ) 1 )( 1 (211y xxyz + ++Nhn v vi v ca ba BT trn ta c: P = xyz 81 Du = xy ra khi 21 z y x.Vy GTLN ca biu thc P l 81. t c khi 21 z y x.V d 1.5 . Cho 0 < x < 1, Tm GTNN ca biu thc:Q = x x413+.Gi iTa c: P = 2) 3 2 ( 7) 1 ( 4.132 7) 1 ( 413+ + ++ xxxxxxxx GTNN ca biu thc:Q = x x413+ l 2) 3 2 ( +. t c khi 2) 1 3 () 1 ( 413 xxxxx(t P = cxx bxax++) 1 ( 413 ri ng nht h s suy ra a = b = 1; c = 7)V d 1.6 . Cho x, y, z, t > 0. Tm GTNN ca biu thc: M =t xx zx zz yz yy ty tt x+++++++.Gi iGv: Nguyn Vn TTrng THCS Thanh M56Gio n BDHSG Ton 8 Nm hc : 2011-2012 p dng bt ng thc C-Si ta c:b a b a + +4 1 1vi a, b > 0.Ta c : M = M + 4 4 = 4 1 1 1 1 ,_

+++ ,_

+++

,_

+++

,_

++t xx zx zz yz yy ty tt x= 4 +++++++++++t xt zx zx yz yz ty ty x= 41 1) (1 1) (

,_

++++ +

,_

++++t x z yt zx z y ty x

0 4) ( 4 ) ( 4 + + ++++ + ++t z y xt zt z y xy xVy GTNN ca biu thc: M =t xx zx zz yz yy ty tt x+++++++ l 0. t c khi x = y v z = t.2. S dng BT Bunhiacopski (BCS)V d 2.1. Cho x, y, z tha mn: xy + yz + zx = 1. Tm GTNN ca biu thc: A = x4 + y4 + z4Giip dng BT Bunhia Copski ta c:2 2 2 2 2 2 2 2 2 2 2) ( ) )( ( ) ( 1 z y x z y x z y x zx yz xy + + + + + + + + ) )( 1 1 1 ( ) ( 14 4 4 2 2 2 2z y x z y x + + + + + + P 31 Gi tr nh nht ca biu thc P l 31 t c khi 33 z y xV d 2.2. Tm GTNN caP = c b acb a cba c ba ++ ++ +16 9 4 trong a, b, c l di ba cnh ca mt tam gic.GiiP = 2292121921416 9 4 ,_

+ ++ ,_

+ ++ ,_

+ + ++ ++ + c b acb a cba c bac b acb a cba c ba 229 16 9 4.2 ,_

++ ++ +

,_

+ +c b a b a c a c bc b a 26229 81.2 229) ( ) ( ) () 4 3 2 (.22 + ++ + 1]1

+ + + + ++ +

,_

+ +c b ac b ac b a b a c a c bc b aVy gi tr nh nht ca biu thc P l 26 t c khi 5 6 7c b a V d 2.3.Tm gi tr nh nht ca Q = c acb cba ba ++ ++ + 1 1 1. Trong 0 , , > c b a v 1 + + c b a.GiiGv: Nguyn Vn TTrng THCS Thanh M57Gio n BDHSG Ton 8 Nm hc : 2011-2012 Ta c: Q ac bca bc abc b ab a c a c b c b ac b ab aca cbc ba4 1) ( 3) () 2 ( ) 2 ( ) 2 () (2 2 222 2 + ++ ++ + + + ++ ++++++Vy gi tr nh nht ca biu thc Q l 1 t c khi 31 c b a V d 2.4 Cho a, b, c > 0 tha mn a + b + c = 1. Tm GTNN ca P = ca bc ab c b a1 1 1 12 2 2+ + ++ +Gii+ + + +ca bc ab ca bc ab9 1 1 1 P ca bc ab c b a + +++ +9 12 2 2

ca bc ab ca bc ab c b a + ++

,_

+ +++ +7) ( 24 12 2 2 30) (2199) () 2 1 (2 22+ ++ + +++ ++c b a ca bc ab c b aVy gi tr nh nht ca biu thc P l 30 t c khi a = b = c = 31 Tit 21-22I.BT NG THC C SI V CC H QU1.Chnh minh :(Vi a , b 0) (BT C-si)Gii: ( a b ) = a - 2ab + b 0 a + b 2ab .ng thc xy ra khi a = b2.Chng minh:. (Vi a , b 0) Gii:( a+b ) = (a - 2ab + b )+ 4ab = (a-b) + 4ab 0 + 4ab ( a + b ) 4ab .ng thc xy ra khi a = b.3.Chng minh: (Vi a , b 0) Gii:2(a + b) ( a+b ) = a-2ab+b = (a-b)02(a + b) ( a+b ). ng thc xy ra khi a = b.4.Chng minh:.(Vi a.b > 0)Gii:+= .Do ab 2 .Hay+2 . ng thc xy ra khi a = b5.Chng minh:.(Vi a.b < 0)Gii:Gv: Nguyn Vn TTrng THCS Thanh M58Gio n BDHSG Ton 8 Nm hc : 2011-2012 +=- .Do2 - -2. Hay +- 2. ng thc xy ra khi a = -b.6.Chng minh:. (Vi a , b > 0) Gii:+ - == 0 +. ng thc xy ra khi a = b.7.Chng minh rng: .Gii: 2(a +b +c) 2(ab+bc+ca)=(a-b) +(b-c) +(c-a)0 2(a +b +c) 2(ab+bc+ca) .Hay a +b +c ab+bc+ca . ng thc xy ra khi a = b;b = c;c = a a = b= c.Gv: Nguyn Vn TTrng THCS Thanh M59Gio n BDHSG Ton 8 Nm hc : 2011-2012 Tit 23-26 0 A B A B Cn lu tnh cht:02 A ng thc xy ra khi v ch khi A = 0 C th nhn hai v bt ng thc vi mt s khc 0 thch hpB.Bi tp vn dng:Chng minh cc bt ng thc sau1.a2 + 4b2 + 4c2 4ab - 4ac + 8bc 2.( ) e d c b a e d c b a + + + + + + +2 2 2 2 23.( )( )( )( ) 1 10 6 4 3 1 + x x x x4. a2 + 4b2 + 3c2 > 2a + 12b + 6c 145. 10a2 + 5b2 +12ab + 4a - 6b + 13 06. a2 + 9b2 + c2 + 219 > 2a + 12b + 4c7. a2 4ab + 5b2 2b + 5 48. x2 xy + y2 09. x2 + xy + y2 -3x 3y + 3 010. x2 + xy + y2 -5x - 4y + 7 011. x4 + x3y + xy3 +y4 012. x5 + x4y + xy4 +y5 0 vi x + y 013. a4 + b4 +c4 a2b2 + b2c2 + c2a214. (a2 + b2).(a2 + 1) 4a2b15. ac +bd bc + advi ( a b ;c d )16.22 22 2

