HI CC TRNG THPT CHUYNKHU VC DUYN HI V NG BNG BC B K YUHI THO KHOA HC, LN TH III MN TON HC(TI LIU LU HNH NI B) H NAM, THNG 11 NM 2010 ===========================================================4HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 MC LC STTNI DUNGTRANG 1 LI NI U 5 2 MT S DNG PHNG TRNH V T CHO HC SINH GII Nguyn Anh Tun (THPT chuyn Bc Giang) 6 3 LM NGC BT NG THC Nguyn c Vang (THPT chuyn Bc Ninh) 27 4 CHNG MINH BT NG THC BNG CCH S DNG BT NG THC SP XP LI V BT NG THC CHEBYSHEVo Quc Huy, T Ton Tin, Trng THPT Chuyn Bin Ha H Nam 31 5 TNH TUN HON TRONG DY S NGUYN Ng Th Hi, trng THPT chuyn Nguyn Tri, Hi Dng 43 6 NH L PASCAL V NG DNG L c Thnh, THPT Chuyn Trn Ph Hi Phng 47 7 HM S HC V MT S BI TON V HM S HC Trng THPT Chuyn Hng Yn 56 8 MT S BI TON S HC TRONG CC K THI OLYMPIC TON Trn Xun ng (THPT Chuyn L Hng Phong Nam nh) 67 9 NH L LAGRANGE V NG DNG ng nh Sn, Chuyn Lng Vn Ty Ninh Bnh 73 10 T S KP V PHP CHIU XUYN TM Trng THPT chuyn Thi Bnh Thi Bnh 93 11 MT S DNG TON V DY S V GII HN Trn Ngc Thng - THPT Chuyn Vnh Phc 105 12 S DNG CNG C S PHC GII CC BI TON HNH HC PHNG Trng THPT chuyn H Long 123 13 BT BIN TRONG CC BI TON L THUYT TR CHI Phm Minh Phng, trng THPT chuyn i hc S phm H Ni 130 ===========================================================5HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 DI TRUYN HC LI NI U Hi cc trng chuyn vng Duyn Hi Bc B n nay c 12 trng thamgia.Trongcnhiu trngctruynthnglunm, cthnh tch cao trong cc k thi hc sinh gii Quc gia v Quc t mn Ton. Nm nay, ln th 3 hi tho khoa hc. Vi cng v l n v ng cai, chngtinhnc12bivitvccchuynchuynsuchohc sinhgiiTon.l cc chuyntmhuytca cc thycdychuyn Ton ca cc trng chuyn trong hi. Xin trn trng gii thiu cc bi vit ca cc thy c trongkyu mn Ton cahitrongdphithokhoahclnth3.Hyvngrngcun k yu ny s mt ti liu tham kho cho cc thy c! T TON - TIN TRNG THPT CHUYN BIN HO - H NAM ===========================================================6HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 MT S DNG PHNG TRNH V T CHO HC SINH GII Nguyn Anh Tun (THPT chuyn Bc Giang)

Li m u Ton hc c mt v p li cun v quyn r, ai am m th mi mi am m Trongv p y huyn b th cc bi ton linquan n Phng trnh v t (chacn thc) - c nt p tht s xao xuyn v quyn r. C l v l do m trong cc k thi HSG cc nc, thi HSG Quc gia (VMO) ca chngta,bitonlinquannPhngtrnhvtthngcmtthchthcccnh Tonhctnglaividungnhanmunhnh,munv.RithcntrongcckthiHSG cp tnh, thi HSG cp thnh ph, thi i hc, thi Tht l iu th v ! Chuyn : Mt s dng phng trnh v t cho hc sinh gii ti vit vi mong mun phn no gip cc Thy cgio dy Ton, cc em hc sinh ph thng trong cc i tuyn thi hc sinhgii Ton cth tm thy nhiu iu b ch v nhiu iu th v i vi dng ton ny. Trong Chuyn c c nhng bi vi cp gii tr cho hc sinh gii (rn luyn phn x nhanh). ivivicgiiphngtrnhvtthhuhtccphngphpgii,ccphng phpbinihayuctrongcunChuynny.Cchphntchnhndngmt phng trnh v chn la phng php gii thch hp l kh v a dng. c kh nng ny chng ta phi gii quyt nhiu phng trnh v t rt ra nhng nhn xt, kinh nghim v hay hn na l mt vi thut gii ton, cng nh lu rng mtbi ton c th c nhiu cch gii khc nhau. Ti vit Chuyn ny vi mt tinh thn trch nhim cao. Ti hy vng rng Chuyn s li trong lng Thy c v cc em hc sinh mt n tng tt p. Vimivdtrongtngphngphpgii,ngiccthtsngtcchomnh nhngbitonvinhngconsmmnhyuthch.TuynhinChuynchcchns khng th trnh khi nhng iu khng mong mun. Ti rt mong nhn c s ng vin vnhngkinnggpchnthnh caQu Thycv cc emhc sinhChuyn tip tc c hon thin hn. Ti xin chn thnh cm n! ===========================================================7HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 1. MT S PHNG PHP GII PHNG TRNH V T 1. MT S QUY C KHI C CHUYN 1.1 Vt: V tri ca phng trnh. Vt2: Bnh phng ca v tri phng trnh. 1.2 Vp: V phi ca phng trnh. Vp2: Bnh phng ca v phi phng trnh. 1.3 Vt (1) : V tri ca phng trnh(1) . 1.4 Vp (1) : V phi ca phng trnh(1) . 1.5 k, k: iu kin.1.6 BT: Bt ng thc. HSG, HSG: Hc sinh gii. 1.7 VMO, VMO: Thi hc sinh gii Vit Nam, CMO: Thi hc sinh gii Canada.

2. PHNG PHP T N PH 2.1 Mt s lu Khigiiphngtrnhvtbngphngphptnphtacthgpccdng nh:2.1.1tnphaphngtrnhchovphngtrnhiskhngcncha cn thc vi n mi l n ph. 2.1.2 t n ph m vn cn n chnh, ta c th tnh n ny theo n kia. 2.1.3 t n ph a phng trnh v h hai phng trnh vi hai n l hai n ph, cng c th hai n gm mt n chnh v mt n ph, thng khi ta c mt h i xng. 2.1.4tnphcphngtrnhchainph,tabinivphngtrnh tch vi v phi bng 0. Thng gii phng trnh ta hay bin i tng ng, nu bin i h qu th nh phi th li nghim. 2.2 Mt s v d V d 1. Gii cc phng trnh sau: 1)218 18 17 8 2 0 x x x x x = . 2)2 4 233 1 13x x x x + = + + . 3)221 12 2 4 x xx x| | + = + |\ . 4)2 22 1 2 1 1 x x x x + + = . Hngdn(HD):1)tx y = vi0 y .Khiphngtrnhchotrthnh 2 2(3 4 2)(6 2 1) 0 y y y y + + = ,suyra 2(3 4 2) 0 y y = ,tac 2 103y+= .T phng trnh c nghim l14 4 109x+= .2) Ta c 4 2 2 2 2 2 21 ( 1) ( 1)( 1) 0 x x x x x x x x + + = + = + + + > , vi mi x. Mt khc 2 2 23 1 2( 1) ( 1) x x x x x x + = + + + . t 2211x xyx x +=+ + (c th vit k0 y hoc chnh xc hn l 333y ), ta c ===========================================================8HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 2 232 1 0 6 3 3 03y y y y = = + = , ta c 33y =(loi 32y = ).T phng trnh c nghim l1 x = . 3) Ta thy0 x | | + > |\ | || | | | + = + | || |\ \ \ .t1x yx+ = , ta c 2 2 22 4(1)4 ( 2) 2 5 2( 2) (4 ) (2)yy y y < + = . Xt 2 2(2) 9 2 4 5 y y y = +4 3 28 28 40 16 0 y y y y + + = (dohaivkhng m). 3 22( 2)( 6 16 8) 0( 2)(( 2)( 4 8) 8) 0y y y yy y y y + = + + = Dn n2 y = (do 2(( 2)( 4 8) 8) 0 y y y + + >vi miy tha mn (1)). T phng trnh c nghim l1 x = . Nhn xt: Bi ton ny ta c th gii bng Phng php nh gi trong phn sau.4) Ta c phng trnh tng ng vi 2 21 1 2 2 1 x x x x = 4 2 2 2 2 3 21 1 4 4 (1 ) 4 4 1 8 1 x x x x x x x x x = + + +

2 2 22 2 2(1 4 1 8 1 ) 001 4 1 8 1 0(1)x x x xxx x x + == + = Xt (1), t 21 y x = , suy ra0 y v 2 21 x y = .Ta c 2 31 4 8 (1 ) 0 8 4 1 0 y y y y y + = =

2(2 1)(4 2 1) 0 y y y + =

1 54y+ = . T suy ra 5 58x= .Th li ta c nghim ca phng trnh l0 x =v 5 58x= . Nhn xt: Bi ton ny ta c th gii bng Phng php lng gic trong phn sau.V d 2. Gii phng trnh 2 23 1 ( 3) 1 x x x x + + = + + . HD: t 21 x y + = , vi1 y . Khi ta c 23 ( 3) y x x y + = + ( 3)( ) 0 y y x = . Dn n3 y =vy x = . T phng trnh c nghim l2 x = . ===========================================================9HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 V d 3. Gii phng trnh 8 3 8 417 2 1 1 x x = . HD:t 8 417 x y = vi0 y v 3 82 1 x z = .Khitach4 3 4 31 12 33 2 ( 1) 33y z z yy z y y = = + = + = . Xt 4 3 3 22 ( 1) 33 ( 2)(2 5 7 17) 0 y y y y y y + = + + + = . Suy ra c y - 2 = 0. T nghim ca phng trnh l x= 1 v x= -1. V d 4. Gii cc phng trnh sau:1)2 24 2 3 4 x x x x + = + . 2)3 2 3481 8 2 23x x x x = + . HD: 1) t 24 x y = , vi0 2 y .Khi ta c h 2 22 34x y xyx y+ = + + =. Th hoc li t; x y S xy P + = =ri gii tip ta c nghim ca phng trnh l 0 x = ;2 x =v 2 143x = . 2) t 3 2 3481 8 2 3 3 23x y x y y y + = = + .Khi ta c h 3 23 243 2343 23x y y yy x x x= += +.Xt hiu hai phng trnh dn nx y = (do 2 2 21 1 1 1( ) ( 2) ( 2) 02 2 2 3x y x y + + + + > ). Thay vo h v gii phng trnh ta c 3 2 60;3x x= = . V d 5. Gii phng trnh 2 25 14 9 20 5 1 x x x x x + + = + . HD: k5 x . Vi iu kin ta bin i phng trnh cho nh sau: 2 22 25 14 9 20 5 15 14 9 20 25( 1) 10 ( 1)( 4)( 5)+ + = + + + + = + + + + + x x x x xx x x x x x x x 22 5 2 5 ( 1)( 5) 4 + = + + x x x x x2( 1)( 5) 3( 4) 5 ( 1)( 5) 4 + + + = + + x x x x x xt( 1)( 5) ; 4 x x y x z + = + = , vi0; 3 y z . ===========================================================10HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Ta c2 22 3 5 ( )(2 3 ) 0 y z yz y z y z + = = , t ta c 32y zy z=

=. Nuy z =th ta c 5 612x+=(do 5 x ). Nu 32y z =th ta c 78;4x x = = . Vy phng trnh c ba nghim trn. V d 6. Gii phng trnh 24 97 728xx x++ = , vi0 x > . Nhnxt:Dngphngtrnhnytathngt 4 928xay b+= + ,saubnh phng ln ri ta c bin i v h i xng vi hai n, x y . T ta s bit c gi tr ca a, b. Vi bi ton ny ta tm c 11;2a b = = . (Nu a = 1 v b = 0 m gii cth l phng trnh qu n gin, ta khng xt y). HD: t 4 9 128 2xy+= + , do0 x >nn 4 9 9 128 28 2x +> > , t 0 y > . Ta c h 2217 7217 72, 0x x yy y xx y+ = ++ = +> . Gii h bnh thng theo dng ta c 6 5014x += . V d 7. Gii phng trnh 3 2 32 2 x x = . Nhnxt:Khigiimtphngtrnhkhngphilcnocngcnghimthc,c nhng phng trnh v nghim nhng khi cho hc sinh lm bi ta cng kim tra c nng lc ca hc sinh khi trnh by li gii bi ton . Chng hn nh bi ton trong v d ny. HD:t 3 2 32 2 x x = =yvi0 y .Khitach 2 33 222x yx y = += vt phng trnh ban u ta c2 x . Xt hiu hai phng trnh ca h ta c phng trnh 2 2( )( ) 0 x y x xy y x y + + + = . Vix y = th 3 22 x x = , dn n v nghim. Cn 2 2 2( )(1 ) 0 x xy y x y y x x y + + = + > vimi0 y v2 x .Doh v nghim hay phng trnh cho v nghim. 2.3 Mt s bi tp tng t Bi 1. Gii cc phng trnh sau: 1)2 22 2 2 x x x x + = . (HD: t2 ; 0 y x y = , ta c 2 2( 1)( 1)(2 4) 0 y y y y y + = . ===========================================================11HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 T 5 1 33 11; ;2 8y y y += = =v c nghim ca phng trnh l5 1 33 11; ;2 8x x x+ += = = ). 2)2 32 5 1 7 1 x x x + = . (HD:Tphngtrnhsuyra1 x .t 211x xyx+ +=,bnhphngdnn3 2 3 y + . Phng trnh tr thnh22 7 3 0 y y + = , ta c 3 y = . T 4 6 x = ).