,_

++ b a b a17.22 2 23 3

,_

+ ++ + c b a c b a18.bccaabaccbba+ + + +(vi a b c > 0)19.ababb a+ +912 ( Vi a,b > 0)20.c b a abccabbca 1 1 1+ + + +(Via,b,c > 0)===========o0o===========Gv: Nguyn Vn TTrng THCS Thanh M60Gio n BDHSG Ton 8 Nm hc : 2011-2012 HNG DN: Bi 1: Gi VT ca bt ng thc l A v VP ca bt ng thc l B (Nu khng ni g thm qui c ny c dng cho cc bi tp khc).Vi cc BT c du ; th cn tm iu kin ca cc bin ng thc xy ra.A B =( )22 2 b c a +Bi 2:4A 4B =( ) ( ) ( ) ( )2 2 2 22 2 2 2 e a d a c a b a + + + Bi 3:A 1 =( )( )( )( ) 9 6 4 3 1 + x x x x=( )23 + YBi 4:A B =( ) ( ) ( ) 1 1 3 3 2 12 2 2+ + + c b aBi 5:A = ( a 1)2 + (3a 2b)2 + (b + 3)2Bi 6:AB= ( a 1)2 +(3b 2)2 + (c - 2)2 +21Bi 7:A B =( ) ( )2 21 2 + b b aBi 8:x2 xy + y2 = 43222y yx + ,_

Bi 9:.x2 xy + y2 -3x 3y + 3 =( ) ( )( ) ( )2 21 1 1 1 + y y x x .Bin i tip nh bi 8Bi 10: Tng t bi 9Bi 11:x4 + x3y + xy3 +y4 = ( )( )2 2 2y x y xy x + + Bi 12: Tng t bi 11Bi 13: Xem v d 7Bi 14: A B = (a2 + b2).(a2 + 1) - 4a2bBi 15: A - B = ac + bd - bc - advi ( a b ;c d )= ( )( ) b a d c Bi 16:A - B = ( ) ( )422 2 2b a b a + +. Bi 17: Xem bi tp 16Bi 18:A - B =(a-c)(b-a)( .(Vi a bc 0)Bi 19:A - B = ( ) ( )abb a a b+ + 93 32 2

( Vi a,b > 0)Bi 20:A - B = ( ) ( ) ( )abcab ac ac bc bc ab2 2 2 + + (Via,b,c > 0)===========o0o===========Gv: Nguyn Vn TTrng THCS Thanh M61Gio n BDHSG Ton 8 Nm hc : 2011-2012 Tit 27-30TM GI TR LN NHT - GI TR NH NHT I: DNG ----------------------------------------------------------------------------------------------- Nu a > 0 : 2224ac-bax+ bx +c = 4a 2bP a xa _ + + , Suy ra 24ac-b = 4aMinPKhi bx=-2a Nu a < 0 :2224 a c+bax+ bx +c = 4 a 2bP a xa _ , Suy ra 24 a c+bax4 aM P Khi bx=2 aMt s v d:1. Tm GTNN ca A = 2x2 + 5x + 7Gii:A = 2x2 + 5x + 7 = 25 25 252( 2. ) 74 16 16x x + + += 2 2 25 25 56 25 5 31 52( ) 7 2( ) 2( )4 8 8 4 8 4x x x + + + + + +. Suy ra 31 58 4MinA Khi x .2. Tm GTLN ca A = -2x2 + 5x + 7Gii: A= -2x2 + 5x + 7 = -25 25 252( 2. ) 74 16 16x x + +=

2 2 25 25 56 25 5 81 52( ) 7 2( ) 2( )4 8 8 4 8 4x x x+ + + . Suy ra 81 58 4MinA Khi x .3. Tm GTNN ca B = 3x + y - 8x + 2xy + 16.Gii:B = 3x + y - 8x + 2xy + 16= 2(x - 2) + (x + y) + 8 8.MinB = 8 khi : . 4. Tm GTLN ca C = -3x - y + 8x - 2xy + 2.Gii: C = -3x - y + 8x - 2xy + 2 = 10 - 10. GTLNC = 10khi: .BI TP:5. Tm GTNN25 2008 A x x +6. Tm GTLN B = 1 + 3x - x27. Tm GTLN D = 22007 5 x x 8.Tm GTNN ca F = x4 + 2x3 + 3x2 + 2x + 1.9. Tm GTNN ca G = 4 3 210 25 12 x x x + +10. Tm GTNN ca M = x + 2y - 2xy + 2x - 10y.11. Tm GTNNC = ( ) 5 1 3 4 1 32+ x x12.Tm GTNN ca N = (x +1) + ( x - 3) 13. Tm GTNN ca K=x + y - xy +x + yGv: Nguyn Vn TTrng THCS Thanh M62Gio n BDHSG Ton 8 Nm hc : 2011-2012 HNG DN5.A = x - 5x + 2008 = (x - 2,5)2 + 2001,75 MinA = 2001,75 khi x = 2,56. B = 1 + 3x - x2 = -1,25 - ( x - 1,5)2

7.D = 2007 - x - 5x = 2004,5 - ( x + 2,5)2 8.F = x4 + 2x3 + 3x2 + 2x + 1 =(x +x+1) =.9.G = x - 10x +25x + 12 =x(x - 5) + 1210. M = x + 2y - 2xy + 2x - 10y = (x - y + 1) + (y - 4) -16.11.C = ( ) 5 1 3 4 1 32+ x x * Nu x . C = (3x - 3) + 1 * Nu x 0). (BT C-si)2. ( ) ab b a 42 +3. ( ) ( )2 2 22 b a b a + +4.0 , ; 2 > + b aabba5.0 , ;4 1 1>+ + b ab a b a6. ca bc ab c b a + + + +2 2 27.( ) ( )( )2 2 2 2 2y x b a by ax + + + ( Bu nhi a cop xki)8.( )y xb aybxa++ +2 2 29.( )z y xc b azcybxa+ ++ + + +2 2 2 2V d 9:Chng minh c b abcaabccab+ + + + (Via,b,c > 0)Gii:2A - 2B = c b abcaabccab2 2 2 2 2 2 + +=