Bi 2. Gii phng trnh 2 2(4 1) 1 2 2 1 x x x x + = + + . (HD: t 21 x y + = , vi1 y . T ta c 12 12y y x = = . Phng trnh c nghim 43x = ). Bi 3. Gii cc phng trnh sau: 1) 3(2 2) 2 6 x x x + = + + . (HD: t3 2 , 6 x y x z = + = , vi0; 0 y z .Ta c3 4 x y z = + = . T phng trnh c 2 nghim 11 3 53;2x x= = ). 2)42 2(1 ) 2 1 x x + + = . (HD: k0 2 1 x . t 42 2(1 ) 2 2 1 x y y x + = = v4 4 42 2 x z z x = =vi0; 0 y z . Suyra 42 42( ) 1(1)2 1(2)y zy z+ =+ = .T(1)thay 412y z = vo(2)tac 2 2 241( 1) ( ) 02z z + + = . Xt hiu hai bnh phng suy ra 44 3 214 22z= .T ta c nghim ca phng trnh l 4444 3 2122x| | | |= | | |\ ). Bi 4. Gii phng trnh 21000 1 8000 1000 x x x + = . (HD: t 1 1 8000x + + =2y , ta c 222000(*)2000x x yy y x = = . ===========================================================12HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 T(*)suy ra( )( 1999) 0 x y x y + + =v , do 1999 0 x y + + > . Suy rax y = , ta c nghim2001 x = , loi0 x = ). Bi 5. Gii cc phng trnh sau: 1)321 22 5xx+=+. (HD: t 21 0; 1 y x z x x = + = + , ta c22 255 2( ) 2 2y yyz y zz z| |= + = + |\ 25 12 2 0 22y y y yz z z z| | + = = = |\ . Nu2yz=ta c 21 2 1 x x x + = +214 5 3 0xx x + = (v nghim). Nu 12yz=ta c 22 1 1 x x x + = +15 375 3722xxx == (tha mn)). 2)2 32 5 2 4 2( 21 20 x x x x + = . (HD: k 4 15xx

. t 22 8 10 x x y =v4 x z + = , vi0; 0 y z .Khi ta c( )( 3 ) 0 y z y z = . T phng trnh c bn nghim l 9 1934x=v 17 3 734x= ). Bi 6. Gii cc phng trnh sau: 1)24 3 5 x x x = + . (HD: t5 2 x y + = , ta c 5 291;2x x+= = ). 2)232 42xx x++ = , vi 1 x . (HD: t 312xy+= + ,c 3 1714x += < (loi), nu1 x th 3 174x += ). 3)2427 183x x x + = + , vi0 x > . (HD: Tng t, ta c 5 3718x += ). 3. PHNG PHP NH GI 3.1 Mt s lu Khigiiphngtrnhvt(chnghn( ) ( ) f x g x = )bngphngphpnhgi, thng l ta ch ra phng trnh ch c mt nghim (nghim duy nht).Ta thng s dng ===========================================================13HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 ccbtngthccinCsi,Bunhiacopxki,avtrivtngbnhphngccbiu thc,ngthivphibng0.Tacngcthsdngtnhniucahms(cth thy ngay hoc s dng o hm xt s bin thin ca hm s) nh gi mt cch hp l.Thngtanhginhsau: ( ) ( )( ) ( ) ( ) ( )( ) ( )f x g xf x C C f x g x Cg x C C= = = ,hocnhgi ( ) ( ) f x g x cng nh l( ) ( ) f x g x Ngoi ra i vi bi c th no ta s c cch nh gi khc. Cng c mt s phng trnh v t c nhiu hn mt n m ta gii bng phng php nh gi. 3.2 Mt s v d V d 1. Gii phng trnh 24 1 4 1 1 x x + = . HD: Bi ton ny c trong thi vo i hc Bch Khoa v HQG nm 2001. Bi ny c nhiu cch gii, p n s dng o hm. Ta c th lm n gin nh sau: Ta thy 12x =l nghim ca phng trnh. Nu 12x >th Vt > 1 = Vp. Nu 12x th Vt(1) > 1 > Vp(1). Nu3 x = Vp (phng trnh khng c nghim). Nu0 x > th ta xt tam gic vung ABC vi 090 A = , AB = 4; AC = 3.Gi AD l phn gic ca gc A, ly M thuc tia AD.tAM=x,xt 2 29 3 2. ACM CM x x = + vxt 2 216 4 2. ABM BM x x = + . T suy ra Vt =5 CM BM BC + = . Du ng thc xy ra khiM D ,hay ===========================================================20HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 2 22 2341 6 91 6 1 6 . 9 4 8 2 . 9 1 6 . 9 3 6 2 .7 1 2 2 . 01 2 27C MB MC M B Mx x x xx xx= = + = + = = Vy phng trnh c nghim l 12 27x = . V d 2. Gii phng trnh 2 2 2 4 44 4 1 2 3 5 16 x x x y y y x + + + + = + . Nhn xt: Bi ton ny khng kh, ch kim tra tnh cn thn ca hc sinh m thi v sau khi t iu kin tm c gi tr ca x. Tuy nhin nu hc sinh hc hi ht s ngi nhn m khng lm c bi. HD:tkchophngtrnhxcnhtasc2 x = .Khiphngtrnhtr thnh1 2 y y = , suy ra 32y = . Vy phng trnh c mt nghim l 3( ; ) 2;2x y| |= |\ .V d 3. Gii phng trnh 3 2 3 2 37 1 8 8 1 2 x x x x x + + = . HD: t 3 2 3 2 37 1; 8; 8 1 y x z x x t x x = + = = ,suy ra2 y z t + + =v3 3 38 y z t + + = (1). Mt khc( )38 y z t + + =(2). T (1) v (2) ta c 3 3 3 3( ) ( ) 3( )( )( ) 0 y z t y z t y z z t t y + + + + = + + + =

0 (3)0 (4)0 (5)y z y zz t z tt y t y+ = = + = = + = = . Xt (3) ta c1 9 x x = = , xt (4) c1 x =v (5) c0 1 x x = = . Vy tp nghim ca phng trnh l{ } 1; 0;1; 9 S= . V d 4. Gii phng trnh 2 24 20 4 29 97 x x x x + + + + = . HD: Trong mt phng ta xt hai vc t( 2; 4) a x = r v( 2; 5) b x = r. Khi ta c( 4; 5) a b + = r r, suy ra97 a b + =r r v ta cng c 24 20 a x x = +r, 24 29 b x x = + +r. Phng trnh tr thnha b a b + = +r r r r, ngthc xy ra khiar vbr cngchiu 2 24 5x x = . T ta c phng trnh c mt nghim l 29x = . V d 5. Gii phng trnh 2 2 4 21 2 1 2 2( 1) (2 4 1) x x x x x x x + + = + . HD: t 2 22 1 ( 1) y x x x = = , suy ra 2 20 1( 1) 1yx y = . ===========================================================21HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Ta c 2 2 21 1 2(1 ) (1 2 )(1) y y y y + + = . Mt khc 2 21 1 1 1 2 (2) y y y y + + + . T (1) v (2), suy ra 2 2 2 22(1 ) (1 2 ) 2 y y y t 2y z = , ta c0 1 z v 2 22(1 ) (1 2 ) 2 (4 10 7) 0 z z z z z z + 0 z (do 24 10 7 0 z z + > ). Do 0 z = , suy ra0 y =hay 22 0 x x =02xx=

=. Vy phng trnh c nghim l0 x =v2 x = . 2.MTSBITONTHILPITUYNHCSINHGIITNH BC GIANG Chn i tuyn ca tnh Bc Giang thi hc sinh gii quc gia cng c nhng bi ton gii phng trnh v t. Sau y l mt s bi.

Bi 1 (Lp i tuyn HSG quc giatnh Bc Giang nm hc 2004 2005) Gii phng trnh 3 2 3 32 11 4 4 14 5 13 2 x x x x x x + + = + . Bi 2 (Kim tra i tuyn HSG quc giatnh Bc Giang nm hc 2004 2005) Gii phng trnh 3 2 3 3 3 22 2 3 1 2 3 1 x x x x x x + + = . Bi 3 (Lp tin i tuyn HSG quc giatnh Bc Giang nm hc 2006 2007) Gii phng trnh 48 4 2 3 3 x x x x + + + = + + . Bi 4 (D tuyn ton QG gi B GD-T ca Bc Giang nm hc 2006 2007) Gii phng trnh 2 2 22 3 2 1 3 3 x x x x x x + = + + . Bi 5. (Kim tra i tuyn HSG quc giatnh Bc Giang nm hc 2007 2008) Gii phng trnh 222007 2008 20092007x x xx x +=+. Bi 6. (Gio s dy i tuyn ton tnh Bc Giang nm hc 2004 2005) Gii cc phng trnh sau: 1) 21 3 2 1 x x x x + + = + . 4) 21 582xx+ = . 2) 3 47 80 x x x + + = + . 5) 4328x x = + . 3) 3 31 2(2 1) x x + = . 6) 2 32 4 3 4 x x x x + + = + . ===========================================================22HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 3. MT S BI TON THI HC SINH GII CA MT S QUC GIA

Thc t bi ton gii phng trnh v t trong k thi hc sinh gii quc gia l khng kh.Tuynhinlmcviclnthtrchtphilmttvicnh,dohcsinh mun ot gii t khuyn khch tr ln phi lm tt bi ton ny. D bit vy nhng khng phi hc sinh xut sc no cng vt qua c. Bi 1 (1995 - Bng A. VMO)Gii phng trnh 3 2 43 8 40 8 4 4 0 x x x x + + = . HD: k1 x . Khi xt 3 2( ) 3 8 40 f x x x x = +v 4( ) 8 4 4 g x x = +trn on [ ) 1; + . Ta c( ) ( ) f x g x = . p dng BT C-si cho bn s khng m, ta c 4 4 4 4 4 441( ) 2 .2 .2 (4 4) (2 2 2 (4 4)) 13(1)4g x x x x = + + + + + = + . ng thc xy ra khi v ch khi 44 4 2 3 x x + = = . Mt khc 3 2 23 8 40 13 ( 3)( 9) 0 x x x x x x + + 2( 3) ( 3) 0(2) x x + . ng thc xy ra khi v ch khi3 x = . T (1) v (2), ta c( ) 13 ( ) g x x f x + . C hai ng thc u xy ra khi3 x = , tha mn iu kin. Vy phng trnh c nghim duy nht l3 x = .

Nhn xt: Ta c th s dng o hm xt s bin thin ca cc hm s( ) f xv ( ) g xtrn on[ ) 1; + , ta c [ ) 1:min ( ) (3) 13 f x f += =v [ ) 1:max ( ) (3) 13 g x g += = . Hoctactht 44 4 x y + = ,vi0 y saudngohmkhostsbin thincahms 12 8 4( ) 24 16 512 2816 f y y y y y = + + ( '( ) 2( 2). ( ) f y y h y = vi ( ) 0 h y > ).Bi 2 (1995 - Bng B. VMO)Gii phng trnh 2 32 11 21 3 4 4 0 x x x + = . HD: t 34 4 x y = . Khi 344yx+=v suy ra 6 328 166y yx+ += . T ta c phng trnh6 3 3 6 31 11( 8 16) ( 4) 3 21 0 14 24 96 0(1)8 4y y y y y y y + + + + = + =

2 4 3 2( 2) ( 4 12 18 14) 0(2) y y y y y + + + + = . Do0 y th Vt(1) dng, do ta xt0 y > , khi 4 3 24 12 18 14 0 y y y y + + + + > . Nn t (2) ta thy2 y =hay 34 4 2 x = , ta c3 x = .Th li ng. Vy phng trnh c nghim duy nht l3 x = . Bi 3 (2002 - Bng A. VMO)Gii phng trnh4 3 10 3 2 x x = . HD: Cch 1 (p n) ===========================================================23HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 k 74 1027 3x . Vi iu kin phng trnh cho tng ng vi phng trnh

2 2 24 3 10 3 4 4 9(10 3 ) (4 ) x x x x x x = + = 4 3 228 16 27 29 0( 3)( 2)( 7 15) 0x x x xx x x x + + = + + = 3 x =(do k v 27 15 0 x x + >vi mixtha mn k) Vy phng trnh c nghim duy nht l3 x = . Cch2:t10 3x y = ,suyra 403y (1)v 2 210 42 03 3y yx x = = >vi mi y tha mn (1). Khi ta c 2 4 24 8 164 3 4 33 9y y yy y + = =

4 328 27 20 0( 1)( 4)( 3 5) 0y y yy y y x + = + + = 1 y = . Hay ta c10 3 1 x = 3 x = .Vy phng trnh c nghim duy nht l3 x = . Bi 4 (1998-CMO)Gii phng trnh 1 11 x xx x= + . Nhnxt: y l bi ton thi hc sinh gii ca Canada, c th ni l n gin, nh nhng vi hc sinh tinh nhng cng y cm by vi mi hc sinh. Tht vy, t k xc nh ca phng trnh ta phi dn n c1 x > . Vi k , phng trnh tng ng vi 1 11 x xx x =

2 21 11 x xx x| | | | = || ||\ \ (dohaivkhngmvimi 1 x > )

2 2( 1) 2 ( 1) 0 x x x x + =

2 2( 1 ) 0 x x =

21 0 x x = . T suy ra 1 52x+= . Cng c th t 2 2( 1) 2 ( 1) 0 x x x x + = , chuyn 22 ( 1) x x sang v phi ri bnh phng hai v, sau t 12x y =ta c phng trnh trng phng n 12y > , giiphng trnh ny tm c 52y = . T suy ra 1 52x+=nhng cch ny hi di. Vy phng trnh c nghim duy nht l 1 52x+= . ===========================================================24HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 4. MT S BI TP T LM Sau y l mt s bi tp t lm m chng ta c th s dng cc phng php trn. Bi 1. Gii cc phng trnh sau: 1)2 2 21 1 2 x x x x x x + + + = + .2)2 21 1 (1 2 1 ) x x x + = + .3)221 21x x xx x +=+. 4)22 4 2 5 1 x x x x + = . 5)3 2 3 2 3 33 2001 3 7 2002 6 2003 2002 x x x x x + + = . Bi 2. Gii cc phng trnh sau: 1)2 2 22 3 2 1 3 3 x x x x x x + = + + .2)42 6065 7 x x+ = .3) ( 2) 1 2 2 0 x x x + = . 4)3 3 3 33 1 5 2 9 4 3 0 x x x x + + + = . 5)2 24 4 10 8 6 10 x x x x = . Bi 3. Gii cc phng trnh sau: 1)2(2004 )(1 1 ) x x x = + .2) 3 3 x x x = + .3) 5 5 x x x x = . 4)4 3 316 5 6 4 x x x + = + . 5)3 2 33 2 ( 2) 6 0 x x x x + + = . Bi 4. Gii cc phng trnh sau: 1)2 35 1 9 2 3 1 x x x x + = + .2)2428 272. 27 24 1 63 2x x x + + = + + .3)13 1 9 1 16 x x x + + = . 4)3 386 5 1 x x + = . 5)3 2 32 ( 4) 7 3 28 0 x x x x x + = . Bi 5. Gii cc phng trnh sau: 1)2 222 2 2 2x xx x+ + =+ + .2)22 2 4 4 2 9 16 x x x + + = + . ===========================================================25HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 3)2 32 5 2 4 2( 21 20) x x x x + = . 4)33 2 x x x = + . 5)24 3 2 312 2 2 1 ( )xx x x x x xx+ + + = + . Bi 6. Gii cc phng trnh sau: 1)3 3 36 6 6 x x + + = .2)4 1 52 x x xx x x+ = + .3)2 4 3 22 4 7 4 3 2 7 x x x x x x + + = + + . 4)2 2 4 61 1 1 1 x x x x + + + = . 5)22213x x| | = |\ . Bi 7. Gii cc phng trnh sau: 1)( ) ( )2 23 2 1 1 1 3 8 2 1 x x x x + = + + + . 2)2 32( 2) 5 1 x x + = + . 3)6 4 2 264 112 56 7 2 1 x x x x + = .4)( )2 3 3 21 1 (1 ) (1 ) 2 1 x x x x + + = + .5)( )22 3 32 11 1 (1 ) (1 )3 3xx x x+ + = + . Bi 8. Gii cc phng trnh sau: 1)3 36 6 4 4 0 x x + = .2)2 32( 3 2) 3 8 x x x + = + . 3)6 2 3 31 1 1 x x x + = . 4)2 2 315 3 8 2 x x x + = + + .5)2 3 3 2 44 4 4(1 ) (1 ) 1 (1 ) x x x x x x x x + + = + + . Bi 9. Gii cc phng trnh sau: 1)3 31 3 3 1 x x + = .2)235121xxx+ =. 3)2 32 11 21 3 4 4 0 x x x + = . 4)4 3 2 24 6 4 2 10 2 x x x x x x + + + + + + = .5)2 2 22 2 2321 1 4 4(2 3)x x x x xx x+ + + + =+. Bi 10. Gii cc phng trnh sau: ===========================================================26HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 1)2311xxx+ =+.2) ( 1) 1 5 1 4 4 0 x x x x + + = .3)4 2 2 210 14 19 (5 38) 2 x x x x + = . 4)2 2( 1) 2 3 1 x x x x + + = + .5)2 211 1 22x x x = . Bi 11. Gii cc phng trnh sau: 1)1 31 04 2xx x+ =+ +.2)33 2 0 x x x + = .3)3 38 4 6 1 1 0 x x x + = . 4)( )2 2 23 2 2 2 1 0 x x x x + + + = .5)2 23 5 12 5 0 x x x + + + = . Bi 12. Gii cc phng trnh sau: 1)2 32( 8) 5 8 x x + = + .2)24 3 4 3 10 3 x x x = .3) ( 3) (4 )(12 ) 28 x x x x + + = . 4)2 2 2 32 1 6 9 6 ( 1)(9 ) 38 10 2 x x x x x x x + + + + = + . 5)2 2 27 22 28 7 8 13 31 14 4 3 3( 2) x x x x x x x + + + + + + + = + . Bi 13. Gii cc phng trnh sau:

1)4 2 2 2 2 2 2214 16 9 2 2 x y x y x y y xx| | + + = + |\ . 2)2 2 2 2 3 21 1 1 12 ... 2 3 3 14 4 4 4x x x x x x x x + + + + + + = + + + .Trong biu thc v tri c tt c 2008 du cn thc bc hai.

===========================================================27HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 LM NGC BT NG THC Nguyn c Vang (THPT chuyn Bc Ninh) Trong bo ton s 377(thng 11 nm 2008) c bi ton sau: Tm s thc k nh nht sao cho vi mi b s thc khng m x, y, z ta lun c:{ } x z z y y x Max k xyzz y x + + +, , .33 . Bt chc cch lm y, ti khai thc mt s bt ng thc quen bit, bng cch thm vo v b mt lng ng bc ti thiu lm thay i s chnh lch.Bi 1. Tm s thc k nh nht sao cho bt ng thc sau ng vi mi x, y khng m: 2 2 2 2. 2 y x k xy y x + +.Bi 2. Tm s thc k nh nht sao cho bt ng thc sau ng vi mi x, y khng m: y x k y x y x + + + . ) ( 22 2. Bi3. Tm s thc k nh nht sao cho bt ng thc sau ng vi mi x, y khng m: { } x z z y y x Max k z y x z y x + + + + + , , . ) ( 32 2 2. Bi 4. Tm s thc k nh nht sao cho bt ng thc sau ng vi mi x, y:

4 4 4 4 4y x . k )2y x( 2 y x ++ +Bi 5. Tm s thc k nh nht sao cho bt ng thc sau ng vi mi x, y khng m:

n n n n ny x ky xy x ++ + . )2( 2 (vi n l s nguyn dng) Bi6. Tm s thc k nh nht sao cho bt ng thc sau ng vi mi x, y, z: { }2 2 2 2 2 2 2 2 2 2, , . max ) ( ) ( 3 x z z y y x k z y x z y x + + + + +Bi7. Tm s thc k nh nht sao cho bt ng thc sau ng vi mi x, y, z: { }2 2 22 12 2221. max ) ... ( ) ... (j i n nx x k x x x x x x n + + + + + + +Bi8. Tm s thc k nh nht sao cho bt ng thc sau ng vi mi x, y khng m: q knnnkkx x Max k x x n x + =. ...11. Bi9. Tm s thc k nh nht sao cho bt ng thc sau ng vi mi x, y ((

2; 0 ===========================================================28HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 ) ( sin . cos cos2cos2y x k y xy x + + + Bi 10. Tm s thc k nh nht sao cho bt ng thc sau ng vi mi a, b khng m: 2 2. )2( 2 ) ( ) ( b a kb af b f a f ++ +trong f(x) = x2 + 2x +3. HNG DN GII Bi 1.+) Gi s bt ng thc2 2 2 2. 2 y x k xy y x + + (*) ng vi mi x, y khng m. Cho x = 0, y = 1 suy ra1 k . +) Ta chng minh 2 2 2 22 . , , : 0 x y x y x y x y x y + + . Tht vy, BT trn tng ng viy . x y2BT ny ng v0 y x . Vy s thc k nh nht cn tm l. 1 k0=Bi 2.+) Gi s bt ng thc y x k y x y x + + + . ) ( 22 2(*) ng vi mi x, y khng m. Cho x = 0, y = 1 suy ra1 2 k . +) Ta chng minh 2 22( ) . ( 2 1)( ), , : 0 x y x y x y x y x y + + + . Tht vy, BT trn tng ng vixy y y ). 2 2 ( x 2 ) y x ( 22 2 2 + +BT ny ng v0 y x . Vy s thc k nh nht cn tm l. 1 2 k0 =Bi 3.+) Gi s bt ng thc{ } x z z y y x Max k z y x z y x + + + + + , , . ) ( 32 2 2 (*) ng vi mi x, y khng m. Cho x = 1, y = z = 0 suy ra1 3 k . +) Ta chng minh 2 2 23( ) ( 3 1)( ); , , : 0 x y z x y z x z x y z x y z + + + + + . Tht vy, BT trn tng ng vi zx ) 3 3 2 ( yz ) 3 2 ( y . x 3 z ) 1 3 ( 2 y2 2 + + + ===========================================================29HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 BT ny ng v + 222 2z ) 3 3 2 ( zx ) 3 3 2 (z ) 3 2 ( yz ) 3 2 (z ) 1 3 ( y xy 3. Vy s thc k nh nht cn tm l. 1 3 k0 =Cch 2: t f(x;y;z) =0 z y x ); z x )( 1 3 ( z y x ) z y x ( 32 2 2 + + . Dng o hm, ch ra c. 0 ) z ; z ; z ( f ) ; z ; y ; y ( f ) z ; y ; x ( f = Bi 4.+) Gi s bt ng thc q k n nx x kMax x x x x n + + + + + . ... ) ... (12 21(*) ng vi khi. 0 x ... x xn 2 1 Cho1 n k 0 x ... x , 1 xn 2 1 = = = . +) BT) x x )( 1 n ( x ... x ) x ... x ( nn 1 n 12n21 + + + + + , vi . 0 x ... x xn 2 1 chng minh c bng cch dn bin nh cch 2 ca bi 3. Vy s thc k nh nht cn tm l. 1 n k0 =Bi5.+) Gi s bt ng thc 4 4 4 4 4y x . k )2y x( 2 y x ++ + ng vi mi x, y khng m. Cho x = 0, y = 1 suy ra 87k . +) Dng o hm, ta chng minh c:) y x .(87)2y x( 2 y x4 4 4 4 4 ++ + , vi 0 y x Vy s thc k nh nht cn tm l.87k0=Bi6.+) Gi s bt ng thc ) x z , z y , y x ( Max . k ) z y x ( ) z y x ( 32 2 2 2 2 2 2 2 2 2 + + + + + ng vi x, y, z khng m. Cho x = 1, y = z = 0 suy ra2 k . +) Dng o hm, ta chng minh c) z x ( 2 ) z y x ( ) z y x ( 32 2 2 2 2 2 + + + + +vi0 z y xVy s thc k nh nht cn tm l. 2 k0=Bi 7 ===========================================================30HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 +) Cho x1 = 1, x2 = = xn = 01 k n+) t f(x1;x2;;xn) = n(x12 + +x2n) (x1++xn)2 (n - 1)(x21 xn2), vi 0 ....2 1 nx x x . Khi 0 ) x ;...; f(x ) x ;...; x ; x ; f(x ) x ;...; x ; f(x n n n 3 2 2 n 2 1 . Vy s thc k nh nht cn tm l. 10 = n kBi 8 +) Gi s bt ng thc q knnnkkx x Max k x x n x + =. ...11