,_

+ + ,_

+ + ,_

+ 2 2 2baabcaccabbccba p dng bt ng thc 0 , ; 2 > + b aabba.Ta c:2A - 2B ( ) ( ) ( ) 0 2 2 2 2 2 2 + + c b a.Vy A B.ng thc xy ra khi a = b = c > 0V d 10: Cho cc s dng x , y tho mn x + y = 1. Chng minh rng :82 12 2++y x xy.Gii:2 2 2 2 2 2 2 22421212222 2 1y xy x y x xy y x xy y x xy + +

,_

++ ++ ++Gv: Nguyn Vn TTrng THCS Thanh M63Gio n BDHSG Ton 8 Nm hc : 2011-2012 ( )882+y x.ng thc xy ra khi 21 y xV d 11: Chng minh bt ng thc : abbccaaccbba+ + + +222222Gii: cacbbacbba. 2 . 22222 +; abaccbaccb. 2 . . 22222 +; bcbaacbaac. 2 . . 22222 +Cng tng v ba bt ng thc trn ta c:abbccaaccbbaabbccaaccbba+ + + +

,_

+ +

,_

+ +2222222222222 2ng thc xy ra khi a = b = c..Bi tp:1. Cho a,b,c l 3 s dng.Chng minh rng( ) 91 1 1 ,_

+ + + +c b ac b a2. Cho cc s dng a,b,c bit a.b.c = 1. Chng minh rng: (a + 1)(b + 1)(c + 1)83. Cho cc sa,b bit a + b = 1. Chng minh rnga)a + b b)a + b 4. Cho 3 s dng a,b,c v a + b + c = 1. Chng minh: ++9 5.Cho x , y , z0v x + y + z 3 . Chng minh rng:+ + + +6. Cho 2 s dng a , b c tng bng 1 .Chng minh rng a.+ 6 b. + 147. Cho 2 s dng a , b c tng bng 1 .Chng minh rng (a +) + (b +) 8. Chng minhbt ng thc sau vi mi a,b,c>0,212121313131b a c a c b c b a a c c b b a + +++ +++ ++++++9. Cho a,b,cl 3 s dng. Chng minh : c b a abcacbbca 1 1 1+ + + +.10. Cho a,b,c l 3 s dng. Chng minh rng :22 2 2c b aa bcc abc ba + ++++++.11.Chng minh: a + bvi a + b 112. Chng minh:23+++++ b aca cbc ba Vi a,b,c > 013. Chng minh: ( ) c b a abc c b a + + + +4 4 414. Bi 28: Cho ; 0 ; 0 ; 0 z y xChng minh rng :(x + y).(y + z).(z + x) 8xyz 15. Cho A = 1 31...2 211 21...2111++ +++++ ++++ n n n n n Chng minh rng1 > AHNG DN:Gv: Nguyn Vn TTrng THCS Thanh M64Gio n BDHSG Ton 8 Nm hc : 2011-2012 1. A = 9 2 2 2 3 3 + + + ,_

+ + ,_

+ + ,_

+ +accbaccaabba 2. p dng (a + 1)2a3. a)A - B = a + b - =2( a + b) - (a + b)0. b) p dng cu a.4. Xem bi 15. + + ++ =+ +=. + + =6.A = += ( + )++ = 6 ( v 2ab (a+b) )B = + = 3(+) +7. (a +) ++ (b +) + =+ 5(a +) + 5(b +)=5( a + b) + 5( +) 5( a + b) + 5. = 25Suy ra: (a +)+ (b +)8. + ;+ ;+ Cng theo v 3 BT trn ta c pcm9. Ta c:+ =(+)2.a bccba abcacb 1. 21