ng khi x1, , xn khng m. Cho x1 = x2 = = xn-1 =1, xn = 0 suy ra1 n k . +) Ta chng minh) x x )( 1 n ( x ... x n xn 1nn 1n1 kk + = vi0 x ... x xn 2 1 Tht vy, BT trn tng ng vi1nn 1 n1 n2 kkx ) 2 n ( x ... x n x . n x + += BT ng v + +nn 2 1 n1 1 n 2x ... x . x n nxx ) 2 n ( x ... x Vy s thc k nh nht cn tm l. 1 n k0 =Bi 9 Vi 02 y x, ta c: 222 2 2 222cos (cos cos ) 4cos .sin cos2 2 4 2sin ( )4sin .cos 4cos .cos2 2 4 21 4cos .c8x y x y x y x yx yx y x y x y x yx y+ + + += = 2 23 2 2os4=+ 1 n k Bi 10 +)Vi a > b > 0ta c 21) ( 2)2( 2 ) ( ) (2 2+=+ +b ab ab ab af b f a f +) D dng chng minh bt ng thc 2 2.21)2( 2 ) ( ) ( b ab af b f a f ++ +vi a, b khng m. Vy gi tr nh nht ca k l: 21= k ===========================================================31HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 CHNG MINH BT NG THC BNG CCH S DNG BT NG THC SP XP LI V BT NG THC CHEBYSHEVo Quc Huy - T Ton Tin Trng THPT Chuyn Bin Ha H Nam Btngthclmtchuynquantrngtrongchngtrnhbidnghcsinh gii Quc gia. Trong cc phng php chng minh bt ng thc th phng php p dng bt ng thc c in thng xuyn c s dng, c rt nhiu bi ton chng minh bt ng thc m li gii cp n vic s dng bt ng thc lin h gia Trung bnh cng - Trungbnhnhn(AM-GM),btngthcCauchySchwarz,btngthcHolder,bt ng thc Schur . Trong khun kh bi vit, ti xin cp n bt ng thc Sp xp li vmtsbitpsdngbtngthcny.Bncnh,bivitcngcpnmt phng php s dng bt ng thc Chebyshev (coi nh h qu ca bt ng thc Sp xp li) nh gi mt s bt ng thc 3 bin dng phn thc. I. Bt ng thc Sp xp li: Gi s 1 2...na a a v 1 2...nb b b ( )*nl hai dy cc s thc. Ta t 1 1 2 21 2 1 1......n nn n nA a b a b a bB a b a b a b= + + += + + + Gi 1 2( , ,..., )nx x xl mt hon v ca 1 2( , ,..., )nb b b , t 1 1 2 2...n nX a x a x a x = + + +Khi ta c bt ng thc sauA X B Dungthcxyrakhivchkhicc ia ttcbngnhauhoccc ib ttcbng nhau. Chng minh: Trc ht ta chng minhA X bng phng php qui np: -Vi1 n = , kt qu l hin nhin. -Gi s bt ng thc ng chon k = , vi1 n k = +ta t 1 k ib x+=v 1 k jx b+= . Tbtngthc( )( )1 10k i k ja a b b+ + tathuc 1 1 1 1 i j k k k j i ka b a b a b a b+ + + ++ + , nh vy trongXta c th thay i ixv 1 kx+ thu c tng ln hn. Sau khi i ta p dng gi thit qui np chokthnh phn u tin ca tngXv suy raA X . BtngthcX B csuyratA X bngcchxtdy 1 1...n nb b b thay cho dy 1 2...nb b b . Vi k hiu nh trn, mt cch ngn gn ta coi A l tng cc ch s cng chiu, B l tng cc ch s o chiu, cn X l tng cc ch s ty . Bt ng thc Sp xp li cho ta: tng cng chiu tng ty tng o chiu. ===========================================================32HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Vic p dng bt ng thc Sp xp li quan trng nht ch bin i bt ng thc cn chng minh v dng c cc v l tng ca tch cc phn t tng ng ca 2 dy m th tcachnglinquanvinhau(cngththocngctht).Chnghnhaidy { } , , a b c v { }3 3 3, , a b c ccngtht,cnvi, , 0 x y z > thhaidy{ } , , x y z v 1 1 1, ,x y z x y z `+ + + ) ngc th t. II. S dng bt ng thc Sp xp li: McdbtngthcSpxplicphtbiuchoccsthcnhngtrongcc bi tp di y gi thit thng cho iu kin cc s dng hoc khng m, iu ny nhm mc ch sp xp 2 dy cng chiu ca gi thit c tha ng. Bn cnh , nu khng c g c bit tc gi xin khng trnh by trng hp xy ra du ng thc ( bi v n hon ton nh pht biu trn, ng thc xy ra khi 1 trong 2 dy l dy dng ), ng thi tc gi xin csdngkhiu thaythcho cyctrongccbitonchngminhbtngthc quay vng ca 3 bin. Ngoi cch p dng bt ng thc Sp xp li, c nhiu bi ton trong s nhng bi di y hon ton c th gii bng nhng phng php khc, v bn cnh s dng bt ng thc Sp xp li ta cn p dng mt s bt ng thc c in khc. Bi ton 1:Cho n s thc( , 2) n n : 1 2, ,...,na a atha mn 1 21 2... 0... 1nna a aa a a+ + + = + + + = Chng minh rng1 212 ...2nna a na+ + + Bi gii:Khiu 1 2, ,...,tj j j v 1 2, ,...,ks s s lccphnt{ } 1, 2,..., n saocho1 2 1 2... 0 ...t kj j j s s sa a a a a a T gi thit ta suy ra 1 21...2tj j ja a a + + + =v 1 21...2ks s sa a a + + + = .Khnggimtnhtngqut,tacthgis 1 22 ... 0na a na + + + (vnutrilita dng php t'i ia a = ). Theo bt ng thc Sp xp li ta c: 1 1 11 1 11 21 2 ... 1 2 ... ( 1) ...11 1 ... 1 ...2k k tk k tn s s s j js s s j ja a na a a ka k a nana a a na na+ + + + + + + + + + + + + + + + Bt ng thc cho c chng minh. ===========================================================33HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Bi ton 2: Cho cc s thc, , 0 a b c tha mn1 a b c + + = . Chng minh rng22a bb c++ Bi gii : Ta c 2 2( ) ( )1a b a b a b c a a bb c b c b c+ + + + += = ++ + + bt ng thc cn chng minh tr thnh ( ) a a ba b cb c+ + ++ (*) Ch l 2 dy{ } , , a b cv 1 1 1, ,b c c a a b `+ + + ) c cng th t, theo bt ng thc Sp xp li ta c 2( ) ( )a ab ca aba b cb c b c b c b c+ + = + ++ + + + . Vy (*) c chng minh. Bi ton 3: Cho, , 0. x y z > Chng minh rng ( ) ( ) ( )2 2 21y x z y x zx y z y z x z x y ((( + + + Bi gii: t( ) ( ) ( )2 2 2, , a x y z b y z x c z x y = + = + = + .Lylgarittnhin2v,abtngthccnchngminhvdng: ln ln x c x a ,btngthcnyngtheobtngthcSpxplivinhnxt rng 2 dy{ } , , x y zv{ } , , a b ccng th t, ng thi hmlntng bin trn( ) 0; +nn dy{ } ln , ln , ln a b ccng c th t nh 2 dy trn, ta suy ra iu phi chng minh. Bi ton 4: Cho tam gic nhnABC . Chng minh rng31 sin sin2A B ===========================================================34HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Bi gii: S dng nh l sin, bt ng thc cn chng minh tng ng vi: ( )22 2233 4 4 0 04R abR R ab R ab RR ab R + (*) ( y a,b,c,Rtheo th t l di cc cnh BC,CA,AB v bn knh ng trn ngoi tip tam gic ABC ). T bt ng thc quen thuc 2 2 2 29 a b c R + + m 2 23 2 ab ab ab a b = + ta thu c: ( )2 23 3R ab c ab (1). Ta s chng minh 22 24 4c abR ab R R ab R + + (2). V2dy{ } , , a b c v 2 2 21 1 1, ,4 4 4 R bc R R ca R R ab R ` + + + )ccngtht nn theo bt ng thc Sp xp li ta c2 2 22 2 2 22 24 4 4 4c a b abR ab R R ab R R ab R R ab R + + + + + Vy (2) c chng minh, kt hp vi (1) ta suy ra (*) c chng minh. Bi ton 5: Cho, , 0. a b c > Chng minh rng2 2 22 2 21 1 11 1 1a b c a b cb c a b c a+ + ++ + + ++ + + Bi gii: Tac ( ) ( )( ) ( )( ) ( ) ( ) ( )( )( )2 2 2 2 2 2 2 2 2 2 2 22 2 2 2 2 22 22 21 1 1 22 1 11 11 1a b a b a b ab a b ab a b ab a ba b a b a b a bb a ab b aa b+ + + + + = + + + + ++ + + + + + = = ++ ++ + TheobtngthcSpxpli ( ) ( )2 2 2 2 2 22 2 2 2 2 2 2 211 1 1 1 1a a a a b ab b b b b b b b+= + + =+ + + + + . T ta c: ===========================================================35HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 22222 2 2 22 2 2 21 1 21 1 1 11 2 11 1 1 1a a b ab b a ba b a ab a b b| | | |+ = + + + ||\ \ | | | |+ + + + + + + = +|| ||+ + + +\ \ hay 2211a ab b++ (pcm). Bi ton 6: Cho, , 0 a b c tha mn1 a b c + + = .Chng minh rng2 2 2439a b c abc + + + Bi gii: Ta cn chng minh 2 2 2 34( )( ) 3 ( )9a b c a b c abc a b c + + + + + + +Khai trin ri a bt ng thc v dng( )35 3 3 a abc ab a b + + - Theo bt ng thc Schur: c( )33 a abc ab a b + + (1). - Theo bt ng thc Sp xp li: c 3 2 2 2a a b b c c a + + v 3 2 2 2a ab bc ca + +, cng tng v ta thu c( ) ( )3 32 4 2 a ab a b a ab a b + + (2). Cui cng, cng tng v ca (1) v (2) ta c pcm. Bi ton 7: Cho, , , 0 a b c dtha mn4 a b c d + + + = .Chng minh rng2 2 2 24 a bc b cd c da d ab + + + Bi gii: Gi s( ) , , , p q r sl hon v ca( ) , , , a b c dsao chop q r s . ===========================================================36HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Khi ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )( )( )2 2 2 222 22 1 12 4 4 2a bc b cd c da d ab a abc b bcd c cda d dabp pqr q pqs r prs s qrs pq rs pr qspq rs pr qs p q r sp s q r+ + + = + + + + + + = + + (+ + + + + + | | | | = + + ( ( || \ \ ( (bt ng thc u tin l bt ng thc Sp xp li, hai bt ng thc sau l bt ng thc AM-GM) l4 p q r s a b c d + + + = + + + = , ta c pcm. III.Bt ng thc Chebyshev dng mu s Bt ng thc Chebyshev c in c th coi nh l h qu ca bt ng thc Sp xp li (xembitppdng1phnV),tdngcinnyngitamrngbtngthc Chebyshev theo mt vi hng, sau y l mt dng m rng c nhiu ng dng chng minh bt ng thc: Bt ng thc Chebyshev dng mu s (cn gi l dng Engel) c pht biu nh sau:a)Nu ta c 1 21 21 2......nnna a ax x xx x x hoc 1 21 21 2......nnna a ax x xx x x th ta c ( )1 2 1 21 2 1 2.........n nn nn a a a a a ax x x x x x+ + ++ + + + + + b)Nu ta c 1 21 21 2......nnna a ax x xx x x hoc 1 21 21 2......nnna a ax x xx x x th ta c ( )1 2 1 21 2 1 2.........n nn nn a a a a a ax x x x x x+ + ++ + + + + + (Chng minh 2 kt qu ny bng cch s dng trc tip bt ng thc Chebyshev)Haiktqutrn,kthpvivicthm ccbiuthcphhp,trnnhiuqutrong vic nh gi cc bt ng thc i 3 bin c cha phn thc, mc d chng ch l m rng n gin t bt ng thc Chebyshev. lm r thm, chng ta xt mt vi v d sau: Bi ton 1: Cho, , 0 a b c >tha mn1 1 111 1 1 a b b c c a+ + + + + + + + ===========================================================37HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Chng minh rng a b c ab bc ca + + + + Li gii: Ta c 1 1 11 1 1a b ca b b c c a ab ac a bc ba b ca cb c+ + = + ++ + + + + + + + + + + + Khnggimtnhtngqut,gisc b a ,thth c ca ca b bc ba a ab ac + + + + + +Lic a b cab ac a bc ba b ca cb c + + + + + +lunng(vbtngthcny a b c ) Do theo a) th ( )( )1 1 11 1 132a b ca b b c c a ab ac a bc ba b ca cb ca b ca b c ab bc ca+ + = + ++ + + + + + + + + + + ++ ++ + + + + Kt hp vi gi thit ta suy ra ( )( )312a b ca b c ab bc caa b c ab bc ca+ + + + + ++ + + + + Ta c iu phi chng minh, ng thc xy ra khi1 a b c = = = . Bi ton 2: Cho, , 0 a b c >tha mn3 a b c + + = .Chng minh rng 4 9 4 9 4 913ab bc caab a b bc b c ca c a+ + ++ + + + + + + + Li gii: Khngmttnhtngqut,gisa b c ,khitbtngthclunng ( )( ) 1 0 a b c + tasuyraab a b ac a c + + + + .Tngttathuc ab a b ac a c bc b c + + + + + +Cngtbtngthclunng( )( )( ) 3 3 0 b c a a + tathuc 4 9 4 9 ab caab a b ca c a+ ++ + + +, tng t ta c 4 9 4 9 4 9 ab ca bcab a b ca c a bc b c+ + + + + + + + + ===========================================================38HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Dotheob)VTcabtngthccnchngminh ( )( )( ) 12 81 12 812 6ab bc ca ab bc caab bc ca a b c ab bc ca+ + + + + + =+ + + + + + + + Ta ch cn chng minh ( )3 4 2713 36ababab+ + bt ng thc cui cng ny ng do ( )233a b cab bc ca+ ++ + = . Bi ton 3: Cho tam gic nhnABC . Chng minh rng 1 1 1 31 tan tan 1 tan tan 1 tan tan 1 2 3 A B B C C A+ + + + + + + + + Li gii: Tng t Bi ton 1 ta c bt ng thc( )( )1 1 11 tan tan 1 tan tan 1 tan tan3 tan tan tantan tan tan 2 tan tan tan tan tan tanA B B C C AA B CA B C A B B C C A+ ++ + + + + ++ ++ + + + + T bt ng thc( ) 3 ab bc ca abc a b c + + + +v ng thctan tan A A = c ( )( )( )( )3 tan tan tantan tan tan 2 tan tan tan tan tan tan3 32 tan tan tan tan tan tan3 tan tan11 2tan tan tantan3 31 2 33 tan1 2tanA B CA B C A B B C C AA B B C C AA AA B CAAA+ ++ + + + += + ++++ += =++ Ta c iu phi chng minh. Bi ton 4: Cho, , 0 a b c >tha mn 2 2 21 a b c + + = .Chng minh rng ( )91 1 1 2a b b c c aab bc ca a b c+ + ++ + + + + + + ===========================================================39HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Li gii: Bt ng thc cn chng minh tng ng vi ( )( )2 2 22 2 22 3 2 23 0 01a b a b c a b c ab ac bca b c ab ab+ + + | | + + |+ + + +\ Khng gim tnh tng qut, gi s1 1 1 a b c ab ac bc + + +Ta cn tip tc kim tra bt ng thc( ) ( ) ( ) ( ) ( ) ( )( )( )( ) ( ) ( ) ( ) ( )( ) ( )( )2 2 2 2 2 22 2 3 3 3 2 2 23 2 2 2 22 2 3 2 2 2 2 22 23 2 2 3 2 21 12 3 2 02 2 3 3 3 2 2 02 1 2 1 2 3 3 02 2a b c ab ac bc a c b ac ab bcab acb c a b c a b c abc b c a b c a b cb c b c a a abc ab abc ac a b a cb c b a c a abc a a a b c ab acb c b c b c+ + + + + + + + + + + + + + + + + + + + + +( )2 22 2 0 abc ab ac + + bt ng thc trn lun ng, tng t ta thu c 2 2 2 2 2 2 2 2 23 2 2 3 2 2 3 2 21 1 1a b c ab ac bc a c b ac ab bc b c a bc ab caab ac bc+ + + + + + + + + Do theo b) ta c ( ) ( )2 2 2 22 2 23 3 2 2 123 2 201 3 3a b c ab ac bc a aba b c ab ac bcab ab ab+ + + + = + + + Ta c iu phi chng minh. Bi ton 5: Cho cc s thc, , x y zsao cho1 x y z + + = .Chng minh rng 2 2 291 1 1 10x y zx y z+ + + + + Li gii: Ta c 2 2 2 2 2 21 1 1 1 1 1x y z x y zx y z x y z+ + + ++ + + + + + Do ta ch cn chng minh bt ng thc trong trng hp, , 0 x y z > . Khng mt tnh tng qut, gi sx y z , khi 2 2 21 1 1 x y z + + + ===========================================================40HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Tip tc, ta kim tra( )( )2 21 01 1x yx y xyx y + + V1 2 1 x y z x y xy xy + + = + tha mn1 1 111 2 1 2 1 2 ab bc ca+ + + + + Chng minh rng3 a b c abc + + Li gii: Ta c 1 1 11 2 1 2 1 2 2 2 2a b cab bc ca a abc b abc c abc+ + = + ++ + + + + + Khng gim tnh tng qut, gi sa b c , th th2 2 2 a abc b abc c abc + + +li c 2 2 2a b cab ac bca abc b abc c abc + + + ( lun ng ) do theo a) ta c( ) 3 1 1 11 2 1 2 1 2 2 2 2 6a b c a b cab bc ca a abc b abc c abc a b c abc+ ++ + = + + + + + + + + + + + kt hp vi gi thit ta suy ra( ) 31 36a b ca b c abca b c abc+ + + + + + + Ta c iu phi chng minh, ng thc xy ra khi1 a b c = = = Bi ton 7: Cho, , 0 a b c >tha mn3 ab bc ca + + = .Chng minh rng ===========================================================41HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 ( ) ( ) ( )2 2 21 1 1 31 1 1 1 2 a b c b c a c a b abc+ + + + + + + + + Li gii: Bt ng thc cn chng minh tng ng vi ( ) ( ) ( )( )( )( )( )( )( )( ) ( ) ( )2 2 21 1 1 1 1 101 2 1 1 2 1 1 2 11 1 101 3 1 3 1 31 1 101 3 1 3 1 3abc a b c abc b c a abc c a ba bc b ac c baa bc b ac c babc ac baabc bc abc ac abc ba + + + + + + + + + + + + + + + + + + + + + Khngmttnhtngqut,gisa b c ,khi ( ) ( ) ( ) 1 3 1 3 1 3 abc ba abc ac abc bc + + + V ta cng c ( ) ( ) ( )1 1 11 3 1 3 1 3bc ac baabc bc abc ac abc ba + + + (v 3 2 2 23 3 1 ab bc ca a b c abc = + + ) Do theo b) ta c( ) ( ) ( )( )( )3 3 1 1 101 3 1 3 1 3 3 9ab bc ca bc ac baabc bc abc ac abc ba abc ab bc ca + + =+ + + + Ta c iu phi chng minh. IV. Mt s bi tp p dng Bi tp 1: (bt ng thc Chebyshev) K hiu, A B ging nh trong bt ng thc Sp xp li. CMR: ( )( )1 2 1 2... ...n na a a b b bA Bn+ + + + + + Bi tp 2: Chon sthcdng 1 2, ,...,nc c c (vi *n ).Takhiu ( ) ( )( ) ( ) ( ) ( )1 22 2 21 2 1 211 2 1 2... , ... ,... , 1 1 ... 1n nnn nRMS c c c n AM c c c nGM c c c HM n c c c (= + + + = + + + = = + + +(

CMR:RHM AM GM HM Bi tp 3: Cho, , 0. a b c >CMR: 12k ka aa b+ ( 2) k Bi tp 4: ===========================================================42HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Cho( ) , , 1; a b c + . CMR: 2 2 2log log log log log logab bc caa bc ab c abcc a b bc ca ab + + + +Bi tp 5: K hiu, , a b cln lt l di cc cnh, , BC CA ABca tam gic nhnABC . CMR:( ) 4cos cos cosa b b c c aa b cC A B+ + ++ + + +Bi tp 6: Cho, , 0 a b c > . Chng minh cc bt ng thc sau: a)( ) ( ) ( )2 2 20a b b c b c c a c a a ba b b c c a + + + + + b)3 3 3 3 3 32 2 2 2 2 2a b b c c aa b cb c c a a b+ + ++ + + ++ + + c)5 5 5 2 2 23 3 3 3 3 32a b c a b ca b b c c a+ ++ + + + +

Bi tp 7: Cho, , 0 a b c >tha mn3 a b c + + = . CMR:1 1 1 39 9 9 8 ab bc ca+ + Bi tp 8: Cho, , , 0 a b c dtha mn4 a b c d + + + = . CMR: 1 1 1 115 5 5 5 abc bcd cda dab+ + + Bi tp 9:Cho, , 0 a b c . CMR: ( )2 2 23 8 8 8 a b c a bc b ca c ab + + + + + + +Bi tp 10:Chng minh rng vi, , 0 a b c v2 0 k ta c bt ng thc2 2 22 2 2 2 2 2 2 2 20a bc b ca c abb c ka a c kb b a kc + + + + + + + + V. Ti liu tham kho: 1.G.H. Hardy, J.E. Littlewood, G. Polya, Bt ng thc. 2.Phm Kim Hng, Sng to bt ng thc ( tp 1). 3.VIMF (Nhiu tc gi), Discovery Inequalities (Third version). ------------------------------------------------------------------------------- ===========================================================43HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 TNH TUN HON TRONG DY S NGUYN Tc gi: Ng Th Hi Gio vin trng THPT chuyn Nguyn Tri, Hi Dng. Dys lmt lnh vc khv rtrng.gii cccbi tonloiny khngch i hi ngi lm Ton phi s dng nhiu kin thc khc nhau ca Ton hc m cn phi c kh nng sng to rt cao. Trong cc bi ton v dy s mt vn c quan tm nhiu l tnh cht s hc ca dy s nh: tnh chia ht, tnh cht nguyn hay tnh chnh phng Chngrtadngvphongph.Trongnhiutrnghp,dyschlvbngoicn bncht bi ton lmt bi s hc. Chnh v l , cc bi ton v s hc nichung, cc bi ton v tnh cht s hc ca dy s ni ring thng xut hin trong cc k thi hc sinh gii quc gia v quc t, v n bao gm nhiu bi ton hay v kh. Trong khun kh ca bi vit ny ti ch cp n mt kha cnh rt nhca dy s nguynl tnh tun hon, hi vng rng y l mtti liutham kho tt cho cc em hcsinh kh v gii. Trc ht ta hy xem nh l su y: nh l: Cho dy s nguyn truy hi cp k ( k l s nguyn dng) ngha l Nu dy b chn th n l dy tun hon k t lc no . Chng minh: Gi s dy b chn bi s nguyn dng M, ngha l. XtccbksCtia bkhcnhaunntrongbutinphac2btrngnhau. Chng hn Ngha l M nn tth ta c Vy dy tun hon vi chu k k t Hqu:Chodysnguynthomn tronglccsnguynvmls nguyn dng ln hn 1. Gi l s d trong php chia cho m. Khi dytun hon. Chng minh: Theogi thit ta c. Theo tnh cht ca ng d thc ta c Theoccxcnhtactcldybchnvtruyhituyntnh cpknntheonhltrndytunhonktlcno,nghalsaocho Chnta c

Vy. Tng t ta c =,, Do dytun hon vi chu k T. ===========================================================44HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Sau y ti s a ra mt s v d in hnh v vic p dng nh l trn. Cc bi ton nu ra y u s dng n tnh tun hon ca dy s d. Gi sl s d trong php chia cho mt s nguyn dngm no . Khi dy b chn v cng c cng cng thc truy hi vi dy nn theo h qu trn n l dy tun hon. Bi 1:ChoVilsdcaphpchiacho 100. Tm s d trong php chiacho 8.Bi gii:Gi l s d trong php chia cho 4. Theo gi thit nn Mt khc tc l dy b chn do dy ny tun hon. Ta tnh c D kim tra tun hon chu k 6, ngha l Li c. Do nn cng tnh chn l suy ra hay Vy M. Do chia ht cho 8. Bi 2:Chody,n=0,1,2,xcnhbiv Chng minh rng: chia ht cho 20 Bi gii: T cng thc truy hi ca dy ta thy( .Gi l s d trong php chia cho 4. Khi Hn na(nn tng t bi 1 dy tun hon chu k 6. Ta c(. V vy tc l chia ht cho 4. Mt khc vi ta c Suy ra Vy.Do chia ht cho 20. Bi 3:Cho dy , n=,1,2,3 xc nh bi Chng minh rng tn ti v s s hng ca dy chia ht cho 2005. Bi gii: ===========================================================45HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Tacchiahtcho2005.Gi lsdtrongphpchiacho2005.T cngthctruyhicadytac ngthidytunhonktlcno,nghal saocho . Chn ta c Vy . Tng t ta cng c Do hay chia ht cho 2005 (pcm) Bi 4: Cho dy , n=,1,2,3 xc nh bi Chng minh rng vi mi s nguyn dng tn tai v s s t nhin sao cho cng chia ht cho. Bi gii: Xt dy , n=,1,2,3 xc nh nh sauTa tnh c Do Gi l s d trong php chia cho m. Khi dytun hon ngha l tn ti s t nhin T>1 sao cho Vychiahtchomvihay chia ht cho m vi . Bi 5: Gi l nghim dng ln nht ca phng trnh Xt dy xc nh theo cng thc sau: Tm s d trong php chia cho 17. Bi gii:t Ta c>0, .Do l hm lien tc trn R nn phng trnh c 3 nghim phn bit: tKhilnghimcaphngtrnhsaiphntuyn tnhthunnhtcptctrnglDotac Haytrong ( s dng nh l Vi-et) V vy Do>0 Suy ra. ===========================================================46HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Licnn

(do. Vy=( . Cho nn=Vy Gil s d trong php chia cho 17. Khi dytun hon v bng tnh ton trctiptac D kim tra tun hon chu k 16, ngha l l Vynn hay chia17d 6. Cui cung ti xin nu thm 2 bi tp khc c th gii theo phng php ny bn c tham kho Bi 1:Cho dy , n=,1,2,3 xc nh bi Chng minh rng: a) Mi s hng ca dy u l s nguyn dng.b) C v s nguyn dng n sao choc 4 ch s tn cng l 2003. c) Khng tn ti s nguyn dng n sao choc 4 ch s tn cng l 2004. Hng dn: Bin i dn n Bi 2:Dys nguyn , n=,1,2,3 xc nh bi Chng minh c v s s hng ca dy chia ht cho 1986. ===========================================================47HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 NH L PASCAL V NG DNG L c Thnh GV THPT Chuyn Trn Ph Hi Phng Trong bi vit chuyn ny ti mun cp n mt nh l c rt nhiu ng dng a dng, l nh l Pascal v lc gic ni tip ng trn. Trong thc t p dng, khi thay i th t cc im, hay l khi xt cc trng hp c bit ta s thu c rt nhiu kt qu khc nhau. Trc ht ta pht biu ni dung nh l: nh l Pascal: Cho cc imA, B, C, D, E, F cng thuc mt ng trn (c th hon i th t). Gi P AB DE, Q BC EF, R CD FA = = = .Khi cc imP, Q, R thng hng. Chng minh: Gi X EF AB, Y AB CD, Z CD EF. = = = pdngnhlMenelauschotamgic XYZiviccngthngBCQ, DEP, FAR , ta c: ( )( )( )CY BX QZ1 1CZ BY QXFZ AX RY1 2FX AY RZEZ PX DY1 3EX PY DZ = = =