,_

+ +b caacb bcaabc 1. 21 ,_

+ +Cng tng v 3 bt ng thc trn ta c pcm. ng thc xy ra khi v ch khia = b = c.(Hy kim tra li)10.p dng BT ( )z y xc b azcybxa+ ++ + + +2 2 2 211. a + b ( a + b ) 12. ( + 1) + ( + 1) + (+ 1) =+ + = (a+b+c) ( + +) (a+b+c) . =Suy ra: 23+++++ b aca cbc ba13.p dng BT v d 6 cho 3 s 4 4 4c b a + + ri tip tc p dng ln na cho 3 s a2b2 + b2c2 + c2a2 ta c pcm.14.p dng BT ( ) xy y x 42 +.Nhn tng tha s ca 3 BT suy ra PCM15.A c 2n + 1 s hng (Kim tra li !).p dng BT 0 , ;4 1 1>+ + b ab a b aVi tng cp s hng thch hp s c pcm Chuyn 9: Phng php tam gic ng dng.I Kin thc c bn1. Ta bit nu 2 tam gic ng dng th suy ra c cc cp gc tng ng bng nhau, cc cp cnh tng ng t l, c bit l t s din tch ca chng bng bnh phng t s ng dng2. chng minh 2 gc bng nhau hay cc cp on thng t l bng pp tam gic ng dng ta c th lm theo cc bc sau : Bc 1 : Xt 2 tam gic c cha 2 gc hay cha cc cp on thng y.Bc 2 : Chng minh 2 tam gic ng dngGv: Nguyn Vn TTrng THCS Thanh M65Gio n BDHSG Ton 8 Nm hc : 2011-2012 Bc 3 : Suy ra cp gc tng ng bng nhau, cp cnh tng ng t l.3. to ra c mt tam gic ng dng vi mt tam gic khc, ngoi cch v mt ng song song vi mt cnh ca tam gic ta cn c th v thm ng ph bng nhiu cch khc, chnghn :- Ni 2 im c sn trong hnh lm xut hin mt tam gic mi- T mt im cho trc, v mt ng thng vung gc vi mt ng thng.- Trn mt tia cho trc, t mt don thng bng mt on thng khc.4. Mt vi ng dng ca pp tam gic ng dnga) Dng pp tam gic ng dng CM 3 im thng hngTa c th CM 2 tam gic ng dng suy ra cc cp gc tng ng bng nhau, t dng cch cng gc dc gc bt dn ti 3 im thng hng.V d 1 : Cho tam gic ABC, cc tia phn gic gc B v gc C ct nhau ti O. Trn cc cnh AB, AC ln lt ly M v N sao cho BM.BC = BO2 ; CN.CB = CO2. CMR 3 im M, O, N thng hng.@ Bg:BM.BC = BO2 BM BOBO BC; BOM vBCO c 1 2B B ; BM BOBO BCnnBOM ~BCO (c.g.c) 1 1O C Chng minh tng t ta cCON ~CBO (c.g.c) 2 2O B Ta c 01 2 3 1 2 3180 O O O C B O + + + + . Suy ra 3 im M, O, N thng hng.Nhn xtiu g gi cho ta dng pp ng dng gii v d trn ? l v trong bi c cho BO l trung bnh nhn ca BM v BC ; CO l trung bnh nhn ca CN v CB, t suy ra c cc cp on thng t l dn ti 2 tam gic ng dng.b) Dng pp tam gic ng dng CM tch ca 2 on thng hoc tng cc tch ca cc cp on thng bng 1 s cho trc.Ta c th CM 2 tam gic ng dng suy ra cccpj cnh tng ng t l, dn ti 2 tch ca cc cp on thng bng nhau.V d 2 :Cho hnh bnh hnh ABCD, gc B nhn. Gi H v K ln lt l hnh chiu ca B trn AD v CD. Chng minh rng DA.DH + DC.DK = DB2 @ Bg:*Tm hng gii :Cc tch DA.DH, DC.DK cha c mi lin quan trc tip no vi nhaucng nh vi DB. V th ta s thay cc tch ny bi cc tch khc bng chng, c lin quan vi nhau cng nh lin quan vi DB. Mun vy phi to ra c nhng cp tam gic ng dng vi iu kin DB phil cnh ca mt trong nhng tam gic nh th.*Li gii: V AI DBIDA ~HDB DA DIDB DH DA.DH = DB.DI(1)IBA ~KDB BA BIDB DK DC BIDB DK DC.DK = DB.BI(2)Cng tng v cc BT (1) v (2) ta c : DA . DH + DC . DK = DB.DI + DB.BI = DB(DI + BI) = DB2(pcm)Gv: Nguyn Vn TTrng THCS Thanh M66AMNC BO321211HB AK C DIGio n BDHSG Ton 8 Nm hc : 2011-2012 Ch : Nu B= 900 th hnh bnh hnh ABCD tr thnh hnh ch nht. Lc p dubgj nh l Pt-ta-go ta c ngay iu phi chng minh.c) Dng pp tam gic ng dng gii bi ton dng hnhi vi 1 s bi ton dng hnh nht l dng hnh tam gic, khi ch bit mt yu t v di, cn li l bit t s gia cc di hoc bit s o cc gc th ta c th ngh n pp tam gic ng dng.V d 3 : Dng tam gic ABC bit A = 600 ; 23ABBC v BC = a.@ Bg:1. Phn tch : Gi s dng c tam gic ABC tho mn biV mt ng thng song song vi BC, ct AB, AC ln lt ti M v NT C v ng thng song song vi AB ct MN ti P. D thy MP = BC = a .AMN ~ABC 23AM ABAN AC VyAMN dng c, t dng c P, C v B.2. Cch dng : - DngAMN sao cho A = 600 ; AM = 2 ; AN = 3- Trn tia MN ly im P sao cho MP = a.- Dng PC // AB (C thuc tia AN)- Dng CB // MN (B thuc tia AM)Tam gic ABC l tam gic phi dng (Phn CM v bin lun t lm)II Bi tp Bi 1: Cho tam gic ABCvung ti A, ng cao AH. Gi M v N ln lt l trung im ca AH v BH. Gi O l giao im ca AN vi CM. Chng minh rng :a) AN CMb) AH2 = 4 MC.MO@ Bg a)ABH ~CAH (g-g) 22AB BH BNCA AH AM hay AB BNCA AM (1)Ta c 1 1B A (2) . T (1) v (2) suy raABN ~CAM (c.g.c) 2 2A C Xt tam gic CAO c 02 290 CAO C CAO A + + 090 O . Vy AN CMb)AOM ~CHM (g.g) AM MOCM MH AM.MH = MC.MO .2 2AH AH= MC.MO hay HA2 = 4 MC.MO.Bi 2 :Cho tam gic ABC, phn gic AE. Chng minh rng AB.AC > AE2.@ Bg AEC B > (t/c gc ngoi caABE). Trn AC ly im F sao cho AEF B AF < ACAEF ~ABE (g-g) AE AFAB AEAB.AF = AE2 AB.AC > AE2.Bi 3 :Gv: Nguyn Vn TTrng THCS Thanh M67AiNiMipAiCiBiai600ACB H NMO221AC E BFGio n BDHSG Ton 8 Nm hc : 2011-2012 Cho tam gic ABC vung ti A. Gi M l mt im di ng trn AC. T C v -ng thng vung gc vi tia BM ct tia BM ti H, ct tia BA ti O. Chng minh rng :a) OA.OB = OC.OHb) Gc OHA c s o khng ic) Tng BM.BH + CM.CA khng i.@ Bg a)BOH ~COA (g-g) OB OHOC OAOA.OB = OC.OHb) OB OHOC OAOA OHOC OB(1)OHA vOBC c O chung(2)T (1) v (2) OHA ~OBC (c.g.c) OHA OBC (khng i)c) V MK BC ;BKM ~BHC (g.g) BM BKBC BH BM.BH = BK.BC (3)CKM ~ CAB (g.g)CM CKCB CA CM.CA = BC.CK (4)Cng tng v ca (3) v (4) ta c BM.BH + CM.CA = BK.BC + BC.CK= BC(BK + CK) = BC2 (khng i).Bi 4 :Cho tam gic ABC cn ti A, ng cao AH. Trn on thng CH v HB ln lt ly hai im M v N sao cho CM = HN. ng thng qua M v vung gc vi BC ct AC ti E. Qua N v ng thng dNE. Chng minh rng khi M di ng trn on thng CH th ng thng d lun lun i qua mt im c nh.@ Bg D thy CH = MN = 12BCHFN ~MNE (g.g) HF HN CMMN ME ME (1)AHC ~EMC (g.g) AH CH CH CMEM CM AH EM (2)T (1) v (2) HF CHMN AH . Do HF = 21 1..2 24BC BCMN CH BCAH AH AH (khng i)V H c nh nn F c nh. Bi 5 :Cho tam gic ABC, 3 ng cao AD, BE, CF. Gi M, N, I, K ln lt l hnh chiu ca D trn AB, AC, BE, CF. Chng minh rng 4 im M, N, I, K thng hng.@ Bg V DM // CF v DI // CA nn BM BD BIBF BC BE MI // FE(1)V DN // BE v DK // AB nn CN CD CKCE CB CF NK // FE(2)AMD ~ADB AM ADAD AB (3)Gv: Nguyn Vn TTrng THCS Thanh M68C K BOAHMC MFN BAEHANEFMC B DIKGio n BDHSG Ton 8 Nm hc : 2011-2012 AND ~ADC AN ADAD AC (4)Chia tng v ca (3) cho (4) ta c AM ACAN ABACF ~ABE AF ACAE AB do AM AFAN AE MN // FE (5)T (1) ; (2) ; (5) suy ra 4 im M, I, N, K thng hng.Bi 6 : Ly cc cnh AB, AC v BC caABC lm cnh y, dng cc tam gic vung cn ABD, ACE, BCF, hai tam gic u dng ra pha ngoiABC cn tam gic th 3 dng trong cng 1 na mt phng b BC vi ABC. Chng minh rng t gic AEFD l hnh bnh hnh (hoc A, E, F, D thng hng)@ Bg Nu BAC 900 ; BAD ~ BCF(2 tam gic vung cn)BD BA BD BFBF BC BA BC . Mt khc DBF ABC ( = 45o + 1B ) BDF ~ BAC (c.g.c) BDF BAC Chng minh tng t cFEC ~BAC FEC BAC Ta c DAE ADF + = (90o + BAC) + (900 - BDF) = 1800 AE // DFChng minh tng t ta c AD // EF. Vy AEFD l hnh bnh hnh. Trng hp nu BAC= 900 th 4 im A, E, F, D thng hng Bi 7 :Cho hnh vung ABCD. Gi M, N ln lt l trung im ca AB, AD. Gi E v F ln lt l giao im ca BN vi MC v AC. Cho bit AB = 30 cm, tnh din tch cc tam gic BEM v AFN.@ Bg ABN =BCM (c.g.c) 1 1B C BN CMABN vung ti A, AB = 30; AN = 15 BN2 = 1125BEM ~BAN 2225 11125 5BEMBANS BMS BN _ ,SBAN = 12.30.15 = 225 SBEM = 225.15 = 45 (cm2).AFN ~CFB 12FN ANFB BC FN = 12BF = 13BNdo SAFN = 13SABN = 13.225 = 75 cm2.Bi 8 : Qua im O nm trong tam gic ABC ta v nhng ng thng song song vi 3 cnh. Cc ng thng ny chia tam gic ABC thnh 3 hnh bnh hnh v 3 tam gic nh. Bit din tch ca cc tam gic l a2 , b2 , c2 .a) Tnh SABCb) Chng minh S a2 + b2 + c2@ Bg a) D thy cc tam gic ODH, EON, FMO ng dng viABC.t SABC = d2 . Ta c 222a DH a DHd BC d BC _ ,(1) 2 222b ON HC b HCd BC BC d BC _ _ , , (2)Gv: Nguyn Vn TTrng THCS Thanh M69CEBDAF1C DNB MAFE11AENFMC H B DOa2b2c2Gio n BDHSG Ton 8 Nm hc : 2011-2012 2 222c MO BD c BDd BC BC d BC _ _ , ,(3)Cng tng v cc ng thc (1) , (2) , (3) ta c1a b c DH HC BDd BC+ + + + a + b + c = d. Vy S = d2 = (a + b + c)2b) S = (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac a2 + b2 + c2 + (a2 + b2) + (b2 + c2) + (c2 + a2) = 3(a2 + b2 + c2)Du = xy ra a = b = c O trng vi trng tm G caABC .