Mt khc, theo tnh cht phng tch ca mt im i vi ng trn ta c: ( ) YC.YD YB.YA, ZF.ZE ZD.ZC, XB.XA XF.XE 4 = = = Nhn (1),(2) v (3) theo v, ta c:( )QZ RY PX CY.BX.FZ.AX.EZ.DY1QX RZ PY CZ.BY.FX.AY.EX.DZQZ RY PX YC.YD ZF.ZE XB.XA15QX RZ PY YB.YA ZD.ZC XF.ZE = = Th (4) vo (5), ta c QZ RY PX1.QX RZ PY =VyP, Q, R thng hng (theo nh l Menelaus). ngthngPQRtrncgilngthngPascalngvibim A, B, C, D, E, F. BngcchhonvccimA, B, C, D, E, Ftathucrtnhiuccngthng Pascal khc nhau, c th ta c ti 60 ng thng Pascal. ZYXRQP ABCDEF ===========================================================48HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Chnghnhnhvbnminhhatrnghpcc im ACEBFD. Ngoi ra khi cho cc im c th trng nhau (khi lcgicsuybinthnhtamgic,tgic,nggic),vd E F th cnhEFtr thnhtiptuyn cangtrnti E , ta cn thu thm c rt nhiu cc ng thng Pascal khc na. Hnhvdiyminhhatrnghpccim ABCDEE, ABCCDD, AABBCC: Tip theo ta a ra cc bi ton ng dng nh l Pascal: Bi ton 1: (nh l Newton) Mt ng trn ni tip t gicABCD ln lt tip xc vi cc cnhAB, BC, CD, DA tiE, F, G, H.Khi cc ng thngAC, EG, BD, FH ng quy. RQYPABCDERQPADBCQRPBCAPQRABCDEF ===========================================================49HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Li gii: GiO EG FH, X EH FG = = . VDlgiaoimcacctiptuynvingtrntiG, H, pdngnh l Pascal cho cc imE, G, G, F, H, H , ta c: EG FH O,GG HH D,GF HE X. = = = Suy raO, D, Xthng hng. pdngnhlPascalchoccim E, E, H, F, F, G, ta c: EE FF B,EH FG X,HF GE O. = = = Suy raB, X, Othng hng. T ta cB, O, Dthng hng. VyEG, FH, BD ng quy tiO. Chng minh tng t i vi ng thngAC ta c iu phi chng minh. Bi ton 2:Cho tam gicABC ni tip trong mt ng trn. GiD, Eln lt l cc im chnh gia ca cc cungAB, AC;Pl im tu trn cungBC;DP AB Q, PE AC R = = .Chng minh rng ng thngQR cha tmIca ng trn ni tip tam gicABC. Li gii: VD, E lnltlimchnhgiacacc cungAB, AC nnCD, BE theo th t l cc ng phn gic ca gc ACB, ABC. Suy raI CD EB. = pdngnhlPascalchosuim C, D, P, E, B, A,ta c: CD EB I = ; DP BA Q; =PE AC R. =VyQ, I, Rthng hng. Bi ton 3: (Australia 2001) XOCDABGEHFIRQED AB CP ===========================================================50HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Cho tam gic ABC ni tip ng trn (O), ng cao nh A, B, C ln lt ct (O) ti A, B, C. D nm trn (O),DA' BC A", DB' CA B", DC' AB C" = = = . Chngminhrng:A,B,C,trctmHthng hng. Li gii: pdngnhlPascalchosuim A, A', D, C', C, B,ta c: AA' C' C H,A' D CB A",DC' BA C". = = = VyH, A", C" thng hng. Tng t suy ra A, B, C, H thng hng. Bi ton 4: (IMO Shortlist 1991) P thay i trong tam gic ABC c nh. Gi P, P l hnh chiu vung gc ca P trn AC, BC, Q, Q l hnh chiu vung gc ca C trn AP, BP, giX P' Q" P"Q' = .Chng minh rng: X di chuyn trn mt ng c nh. Li gii: Ta c: 0CP' P CP"P CQ' P CQ"P 90 = = = =NnccimC, P', Q", P, Q', P"cngthuc mt ng trn. pdngnhlPascalchosuim C, P', Q", P, Q', P" ta c: CP' PQ' A,P' Q" Q' P" X,Q"P P"C B. = = = VyA, X, B thng hng. Vy X di chuyn trn ng thng AB c nh. Bi ton 5: (Poland 1997) NggicABCDElithamn: 0CD DE, BCD DEA 90 = = = .imFtrongon AB sao cho AF AEBF BC= Chng minh rng: FCE ADE, FEC BDC = = . Li gii: HC"B"A"C'B'A'B CADXQ"Q'P"P'AB CPRQPFAEDCB ===========================================================51HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 GiP AE BC = ,Q,RlnltlgiaoimcaADvBDvingtrnng knh PD,G QC RE = . p dng nh l Pascal cho su imP, C, Q, D, R, E,ta c: PC DR B,CQ RE G,QD EP A. = = = VyA, G, B thng hng. Li c: DAGDBGDAEDBCsin GQDDA GQS AG DG.DA.sin GDQ DA.GQ DA.sin QREDGBG S DB.GRDG.DB.sin GDR sin GRD DB.sin RQCDB GRDGS DA.sin ADE DA.DE.sin ADE AES BCDB.sin BDC DB.DC.sin BDCAG AFF GBG BF = = = = = = = = = = T d dng c FCE ADE, FEC BDC = = . Bi ton 6: Cho tam gic ABC ni tip ng trn (O), A, B, C l trung im BC, CA, AB.ChngminhrngtmngtrnngoitipcctamgicAOA,BOB,COCthng hng. Li gii: GiA,B,CltrungimcaOA, OB,OC.I,J,Kltmccngtrnngoi tip cc tam gic AOA, BOB, COC. Khi IlgiaoimcacctrungtrccaOAv OA, hay chnh l giao im ca BC v tip tuyncangtrn(O;OA)tiA.Tng t vi J, K. pdngnhlPascalchosuim A", A", B", B", C", C" ta c: A"A" B"C" I,A"B" C"C" K,B"B" C"A" J. = = = VyI, J, Kthng hng. Bi ton 7: (China 2005) MtngtrnctcccnhcatamgicABCtheothtticcim 1 2 1 2 1 2D , D , E , E , F , F . 1 1 2 2 1 1 2 2 1 1 2 2D E D F L, E F E D M, FD F E N = = = .Chng minh rng AL, BM, CN ng quy. KJIB"A"C"C'B'A'OBCA ===========================================================52HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Li gii: Gi 1 1 2 2 1 1 2 2 1 1 2 2D F D E P, E D E F Q, FE F D R = = = .p dng nh l Pascal cho su im 2 1 1 1 2 2E , E , D , F , F , Dta c: 2 1 1 21 1 2 21 1 2 2E E FF A,E D F D L,D F D E P. = = = Suy raA, L, Pthng hng. Tng t B, M, Q thng hng, C, N, R thng hng. 2 1 1 2 1 2 2 1 1 2 1 2 2 1 1 2 1 2E E D F CA D F X, F F E D AB E D Y, D D FE BC FE Z = = = = = = p dng nh l Pascal cho su im 1 1 1 2 2 2F , E , D , D , F , Eta c: 1 1 2 21 1 2 21 2 2 1FE D F R,E D F E Q,D D E F Z. = = = Suy raQ, R, Z thng hng. Tng t P, Q, Y thng hng, Z, P, X thng hng. Xt cc tam gic ABC, PQR c:X CA RP, Y AB PQ, Z BC QR = = = . p dng nh l Desargues suy ra cc ng thngAP AL, BQ BM, CR CN ng quy.

Bi ton 8: (nh l Brianchon) LcgicABCDEFngoitipmt ng trn.Khi AD, BE, CF ng quy. Li gii: ZNMRQPLF2F1E2E1D2 D1ABCNPMABCDEFHGLKJI ===========================================================53HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Ta s chng minh nh l ny bng cc v i cc thy rng Pascal v Brianchon l hai kt qu lin hp ca nhau. Gi cc tip im trn cc cnh ln lt l G, H, I, J, K, L. Khi GH, HI, IJ, JK, KL, LG ln lt l i cc ca B, C, D, E, F, A. GiGH JK N, HI KL P, IJ LG=M = = Theo Pascal cho lc gic GHIJKL ta c M, N, P thng hng. MM,N,Plnltl icccaAD,BE,CFnnsuyraAD,BE,CFngquyti cc ca ng thng MNP. Bi ton 9:ChotamgicABC,ccphngicvngcaotinhB,ClBD,CE,BB,CC. ng trn ni tip (I) tip xc vi AB, AC ti N, M.Chng minh rng MN, DE, BC ng quy. Li gii: Gi hnh chiu ca C trn BD l P, hnh chiu ca B trn CE l Q. D chng minh:

0ANMI ICP NMI PMI 1802= = + = Nn M, N, P thng hng. Tng t suyra M, N, P, Q thng hng. pdngnhlPascalchosu imB', C', B, P, Q, C,ta c: B' C' PQ S,C' B QC E,BP CB' D. = = = VyS, E, D thng hng, hay l MN, DE, BC ng quy ti S. Bi ton 10:Cho tam gic ABC ni tip ng trn (O). Tip tuyn ca (O) ti A, B ct nhau ti S. Mt ct tuyn quay quanh S ct CA, CB ti M, N, ct (O) ti P, Q.ChngminhrngM,N,P,Qlhng im iu ha. Li gii: VtiptuynME,MD ca(O)ctSA, SB tiK, L.pdngnhlNewtonchotgic ngoitipSKMLtacBE,AD,SM,KL ng quy. PQSC'B'NMIEDAB CILKDEMN QSABCP ===========================================================54HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 p dng nh l Pascal cho su imA, D, E, E, B, C,ta c: AD EB I,DE BC N',EE CA M. = = = VyI, N', M thng hng, hayN N' , tc lN DE . Do DE l i cc ca M i vi (O) nn M, N, P, Q l hng im iu ha. Bi ton 11: (nh l Steiner) ng thng Pascal ca cc lc gic ABCDEF, ADEBCF, ADCFEB ng quy. Li gii: Gi 1 1 2AB DE P , BC EF Q , AD BC P , = = =2 3 3DE CF Q , AD FE P , CF AB Q . = = =pdngnhlPascalchosuim A, B, C, F, E, D,ta c: 1 3 1 31 2 1 22 3 2 3PQ Q P AB FE P,PQ Q P BC ED Q,Q Q P P CF DA R. = = = = = = VyP, Q, R thng hng. pdngnhlDesarguessuyraccng thng1 1 2 2 3 3PQ , P Q , P Qng quy. HayngthngPascalcacclcgic ABCDEF, ADEBCF, ADCFEB ng quy. Bi ton 12: (nh l Kirkman) ng thng Pascal ca cc lc gic ABFDCE, AEFBDC, ABDFEC ng quy. Ta bit trn l c 60 ng thng Pascal. C 3 ng mt ng quy to ra 20 im Steiner. Trong 20 im Steiner c 4 im mt li nm trn mt ng thng to ra 15 ng thng Plucker. Ngoi ra 60 ng thng Pascal li c 3 ng mt ng quy to ra60imKirkman.MiimSteinerlithnghngvi3imKirkmantrn20ng thng Cayley. Trong 20ng thng Cayley, c 4 ng mt li ng quy to ra 15 im Salmon kt thc xin a ra mt s bi ton khc p dng nh l Pascal: Bi ton 13: (MOSP 2005) Cho t gic ni tip ABCD, phn gic gc A ct phn gic gc B ti E. im P, Q ln lt nm trn AD, BC sao cho PQ i qua E v PQ song song vi CD. Chng minh rngAP BQ PQ + = . RQPQ3P3Q2P2Q1P1A FBCDE ===========================================================55HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Bi ton 14:Cc im P, Q trong tam gic ABC sao cho 0BP CP, BQ CQ, ABP ACQ 180 = = + = . Chng minh rng BAP CAQ = . Bi ton 15: (IMO Shortlist 2007) ChotamgicABCcnh,cctrungim 1 1 1A , B , C caBC,CA,ABtngng. im P thay i trn ng trn ngoi tip tam gic. Cc ng thng 1 1 1PA , PB , PCct li ng trn ti A, B, C tng ng. Gi s cc im A, B, C, A, B, C i mt phn bit v cc ng thng AA, BB, CC to ra mt tam gic. Chng minh rng din tch ca tam gic khng ph thuc vo v tr ca P. Bi ton 16:HaitamgicABC,ABCccngngtrnngoitip.Cccnhcahaitamgic ct nhau ti 6 im to ra mt hnh lc gic.Chng minh rng cc ng cho ca hnh lc gic ng quy. Bi ton 17: (IMO 2010) im P nm trong tam gic ABC viCA CB . Cc ng AP, BP, CP ct li ng trn ngoi tip ti K, L, M. Tip tuyn ca ng trn ngoi tip ti C ct AB S. Gi s SC SP = .Chng minh rngMK ML = . Bi ton 18: (MEMO 2010) ng trn ni tip tam gic ABC tip xc cc cnh BC, CA, AB ti D, E, F tng ng. K l i xng ca D qua tm ng trn ni tip. DE ct FK ti S.Chng minh rng AS song song BC. ===========================================================56HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 HM S HC V MT S BI TON V HM S HC Trng THPT Chuyn Hng Yn Hm s hc l hm s c min xc nh l tp con ca tp s t nhin . K hiu A l tp hp tt c cc s hc. I.Mt s tnh cht chung ca cc hm s hc 1.Tch chp Dirichlet (gi l tch chp) nh ngha 1.Chof vg l hai hm s hc. Tch chp Dirichlet gi l tch chp ca fv, gk hiu lf * g , xc nh bi ( ) ( ) ( ) *, *|N ndng d f n g fn d ||

\|= trong tng ly trn tt c cc s nguyn dngd m chia htn . V d 1. Ta xt hai hm s hc sau nu ( )=01n ( ) 1 = n evi mi. * N n Khi vi miA f ta c ( ) ( ) ( ) ( ) n fdnd f n f an d= ||

\|= |* ) vi mi* N n . * f f = ( ) ( ) ( ) ( ) = ||

\|=n d n dd fdne d f n e f b| |* )vi mi* N n . nh l 1. ViA h g f , ,ta c( ) ( )( ) . * * * ). * * * * ). * * )h f g f h g f iiih g f h g f iif g g f i+ = +== 2. Ton t( ) A f R a f Ta ,v( ) A f Lf nh ngha 2. Cho, R a vi miA f ta xc nh ton tf Tanh sau ( ) ( ) . * N n n n f n f Taa =nh ngha 3. Vi miA f ,ta xc nh ton tLf nh sau ( ) ( ) . * ln N n n n f n Lf =V d 2. Vi hai hm s hc ve , ta c( ) ( ) ( ) = = =aaaT N n n n n n T * . ( ) ( ) . * N n n n n e n e Ta aa = =( ) ( ) . * 0 ln N n n n n L = = ( ) ( ) . * ln ln N n n n n e n Le = =nh l 2. ChoA g f ,ta c( )( ) ( )( ) . ). , * * ). * * * )Lg Lf g f L iiiR a g T f T g f T iiLg f Lf g g f L ia a a+ = + =+ = iv) K hiu kf * l tch chp caf vi chnh n k ln, ngha l nu n =1 nu* N n , n>1 ===========================================================57HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 . * ... * * * ; * ; *1 0f f f f f f fk= = =

k ln Khi vi +Z k th( )( ). * * *1A f Lf kf f Lk k = 3. Hm s Mobiusnh ngha 4. Cho, A f hmA g c gi l nghch o tch chp caf nu = g f * , k hiu. *1 = f gnh l 3. ChoA g f , , ta ci) 1*f tn ti khi v ch khi() . 0 1 fii) Nu tn ti 1*f th 1*f c xc nh duy nht theo quy np nh sau ()()( )()( )> ||

\| ==n ddN n ndnf d ffn fff|11 11. , 2 *11*,111 * c bit nu p l s nguyn t th( )( )().1*21fp fp f = iii) Nu tn ti 1*fth( ) ( ). * * *1Lf f f Lf = iv) Nu tn ti 1 1* , * g fth cng tn ti( )1* *g f v c xc nh nh sau ( ) =1* * g f1 1* * * g fNhn xt: T v d 1 v nh l 3suy ra lun lun tn ti 1* v 1*ev xc nh()()()( )()( ) ( )> > = = ||

\| == = =n ddn ndnd n|11 11. 1 0 *11*, 1 1111 * =1* . i vi hm s hc e vic tm 1*e s kh hn , ta s xt di y. nh ngha 5. Hm Mobiusc nh ngha l nghch o tch chp ca hm e, ngha l1*= e . Nhn xt: T v d 1 v nh ngha hm Mobius ta suy ra kt qu quan trng sau ( ) ( )( ) ( )>== = =n dn nun nun n e d|. 1 0, 1 1* nh l 4. Ta ci) Nu* N n th( ) ( ) =011rn ii) NuA F f ,th( ) ( ) ( ) ( ) . *| |N ndnF d n f d f n Fn d n d ||

\|= = nu n = 1 nu n c phn tch tiu chun l n = p1p2pr, r>1 nu tn ti s nguyn t p sao cho p2|n. ===========================================================58HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 (Cng thc o ngc kin-Liuvin) T nh l trn ta c() ( ) ( ) ( ) ( ) ( ) ( ) ( ) . 1 1 5 . 3 15 ; 0 2 8 ; 1 1 2 ; 1 12 3 1= = = = = = = = 4. Hm nhn tnh nh ngha 6. ChoA f vf khng ng nht bng khng, Hmf c gi l hm nhn tnh nu( ) ( ) ( ) n f m f mn f =vi mi* , N n m tha mn (m,n) =1. Hmf c gi l hm hon ton nhn tnh nu( ) ( ) ( ) n f m f mn f =vi mi * , N n m . K hiu M l tp hp tt c cc hm nhn tnh. Nhn xt: T nh ngha trn, gi s kkp p p n ...2 12 1= l s phn tch tiu chun ca n, ta c kt qu sau Nuf l hm nhn tnh th( ) ( ).1==kiiip f n f Nufl hm hon ton nhn tnh th( ) ( ) ( ) p f p f = vi mi* N v p nguyn t. chng minh hai hm nhn tnh bng nhau ch cn chng minh chng bng nhau trn mi ly tha ca cc s nguyn t. chng minh hai hm hon ton nhn tnh bng nhau ch cn chng minh chng bng nhau trn tp cc s nguyn t. Vifl hm nhn tnh, nu m = n = 1 th (m,n) = 1 suy ra() () (), 1 1 1 f f f = do vy () 0 1= fhoc() 1 1= f Nu() 0 1= f th vi mi n cng c (n,1) = 1( ) ( ) () 1 f n f n f = =0, nnf l hm ngnht bng khng, vy nufl hm nhn tnh th() 1 1= f . Nu 2 1, f fl cc hm nhn tnh, hm tchf ca chng c nh ngha bi:( ) ( ) ( ) *2 1N n n f n f n f = , k hiu 2 1f f f =thfcng l hm nhn tnh. Ta d dng kim tra c - Vi mi* N n hm( )na a f =vi mi* N a l mt hm nhn tnh. - Hml hm nhn tnh. - Hm e l hm hon ton nhn tnh. nh l 5. ( tnh cht c bn ca hm nhn tnh) Nu* , 1 N n n >v kkp p p n ...2 12 1= l s phn tch tiu chun ca n th vi mi hm nhn tnhf ta c ( ) ( ) ( ) ( ) ( ) ( ) ( ) () 1 , ... 1 ... ... 1|1 11 kk kn dp f p f p f p f d f + + + + + + = trong tng v tri ly trn tt c cc c s dng d ca n. Chng minh Nu ta khai trin v phi (1), th ta s c mt tng gm cc s hng c dng ( ) ( ) ( )kkp f p f p f ...2 12 1, trong i i 0vi mi i = 1,2,3,..,k. Theo gi thitf l hm nhn tnh nn( ) ( ) ( ) ( )k kk kp p p f p f p f p f ... ...2 1 2 12 1 2 1=nhng( ) k i p p pi i kk,..., 3 , 2 , 1 , 0 ...2 12 1= chnh l mt c d ca n v mi c d ca n u c dng . V vy v phi ca (1) l tng ca nhng s hng c dng( ) d f , trong d chy khp ch mt ln tt cc c dng ca n, chnh l v tri ca (1) ( pcm). ===========================================================59HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Nhn xt: Vn dng nh l trn i vi hm nhn tnh( )na a f =vi mi* N a v kkp p p a ...2 12 1= l s phn tch tiu chun ca a th ta c( ) ( ). ... 1 ... ... 12|121 11 knknknkn dn n n np p p p p p d + + + + + + + + = Nun = 0 v k hiu d(a) l cc s c dng ca a th ( ) ( )( ) ( ). 1 ... 1 1 12 1| |0+ + + = = = ka d a dd a d Nun = 1 v k hiu( ) a l tng cc c dng ca a th( ) .|=a dd a Do ( ) ( ) ( ) .11...11... 1 ... ... 11111 2121 111= + + + + + + + + =+ +kkk k kppppp p p p p p akk nh l 6. ChoR a vM f ta c i) 1 *ftn tiii)M f Ta v nufl hm hon ton nhn tnh thf Ta cng l hm hon ton nhn tnh. iii)( ) .1 * 1 * = f T f Ta a Chng minh. i) Ta c() 1 1= f suy ra 1 *ftnti (theo nh l 3 ) ii) NuR a v* , N n m tha mn (m,n) = 1th ( ) ( )( ) ( ) ( ) ( ) ( ). n f T m f T n n f m m f mn mn f mn f Ta aa a aa= = = v vy ta cM f Ta . Nufl hm hon ton nhn tnh th( ) ( ) ( ) * , N n m n f m f mn f = suy ra ( ) ( )( ) ( ) ( ) ( ) ( ). n f T m f T n n f m m f mn mn f mn f Ta aa a aa= = =hayf Ta l hm hon ton nhn tnh. iii) DoM f theo i) tn ti 1 *f , v vy ta c( ) = = = a a a aT f f T f T f T1 * 1 ** * ( ) .1 * 1 * = f T f Ta a Da vo cc kt qu trn ta hon ton c th chng minh nh l sau nh l 7. Cho. , A g f i) NuM g f th M g f * ,ii) Nu. * M g th M g f v f iii) NuM f th.1 *M f II.Mt s hm s hc thng gp 1. Tng cc c nh ngha 7. Cho n l s nguyn dng, vi mi s thc ta gi hm l tng ly tha ca cc c dng ca n, ngha l( ) . *|0N n d nn dd = >Khi1 = ta vit ( )> = =n ddN n d n n|01*, ) ( ( ) n cn c gi l tng cc c dng ca n. Khi0 = ta vit ( )> = =n ddN n n d n|00*, 1 ) ( trong d(n) l s cc c dng ca n. ===========================================================60HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 nh l 8. Ta ci). * R e T e = ii) l hm nhn tnh. iii)Vi1 , > n N nv nc phn tch tiu chun l rrp p p n ...2 12 1= ta c cc kt qu sau Nu0 th( )( )=+=ri iippn111 Nu0 = th( ) ( )=+ =riin11 Chng minh i) Nu *N n th( )( ) ( ) ( ) ( ) ( ), *| | | |n d d d e d e T d e Tdne n e T en d n d n d n d = = = = ||

\|= suy ra. * e T e =ii) Doe T e, l hm nhn tnh suy rae T e* l hm nhn tnh theo nh l 7, v vy e T e * = l hm nhn tnh. iii) V l hm nhn tnh nn ta ch cn tnh( )1ipvi, 1 r i Nu0 = th( ) ( ) . 1|01= =iipiki ip d p M iipch c cc c l ii i ip p p,..., , , 12do ( ) ( ) ( ) 110+ = =i i iip d p Nu0 th ( ) ( )11... 1) 1 (201= + + + + = =+= iii i itti ippp p p p piii V vy ta c( )( )= +=+0 1011) 1 (1 nunupppiiiii T ta c kt qu cn chng minh V d3: Tnh( ) ( ) ( ) ( ) ( ). 10 ; 10 ; 18 ; 18 ; 182 0 2 0 Gii ( ) ( )( ) ( )( )( )( ) . 1301 51 51 21 2) 5 . 2 ( 104 ) 1 1 )( 1 1 ( ) 10 ( ) 5 . 2 ( 104551 31 31 21 2) 3 . 2 ( 18. 421 21 21 31 33 . 2 186 ) 1 2 )( 1 1 ( 3 . 2 ) 18 ( 1824242 20 0262422 22 2220 0== == + + = = === === == + + = = = dd 2. S cc c nh ngha 8. Cho k, n l cc s nguyn dng, ta gi hm dk(n) l cch vit n thnh tch ca k nhn t, trong th t ca cc nhn t cng c tnh. Nhn xt ===========================================================61HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 d1(n)= 1;d2(n)= d(n) l s cc c dng ca n. Theo nh ngha th dk+1(n) l s cch vit n nh l tch ca (k+1) nhn thay l s cch vit n c dng = n a1a2ak+1. Nu c nh ak+1|n th 1 + kanc th vit nh l tch ca k nhn t v c |||

\|+1 kkand cch. V vy ta c =+n ik kk i d n d|11 ) ( ) (nh l 9. Cho* N k , ta c i) kke d*=ii) dk l hm nhn tnh iii)Nu ==riiip n1l s phn tch tiu chun ca n th vi mi1 kta c ( )= +=rikk kiC n d111 Chng minh i) Chng minh quy np theo k. Nu k = 1 th d1(n) = 1 = e(n) suy ra d1=e*1. Nu k = 2 th = =n dn d n d|21 ) ( ) (Ta li c( ) , 1 ) ( ) ( * ) (| |2 * = ||

\|= =n d n ddne d e n e e n e . * ) ( ) (2 *2N n n e n d ra suy =V vy ta c d2= e*2. Gi s vi mi1 1 k mta c dm= e*m. Khi ta c( )( ) ( )( ) ) ( ) ( * ) () ( ) ( ) (* 1 *|1 *|1 *|1n e n e e i einei e i d n dk kn ikn ikn ik k= = ||

\|== = haydk = e*k* N k (pcm). ii)V e l hm nhn tnh suy ra dk = e*k l hm nhn tnh (theo nh l 7). iii) S dng kt qu: Num,nl cc s nguyn khng m th ) 1 (110++ +=+ =mn mminn iC CV dk l hm nhn tnh nn ta ch cn chng minh( ) ,11 +=kk kC p d (2)vi p nguyn t, * N . Tht vy ta c Nu k = 1 th( ) , 11101 += = =kkC C p d suy ra (2) ng. Nu k = 2 th( ) ( ) , 11111 2 + += = + = =kkC C p d p d suy ra (2) ng. Gi s vi mik m 1m( ) ,11 +=mm mC p dkhi ta c ( ) ( )kki ikk iik kC C p d p d+= = + + = = = 0 011 1 ( theo (1)) nn (2) ng vik+1.Vy (2) ng vi mi* N k V vy nu ==riiip n1l s phn tch tiu chun ca n th( )= +=rikk kiC n d111 3. Hm le( ) n ===========================================================62HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 nh ngha 9. Cho* N n , ta gi hm( ) n l s cc s nguyn dng nguyn t vi n v nh hn n, tc l ( ) . 11 ) , (1= =n dn dn nh l 10. Ta c i)( ) . ) (|=n kknk n ii). *1e T =iii) l hm nhn tnh iv) Nu n l s nguyn dng th( )|||

\| =n ppn n|11 . Chng minh i) T nhn xt ca nh ngha 5 ta c vi mi n>1 th( ) =0|n dd ( ) ( ) = == = = =ndn kd k n d kndn dn dk k n1|| ) , |( 11 ) , (1. ) ( 1 1 C nh k l l c ca n, ta phi ly tng vi nhng gi tr ca d tha mnn d 1 m d chia ht cho k. Nu d=qk th don d 1 suy ra knq 1v ( ) ( ) ( ) = == = =k nq n k n k n kk nqknk k k n/1 | | |/1. 1 ) ( (pcm) ii). Theo i) ta c( ) ( ) ( ) ( ) ( ) = ||

\|= ||

\|= = e Tkne T kknkne kknk nn k n k n k1 1| | |* e T1* = . iii). Theo ii) ta ce T1* = . Me T1, l hm nhn tnh nnl hm nhn tnh. iv) Vl hm nhn tnh xc nh( ) n ta ch cn tnh gi tr ca( ), p vi * N v p nguyn t. Ta c ( ) ( ) ( ) ( ) ( ) = |||

\| = = + = = = =n p ii ip kpn npp p p p p p p p pkpk pi\1 1 00 |).11 ( ) (11 III. Mt s bi tp v hm s hc Bi 1. Cho* N n , ta c a) =n dn d|. ) ( ( H thc Gauss) b)( )= ||

\|n dnind i|. ) ( Gii a) Doe T1* = v e(n)= 1 vi mi* N n nn ta c ( ) ( ). ) )( ( ) () ( * * ) ( * ) ( ) (1 *1|1| |= = == = ||

\|= e v n n e T dn e T e n edne d dn dn d n d b) Ta c( )( ) ( )( ) ( )( ) ( )= = = = ||