V d 8 :a. Rt gn Biu thc 6 29 12 422 + +a aa aBVi a 23 b. Thc hin php tnh: ( ) a a aaaa a++++ +2228:5 , 0 12 5 , 03 2(a t2.)Gii:a.6 29 12 422 + +a aa aB( )( )( ) 23 22 3 23 22+ ++aaa aab.( ) ( ) a a aaaa aa a aaaa a++++ +++++ +228224 22228:5 , 0 12 5 , 032 3 2( )( ) ( ) ( ) a a aaa a a a aa a 122224 2 24 222+ + + + V d 9 : Thc hin php tnh: xy y xy xy xxy y xA2:2 23 32 22 2 ++ +.( Vi x t y)Gii:( ) ( )( )( ) ( )( )22 22 2 22 23 32 22 22:y xy xxy y x y xy xy x y xxy y xxy y xy xy xxy y xA+ + ++ + ++ + V d 10 : Cho biu thc : 1 212 3 43 4+ + + + +x x x xx x xA.a. Rt gn biu thc A.b. Chng minh rng A khng m vi mi gi tr ca x .Gii:111 212 2 3 43 42 3 43 4+ + + + + ++ + + + +x x x x xx x xx x x xx x xA( ) ( )( ) ( )( )( )( )( )( ) ( )( )( )( )( ) 111 11 11 11 11 11 1222 22 22 232 2 23+++ + + ++ + + ++ + + + + +xxx x xx x xx x xx xx x x x xx x xb.( )( ) 0 0 1 ; 0 1 ;112 222 > + +++ A x xxxAGv: Nguyn Vn TTrng THCS Thanh M70Gio n BDHSG Ton 8 Nm hc : 2011-2012 V d 11 : Tnh gi tr biu thc : 8 7 6 58 7 6 5 + + ++ + +a a a aa a a a vi a = 2007.Gii:( )( )13 132 33 2 132 38 7 6 5 881 2 38 7 6 58 7 6 58 7 6 58 7 6 58 7 6 52007111 11 1 1 1 + + ++ + ++ + ++ + ++ + ++ + ++ + ++ + ++ + ++ + + B aa a aa a a aa a aa a a a aaa a aa a a aa a a aa a a aa a a aa a a aB V d 12 : Tnh gi tr biu thc : 22:25 10252 2 32 + y yyx x xx.Bit x2 + 9y2 - 4xy = 2xy - 3 x.Gii: x2 + 9y2 - 4xy = 2xy - 3 x ( ) 0 3 32 + x y x''1333yxxy x( )( )( )( )( )21 255 522:25 10252 2 2 32+ + + yy yx xx xy yyx x xxC( )( )( ) ( ) 382 . 32 . 851 5 + +x xy xBi tp:13. Chng minh rng Biu thcP =( )( )( )( ) 1 11 12 2 22 2 2+ + + + + +x a a a xx a a a xkhng ph thuc vo x.14. Cho biu thc M = 8 26 3 4 2 222 3 4 5 ++ + x xx x x x x.a. Tm tp xc nh ca M.b. Tnh gi tr ca x M = 0.c. Rt gn M.15. Cho a,b,c l 3 s i mt khc nhau. Chng minh rng :( )( ) ( )( ) ( )( ) a c c b b a b c a cb ac b a ba cc a b ac b++ + + 2 2 216. Cho biu thc : B = 10 9 9 9102 3 4 + ++x x x xxa. Rt gn Bb. Chng minh rng : n8 + 4n7 + 6n6 + 4n5 + n4 16 vi n ZGv: Nguyn Vn TTrng THCS Thanh M71Gio n BDHSG Ton 8 Nm hc : 2011-2012 a. Rt gn biu thc : 996 3 266 3 23 222++ + + ++xxy x xyxyy x xyy xA vi x-3; x3; y-2.b. Cho Biu thc : A = 3 222223:224422x xx xxxxxxx

,_

++.a. Tm iu kin c ngha v Rt gn biu thc A.b. Tm gi tr ca x A > 0.c. Tm gi tr ca A trong trng hp 4 7 x.19.a.Thc hin php tnh:a.A = 16 8 4 21161814121111x x x x x x ++++++++++.b. Rt gn C = 222 22 2991919191aaa aa a++++.20. Cho a,b,c l 3 snhau i mt.Tnh S = ( )( ) ( )( ) ( )( ) b a c baca c b abca c c bab + + .21. Tnh gi tr ca biu thc : 33532++b aa bb ab abit:0 9 & 0 5 3 102 2 2 2 b a ab b a22. Cho a + b + c = 1 v12 2 2 + + c b a .a. Nu czbyax . Chng minh rng xy + yz + zx = 0.b.Nu a3 + b3 + c3 = 1. Tnh gi tr ca a,b,c23. Bi 11 : Cho Biu thc : 1 351 31 2++aaaaA.a. Tnh gi tr ca A khi a = -0,5.b. Tnh gi tr ca A khi : 10a2 + 5a = 3.24.Chng minh nu xyz = 1 th:1111111+ +++ +++ + zx z yz y xy x.25. Chng minh ng thc sau:ab an a bnab bn an ab a abb ab ab aab a3 3 9 63 5 293222 22 22 22+ + + ++26. Thc hin php tnh:

,_

,_

,_

,_

2 2 2 2200811 ...411311211.27. Tnh tng : S(n) = ( )( ) 2 3 1 31...8 . 515 . 21+ + + +n n.28.Rt gn ri tnh gi tr ca biu thc :A = 22 17 12 22 3 + aa a a .Bit a l nghim ca Phng trnh : 1 1 32 + a a.29.Gi a,b,c l di 3 cnh ca tam gic bit rng:8 1 1 1 ,_

+ ,_

+ ,_

+cabcab Chng minh rng tam gic l tam gic u.30.Chng minh rng nu a,b l 2 s dng tha iu kin: a + b = 1 th : ( )321 12 2 3 3+ b aa babba31. Thc hin php tnh:Gv: Nguyn Vn TTrng THCS Thanh M72Gio n BDHSG Ton 8 Nm hc : 2011-2012 A = ( )( ) ( )( ) ( )( ) z x z yxy zz y y xxz yz x y xyz x+ +++ +++ +2 2 232. Rt gn biu thc : A = c b aabc c b+ + + + 3 a3 3 3.33. Chng minh rng biu thc sau lun dng trong TX: B = ( )1]1

,_

++

,_

++xxxxxxxx1111:113 322234. Rt gn ri Tnh gi tr biu thc vi x + y = 2007.A = xy y y x xxy y y x x2 ) 6 ( ) 6 () 3 ( 2 ) 5 ( ) 5 (+ + + + + + + +.35. Cho 3 s a,b,c0 tha mn ng thc: aa c bbb c acc b a + + +.Tnh gi tr biu thc P = ( )( )( )abca c c b b a + + +.36. Cho biu thc : 22222224.24.24y xzy zxx yzx yzz xyz xyA+++. Chng minh rng nu :x + y + z = 0 th A = 1.HNG DN:13.P =( )( )( )( )222 2 22 2 2111 11 1a aa ax a a a xx a a a x+ + ++ + + + + +14.M = 8 26 3 4 2 222 3 4 5 ++ + x xx x x x x. ( ) ( )41 3223+ +xx x15.( )( ) a c b a c a b ac b+ 1 1=( )( ) b a c b c b a ba c+ 1 1= ( )( ) a c c b b c a cb a+ 1 116. a.Rt gn B = ( )( )( ) 1 10 11010 9 9 9102 2 3 4+ + + + ++x x xxx x x xx( )( )( )( )( )( )( )( )

+ 10 ;1 10 1101 10 ;1 1122xx x xxlx xx xb. n8 + 4n7 + 6n6 + 4n5 + n4 ( ) [ ]41 + n n17.996 3 266 3 23 222++ + + ++xxy x xyxyy x xyy xA ( )( )( ) 2 3 30996 3 266 3 23 222+ + ++ + + ++y x x xxy x xyxyy x xyy x18.a.A = 3423:22442223 2222

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++xxx xx xxxxxxx.Gv: Nguyn Vn TTrng THCS Thanh M73Gio n BDHSG Ton 8 Nm hc : 2011-2012 b.A > 0 3 0342> > xxxc.

31 14 7xxx x = 11 2121 A x = 3 A khng xc nh19.a.A = 32 16 8 4 21321161814121111x x x x x x x ++++++++++.b. Rt gn C = 1991919191222 22 2 ++++aaa aa a.20. S = ( )( ) ( )( ) ( )( ) b a c baca c b abca c c bab + + ( ) ( ) ( )( )( )( )( )( )( )( )( )( )1 + + a c c b b aa c c b b aa c c b b aa c ac c b bc b a ab21.T:2 2 2 2 2 210 3 5 0 9 & 0 5 3 10 a b ab b a ab b a (1) Bin i A = 2 22 296 15 333532b ab ab ab aa bb ab a ++(2)Th (1) vo (2); A = - 3 22.Ta + b + c = 1 v12 2 2 + + c b asuy ra:ab + bc + ca = 0(1)a. Nu czbyax suy ra : z y xc b az y xczbyax+ + + ++ + ( )2 2 2 2z y x z y x + + + + Suy raxy + yz + zx = 0.b. p dng( ) ( ) ( )( )( ) a c c b b a c b a c b a + + + + + + + 33 3 3 3Ta3 + b3 + c3 = 1. Suy ra: ( )( )( ) 0 3 + + + a c c b b a T tnh c a , b , c.23. Xem bi 2124.T xyz = 1 Bin iyz yyzyz yyyz yzx z yz y xy x+ +++ +++ ++ +++ +++ +1 1 11111111.25. Chng minh :a bb aab an a bnab bn an ab a abb ab ab aab a++ + + ++3 3 3 9 63 5 293222 22 22 2226.

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2 2 2 2200811 ...411311211.3996199921999.199811998 ... 4 . 3 . 21999 .. 5 . 4 . 3.1998 ... 4 . 3 . 21997 ... 3 . 2 . 1 27.Gv: Nguyn Vn TTrng THCS Thanh M74Gio n BDHSG Ton 8 Nm hc : 2011-2012 ( )( )( ) 2 3 2 2 311 31...81515121312 3 1 31...8 . 515 . 21+

,_

++ + + + + +nnn nn n.28. 1 8 222 17 12 222 3+ + a aaa a aA.

+ 5 2 ; 15 ; 1 3 ; 01 1 32A a aA A a aa a.29.( ) ( ) ( )0 8 1 1 12 2 2++ ,_

+ ,_

+ ,_

+caa cbcc babb acabcab30. Rt gn

( ) ( )( )( )( ) 1 11321 12 22 22 2 3 3+ + + + + + a a b b abb a b ab aa babba31.( )( ) y xyz xxz x y xyz x+++ +2=( )( ) z yzy xyz y y xxz y+++ +2( )( ) z xxz yzz y z xxy z+++ +2. Cng tng v c A = 0.32. A = c b aabc c b+ + + + 3 a3 3 3.( )( ) ca bc ab c b a c b a abc c b + + + + + +2 2 2 3 3 33 a33. TX: 1 t x ;B = 211x +34. A = ( )( )( )( ) y x y xy x y xxy y y x xxy y y x x+ + + + + ++ + + + + + + +61 62 ) 6 ( ) 6 () 3 ( 2 ) 5 ( ) 5 (.35. T:aa c bbb c acc b a + + +.Suy ra: 2 2 2 + + + + + +aa c bbb c acc b aSuy ra: aa c bbb c acc b a + ++ ++ +Suy ra: hoc a + b + c = 0hoca =b = c.P = -1 hoc P = 836. T: x + y + z = 0 suy ra: xyz z y x 33 3 3 + +NMA . ( ) ( )3 3 3 3 3 3 3 3 3 2 2 24 16 63 x z z y y x z y x xyz z y x M + + + + + ( ) ( )3 3 3 3 3 3 3 3 3 2 2 24 2 9 x z z y y x z y x xyz z y x N + + + + + + =========o0o========= KIM TIN QUA MNG VIT NAMGv: Nguyn Vn TTrng THCS Thanh M75Gio n BDHSG Ton 8 Nm hc : 2011-2012 Qu thy c v bn hy dnh thmmt cht thi gian c bi gii thiu sau ca ti v hy tri n ngi ng ti liu nybng cch dng Email v m s ngi giithiu ca ti theo hng dn sau. N s mang li li ch cho chnh thy c v cc bn, ng thi tri n c vi ngi gii thiu mnh: Knh cho qu thy c v cc bn. Li u tin cho php ti c gi ti qu thy c v cc bn li chc tt p nht. Khi thy c v cc bn c bi vit ny ngha l thy c v cc bn c thin hng lm kinh doanh Ngh gio l mt ngh cao qu, c x hi coi trng v tn vinh. Tuy nhin, c l cng nh ti thy rng ng lng ca mnh qu hn hp. Nu khng phi mn hc chnh, v nu khng c dy thm, liu rng tin lng c cho nhng nhu cu ca thy c. Cn cc bn sinh vinvi bao nhiu th phi trang tri, tin gia nh gi, hay i gia s kim tin thm liu c ? Bn thn ti cng l mt gio vin dy mn TON v vy thy c s hiu tin lng mi thng thu v s c bao nhiu. Vy lm cch no kim thm cho mnh 4, 5 triu mi thng ngoi tin lng. Thc t ti thy rng thi gian thy c v cc bn lt web trong mt ngy cng tng i nhiu. Ngoi mc ch kim tm thng tin phc v chuyn mn, cc thy c v cc bn cn su tm, tm hiu thm rt nhiu lnh vc khc.Vy ti sao chng ta khng b ra mi ngy 5 n 10 pht lt web kim cho mnh 4, 5 triu mi thng.iu ny l c th?. Thy c v cc bn hy tin vo iu . Tt nhin mi th u c gi ca n. qu thy c v cc bn nhn c 4, 5 triu mi thng, cn i hi thy c v cc bn s kin tr, chu kh v bit s dng my tnh mt cht. Vy thc cht ca vic ny l vic g v lm nh th no? Qu thy c v cc bn hy c bi vit ca ti, v nu c hng th th hy bt tay vo cng vic ngay thi.Thy c chc nghe nghiu n vic kim tin qua mng. Chc chn l c. Tuy nhin trn internet hin nay c nhiu trang Web kim tin khng uy tn( l nhng trang web nc ngoi, nhng trang web tr th lao rt cao...). Nu l web nc ngoi th chng ta s gp rt nhiu kh khn v mt ngn ng,nhng web tr th lao rt cao u khng uy tn, chng ta hy nhn nhng g tng xng vi cng lao ca chng ta, l s tht. Vit Nam trang web tht s uy tn l : http://satavina.com.Lc u bn thn ti cng thy khng chc chn lm v cch kim tin ny. Nhng gi ti hon ton tin tng, n gin v ti c nhn tin t cng ty.( thy c v cc bn c tch ly c 50.000 thi v yu cu satavina thanh ton bng cch np th in thoi l s tin ngay).Tt nhin thi gian u s tin kim c chng bao nhiu, nhng sau s tin kim c s tng ln. C th thy c v cc bn s ni: l v vn, chng ai t nhin mang tin cho mnh. ng chng ai cho khng thy c v cc bn tin u, chng ta phi lm vic, chng ta phi mang v li nhun cho h. Khi chng ta c qung co,xem video qung co ngha l mang v doanh thu cho Satavina, ng nhin h n cm th chng ta cng phi c cho m n ch, khng th ai di g m lm vic cho h.Vy chng ta s lm nh th no y. Thy cv cc bn lm nh ny nh: 1/ Satavina.com l cng ty nh th no:Gv: Nguyn Vn TTrng THCS Thanh M76Gio n BDHSG Ton 8 Nm hc : 2011-2012 l cng ty c phn hot ng trong nhiu lnh vc, tr s ti ta nh Femixco, Tng 6,231-233 L Thnh Tn, P.Bn Thnh, Q.1, TP. H Ch Minh. GPKD s 0310332710 - do S K Hoch v u T TP.HCM cp. Giy php ICP s 13/GP-STTTT do S Thng Tin & Truyn Thng TP.HCM cp.qun 1 Thnh Ph HCM. Khi thy c l thnh vin ca cng ty, thy c s c hng tin hoa hng t vic c qung co v xem video qung co( tin ny c trch ra t tin thu qung co ca cc cng ty qung co thu trn satavina)2/ Cc bc ng k l thnh vin v cch kim tin: ng k lm thnh vin satavina thy c lm nh sau:Bc 1: Nhp a ch web: http://satavina.com vo trnh duyt web( Dng trnh duyt firefox, khng nn dng trnh duyt explorer) Giao din nh sau:

nhanh chng qu thy c v cc bn c th coppy ng linh sau: http://satavina.com/Register.aspx?hrYmail=hoangngocc2tmy@gmail.com&hrID=66309 ( Thy c v cc bn ch in thng tin ca mnh l c. Tuy nhin, chc nng ng k thnh vin mi ch c m vi ln trong ngy. Mc ch l thy c v cc bn tm hiu k v cng ty trc khi gii thiu bn b )

Bc 2: Click chut vo mc ng k, gc trn bn phi( c th s khng c giao din bc 3 v thi gian ng k khng lin tc trong c ngy, thy c v cc bn phi tht kin tr). Bc 3: Nu c giao din hin ra. thy c khai bo cc thng tin:Gv: Nguyn Vn TTrng THCS Thanh M77Gio n BDHSG Ton 8 Nm hc : 2011-2012 Thy c khai bo c th cc mc nh sau:+ Mail ngi gii thiu( l mail ca ti, ti l thnh vin chnh thc): hoangngocc2tmy@gmail.com+ M s ngi gii thiu( Nhp chnh xc) :66309Hoc qu thy c v cc bn c th coppy Link gii thiu trc tip: http://satavina.com/Register.aspx?hrYmail=hoangngocc2tmy@gmail.com&hrID=66309

+ a ch mail: y l a ch mail ca thy c v cc bn. Khai bo a ch tht cn vo kch hot ti khon nu sai thy c v cc bn khng th l thnh vin chnh thc.+ Nhp li a ch mail:.....+ Mt khu ng nhp: nhp mt khu khi ng nhp trang web satavina.comGv: Nguyn Vn TTrng THCS Thanh M78Gio n BDHSG Ton 8 Nm hc : 2011-2012 + Cc thng tin mc: Thng tin ch ti khon:thy c v cc bn phi nhp chnh xc tuyt i, v thng tin ny ch c nhp 1 ln duy nht, khng sa c. Thng tin ny lin quan n vic giao dch sau ny. Sai s khng giao dch c.+ Nhp m xc nhn: nhp cc ch, s c bn cnh vo trng+ Click vo mc: ti c k hng dn.....+ Click vo: NG KSau khi ng k web s thng bo thnh cng hay khng. Nu thnh cng thy c v cc bn vo hm th khai bo kch hot ti khon. Khi thnh cng qu thy c v cc bn vo web s c y thng tin v cng ty satavina v cch thc kim tin. Hy tin vo li nhun m satavina s mang li cho thy c. Hy bt tay vo vic ng k, chng ta khng mt g, ch mt mt cht thi gian trong ngy m thi. Knh chc qu thy cv cc bn thnh cng. Nu qu thy c c thc mc g trong qu trnh tch ly tin ca mnh hy gi trc tip hoc mail cho ti: Ngi gii thiu:Nguyn Vn T Email ngi gii thiu: hoangngocc2tmy@gmail.com M s ngi gii thiu:66309 Qu thy c v cc bn c th coppy Link gii thiu trc tip: http://satavina.com/Register.aspx?hrYmail=hoangngocc2tmy@gmail.com&hrID=663092/ Cch thc satavina tnh im quy ra tin cho thy c v cc bn:+ im ca thy c v cc bnc tch ly nh vo c qung co v xem video qung co.Nu ch tch ly im t chnh ch cc thy c v cc bnth 1 thng ch c khong 1tr.Nhng tng im thy c cn pht trin mng li bn b ca thy c v cc bn.3/ Cch thc pht trin mng li:- Xem 1 qung co video: 10 im/giy. (c hn 10 video qung co, mi video trung bnh 1 pht)- c 1 tin qung co: 10 im/giy. (hn 5 tin qung co)_Tr li 1 phiu kho st.:100,000 im / 1 bi._Vit bi....Trong 1 ngy bn ch cn dnh t nht 5 pht xem qung co, bn c th kim c: 10x60x5= 3000 im, nh vy bn s kim c 300ng . - Bn gii thiu 10 ngi bn xem qung co (gi l Mc 1 ca bn), 10 ngi ny cng dnh 5 pht xem qung co mi ngy, cng ty cng chi tr cho bn 300ng/ngi.ngy.- Cng tng t nh vy 10 Mc 1 ca bn gii thiu mi ngi 10 ngi th bn c 100 ngi (gi l mc 2 ca bn), cng ty cng chi tr cho bn 300ng/ngi.ngy.- Tng t nh vy, cng ty chi tr n Mc 5 ca bn theo s sau :- Nu bn xy dng n Mc 1, bn c 3.000ng/ngy 90.000 ng/thng.- Nu bn xy dng n Mc 2, bn c 30.000ng/ngy 900.000 ng/thng.- Nu bn xy dng n Mc 3, bn c 300.000ng/ngyGv: Nguyn Vn TTrng THCS Thanh M79Gio n BDHSG Ton 8 Nm hc : 2011-2012 9.000.000 ng/thng.- Nu bn xy dng n Mc 4, bn c 3.000.000ng/ngy 90.000.000 ng/thng.- Nu bn xy dng n Mc 5, bn c 30.000.000ng/ngy 900.000.000 ng/thng.Tuy nhin thy c v cc bn khng nn m t n mc 5. Ch cn c gng 1thng c 1=>10 triu l qu n ri. Nh vy thy c v cc bn thy satavina khng cho khng thy c v cc bn tin ng khng. Vy hy ng k v gii thiu mng li ca mnh ngay i.Lu : Ch khi thy c v cc bn l thnh vin chnh thc th thy c v cc bn mi c php gii thiu ngi khc. Hy gii thiu n ngi khc l bn b thy c v cc bnnh ti gii thiu v hy quan tm n nhng ngi m bn gii thiu v chm sc h( khi l thnh vin thy c v cc bn s c m s ring).Khi gii thiu bn b hy thay ni dung mc thng tin ngi gii thiu l thng tin ca thy c v cc bn. Chc qu thy c v cc bn thnh cng v c th kim c 1 khon tin cho ring mnh. Ngi gii thiu:Nguyn Vn T Email ngi gii thiu: hoangngocc2tmy@gmail.com M s ngi gii thiu:66309 Qu thy c v cc bn c th coppy Link gii thiu trc tip: http://satavina.com/Register.aspx?hrYmail=hoangngocc2tmy@gmail.com&hrID=66309

Website: http://violet.vn/nguyentuc2thanhmyHY KIN NHN BN S THNH CNGChc bn thnh cng! Gv: Nguyn Vn TTrng THCS Thanh M80Gio n BDHSG Ton 8 Nm hc : 2011-2012 Gv: Nguyn Vn TTrng THCS Thanh M81