\|n dn n e e T n e e e T n dind i|1 1. * * * * * ) ( (theo nh l 8) ===========================================================63HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Nun l s nguyn t nu tn ti s nguyn t p sao cho p2|n, nu n c phn tch tiu chun l n = p1p2pr, r 2 nun l s nguyn t nu tn ti s nguyn t p sao cho p2|n, nu n c phn tch tiu chun l n = p1p2pr, r 2 Bi 2. t( ) ( ) =n dN n d n|* . Chng minh rng ( )=101n Gii S dng nh l 4 v cng thc tnh( ) n , ta xt cc trng hp sau Nun l s nguyn t th( ) ( ) () 1 ) ( 1| = = =n d nn d Nu tn ti s nguyn t p sao cho p2|n, thm p n=vi ( ) 1 , , 2 = p m v N suy ra ( ) ( ) ( ) 0 ) (| |= = = p dn d n dp p d nNu n c phn tch tiu chun l n = p1p2pr,( r 2) th ( ) ( ) () < < < == =r i i ir i i ir i ii iriin drrp p p p p p p p p d n1 2 12 12 12 1... 12 11 1 |) ... ( ) ... ( )... ( ) ( ) ( 1 ( ) ( ) ( ) ( ) ( ) ( )( ) ( )) ... 2 () 1 ( 3 22 11 3 2 111 1 ... 1 1 1rr r rrrrr r r rrC C CrC C r C C Cnn+ + + = = M ( ) ( )121 2 11 2 ... 2 = = + + +rrr rr r rn ra suy r rC C C V2 r nn( ) 1 1 r . Ta c2 21 rhay( ) 1 = n Do trong mi trng hp ta u c( )=101n Bi 3 Cho* , N n M f a) Nu n l, chng minh rng( ) ( ) ( ). 1| |/ = n d n dd nd f d fb) Nu n chn, n = 2sm vi1 s , m l, chng minh rng ( ) ( ) ( ) ( ) ( ). 2 2 1| | |/ = m ksn d n dd nk f f d f d fc)Tnh gi tr ca tngS =( ) ( ) ( ) n dd nd f|/. 1 1Gii a)Nu n l, d|n suy ra d l v n|d cng l. Khi ta c( ) 1 1| = d nv ( ) ( ) ( ) = = n d n d n dd nd f d f d f| | ||) ( ) ( 1(pcm). b)Nu n chn, n = 2sm vi1 sv m l th c Vi d|n suy ra d c dng1 ,..., 2 , 1 , 2 = = s i m di hoc, 2 k di= trong i = 0,1,,s v k|m. Ta c ===========================================================64HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( )( )( ) ( ) ( ) ( )( ) ( ) ( )( ) a dung p k f k f m fk f k f m fk f m fk f m f d fm kssi m kisiim ks k msi m kisiisi m kisiisi m ki k nsii m nn dd nkmi si si i + = + + = + = + = ========|10 |11|/10 |110 |21120 |2 |112 /|/2 2 22 1 2 22 1 2 12 1 2 1 1 ( ) ( ) ( )( ) ( ) ( ) = = + ===m ks sn dsi m ksm kisiik v k f f d fk f k f m f| |0 | |11, 1 , 2 ) ( 2 22 2 2 2 suy ra( ) ( ) ( ) ( ) = m ksn d n dd nk f f d f d f| | |/. 2 2 ) ( 1 (pcm) c) Nu n l theo cu a) ta c( ) ( ) ( ) ( ). 2 1 1| |/ = =n d n dd nd f d f SNu n chn, n = 2sm, s 1,m l th theo cu b) ta c ( ) ( ) = =m ksn dd nn dk f f d f d f S| |/|) ( 2 2 ) ( 1 ) (Do ta c =m ksn dk f fd fS||) ( ) 2 ( 2) ( 2 Nhn xt: T bi ton trn ta c cc kt qu sau 1. Nu = f , s dng h thc Gauss (bi 1) ta c ( ) ( )= 01|/ndn dd n2. Nu( ) * N n n n f =th ( )= +n d m ksn dn dd nk ddd| |1||/21Bi 4.Chng minh cc ng thc sau ( ) ( )( )( )( ) . * ) ( ). * ) ( * ) ( ). * ) ( * ) ( )* / ) ( / ) ( )2| |32322 2|2|2N n k d k d dN n n d n d cN n n d n d bN n k n d k k n k d an k n kn k n k |||

\|= = = = Gii a)Theo nh ngha tch chp, ng thc cn chng minh tng ng vi vic chng minhnu n l nu n chn, n = 2sm, s 1,m l nu n l nu n chn nu n l nu n chn, n = 2sm, s 1,m l ===========================================================65HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 ( )( ) ( )( ) n d n d * *2 2 =(1) Do , dl cc hm nhn tnh nnd v d d * * , ,2 2 2 2 cng l hm nhn tnh. V vy chng minh (1) ta ch cn chng minh ( )( ) ( )( ), * *2 2 p d p d =vi p nguyn t v1 Tht vy ta c ( )( ) ( ) ( )( ) () ( ) ( )( ) ( ) 1 2 1 1 11*2 21 2 202 2+ = + + =+ === p p d p dp p d p dii i ( )( ) ( ) ( )() ( ) ( ) ( )( ) ( ) 1 2 1 1 11*1 2 202 2+ = + + =+ === p d p p dp d p p dii i suy ra( )( ) ( )( ). * *2 2 p d p d =(pcm) b) Gi s rrp p p n ...2 12 1=l s phn tch tiu chun ca n. Khi rrp p p n 2 22212...2 1=v( ) ( )=+ =riin d121 2Mt khc theo cu a) ta c( )( ) ( )( ) = =+ = =riiriip d n d1 12 2) 1 2 ( * *1 ( ) ) ( * ) (2 2n d n d = (pcm) c) Do 232* , d d l hm nhn tnh. chng minh ng thc cho ta ch cn chng minh( )( ) p d p d232* ) ( = vi mi p nguyn t v1 . ( )( ) ( ) ( ) ( ) () ( ) ( )( ) , 1 1 21 ) * (. 1 ) (2 2 21222323023232 2+ = + + = + =+ = =+ =+ += C Cp p d p d p p d p dp diii i suy ra( )( ) p d p d232* ) ( = ( ) . * ) ( * ) (232N n n d n d = d) Ta c ( ) ( ) ) ( * / ) ( ) (3|3|3n e d k n e k d k dn k n k= = ( )( ) ( ) ( )( ) n e d n e d k n e k d k dn k n k2 22|2|* * ) / ( ) ( ) ( = =|||

\|=|||

\| Do ng thc cn chng minh tng ng vi phi chng minh ( )( ) ( ) ( ) * * *2 3N n n e d n e d =hay( )( ) ( ) ( ), * *2 3 p e d p e d =vi mi p nguyn t v1 . Tht vy ta c ===========================================================66HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 ( )( ) ( )( ) ( ) ( )( )( ) ( )( )( ) ( )( ) ( )( ) ( )( )( ) ( ) ( )42 142 11 ... 2 11 *42 11 ... 2 11 *2 2 2220202 23 3 3030303 3+ += ||

\| + += + + + + =||

\|+ = ||

\|=+ += + + + + =+ = = = = == = = i ii ii iiii ii p e p d p e di p d p e p d p e d ( )( ) ( ) ( ), * *2 3 p e d p e d =vyta c pcm( )2| |3) (|||

\|= n k n kk d k dBi tp t luyn Bi 1.ChoM f , chng minh rng a)fl hm hon ton nhn tnh khi v ch khi. *1 *f f f = b)( ) ( ) ( ) ( ) p p f d f dn p n d, 1| | = nguyn t. c)( ) ( ) ( ) ( ) p p f d f dn p n d, 1| |2 = nguyn t. d).Nu n l s nguyn dng, k hiu( ) n wl s cc c nguyn t phn bit ca n th ( )( ) n wn dd 2|2=Bi 2. Gi) (0n l k hiu tng cc c l ca s nguyn dngn . Chng minh rng a). ) 1 ( ) (/||0d nd nn d = b) ), 2 / ( 2 ) ( ) (0n n n =vinl s chn. c)) (0n l hm nhn tnh. Bi 3. Chng minh rng a) . * , 2 ) ( N n n n d b) . 2 , ) ( ) ( 2 + n N n n n n c) . * , 2 ) ( ) (2 / ) 1 (N n N k n n d n n nk kkk +d) . * ), ( ) ( N n n d n n ===========================================================67HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 MT S BI TON S HCTRONG CC K THI OLYMPIC TON Trn Xun ng(THPT Chuyn L Hng Phong Nam nh) Trong k thi Olympic ton Quc t ln th 49 c t chc ti Ty Ban Nha c bi ton sau (bi ton 1) m tc gi ca n l Kestutis Cesnavicius (Lithuania) (Litva). Bi ton 1: Chng minh rng tn ti v s s nguyn dng n sao cho 21 n +c c nguyn t ln hn 2 2 n n +Bi ton ny l bi ton kh nht ca ngy thi th nht. Li gii ca bi ton 1 c pht trin t li gii ca cc bi ton n gin hn sau y:Bi ton 2: Chng minh rng tn ti v s s nguyn dng n sao cho n2 + 1 khng l c ca n!. ( thi chn i tuyn ca Innxia d thi Ton Quc t nm 2009) . Li gii ca bi ton 2: B : Tn ti v s s nguyn t dng 4k + 1 (k N*)Chng minh: Gi A l tp hp gm tt cc s nguyn t dng 4k+1 (kN*) , Khi A rngv 5A. Gi s A l tp hu hn. Gi p0 l phn t ln nht ca Ap0 5 .Gi s p1, p2 pn l tt c cc s nguyn t nh hn p0.t2 2 20 14 ... 1na p p p = +khi ? a N*, a > 1. Gi s q l c nguyn tca aq pi ,i {0,1,2 , n}. Mt khc (2p0p1 pn) 2 + 10 (modq)- 1 l s chnh phng (modq) v q l.Suy raqqqq = =|||

\| 2 :211 ) 1 ( 1121 ) 4 (mod 1q c dng 4k + 1 (k N*). Mt khc q> p0. iu ny mu thun vi cch chn p0. Vy tn ti v s s nguyn t dng4k + 1 (kN*). Chngtachuynsangvicgiibiton2.Gisplsnguyntdng4k+1(k N*) 1 1 ) 1 (121 = =|||

\| pp l s chnh phng (modp)np { 0,1,2 . ,p - 1} sao cho 2pn ) (mod 1 p2pn+1: p v np! khng chia ht cho p np ! khng chia ht cho 2pn + 1. Ta c: 2pn+ 1 pnp1 p . V tn ti v ===========================================================68HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 s s nguyn t p dng 4k + 1 (kN*) nn tn ti v s s nguyn dng n sao cho n2+ 1 khng l c ca n! Bi ton 3:Chng minh rng tn ti v s s nguyn dng n sao cho c nguyn t ln nht ca n2+ 1 ln hn 2n (Tp chAnimath ca Php nm 2006) Li gii ca bi ton 3:Gi s p l s nguyn t dng 4k + 1 (k N*)Suy ra 1 1 ) 1 (121 = =|||

\| pp l s chnh phng (modp)x {0,1,2, ,p - 1} sao cho x2 - 1(modp).Ta c: q2 (p- q)2 (modp)(q Z)q {0,1,2, , 21 p} sao cho q2-1 (modp).Tht vy gi s 21 p < x < px 21 + p. t q = p x, ta c: q2 = ( p x)2 x2 - 1 (modp) v 0 < q21 p. Ta c:q2 +1 Mp vp2q+1>2q.Suyracnguyntlnnhtcaq2 +1lnhn2q.Vcvss nguyn t dng 4k + 1(kN*) nn tn ti v s s nguyn dng n sao cho n2 +1c c nguyn t ln hn 2n. Sau y l cc li gii ca bi ton 1Li gii th nht ca bi ton 1: Xt s nguyn t p dng 4k + 1 (k N*) 1 1 ) 1 (121 = =|||

\| pp l s chnh phng (modp)x {0,1,2, p - 1} sao cho x2 - 1(modp). V x2 (p- x)2 (modp) (x Z)x {0,1,2 ,21 p} sao chox2-1 (modp). {0,1,2, ,21 p} sao cho ||

\|21 p2-1 (modp) t m = ||

\|21 pm {0,1,2, , 21 p} v m2-1 (modp) Gi s p > 20. N?u 0 +43 1 4p0 < 2 +1 +21 1 4p (2 +1)2 +43 1 4pp>2m+m 2 .Vm2+1M pnnm2p-1m 1 p .Vtn tivssnguyntpdng4k+1(kN*)nntntivss nguyn dng n sao cho c nguyn t ln nht ca n2 + 1 ln hn 2 2 n n + . Li gii th 2 ca bi ton 1: Gi s n l s nguyn, n24. Gi s p l c nguyn t ca(n!)2 + 1. Hin nhin p > n. Gi sx (0, 2p) l s d trong php chia n ! hoc n! cho p. Khi ? 0 < x< p x < p. Ta c? x2 + 1 chia ht cho p. Tht vy tn ti m Z sao cho n! = mp + x hoc n! = mp + x. Trong c hai trng hp ta u c (n!)2+1 = (mp+x)2 +1 x2 +1 = (n!)2 + 1 m2p2 2mpxx2+1M p . T suy ra p l c ca p2 - 2px + 4x2 + 4 = (p 2x)2 + 4p (p 2x)2 + 4p 2x +4 pp - 4 2x +4 p - 4 2x +20 4> 2x p 2x +4 p > 2x +x 2 T y suy ra iu phi chng minh Bi ton sau l bi ton tng qut ca bi ton 1Bi ton 4: Chng minh rng tn ti v s s nguyn dng n sao cho n2 + 1 c c nguyn t ln hn 2n + 2 nBi ton 5: Chng minh rng vi mi s nguyn n3, tn ti cp s nguyn dng l (xn, yn) sao cho nn ny x 2 72 2= +( thi Olympic Ton ca Bungari nm 1996) Li gii: Vi n = 3 , chn x3 = y3 = 1Gisvin3,tnticpsnguyndngl(xn,yn)saocho nn ny x 2 72 2= + .Ta chng minh rng mi cp . (X=27,2n nn ny xYy x =+) , (X= 27,2n nn ny xYy x +=) tho mn1 2 22 7+= +nY XTht vy 2 22727||

\|+||

\| n n n ny x y x m= 2 (2 27n ny x + ) = 2. 2n = 2n+1 V xn , yn l nn xn = 2k + 1, yn = 2l + 1 (k, l Z)12+ + =+ l ky xn n ===========================================================70HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 v l ky xn n =2. iu chng t rng mt trong cc s 2,2n nn ny x y x + l l . V vy vi n +1 tn ti cc s t nhin l xn+1 v yn+1 tho mn nn ny x 2 72121= ++ ++1 Bi ton 6 : Chng minh rng vi mi s nguyn dng n, phng trnh x2 + 15y2 = 4n c t nht n nghim t nhin (x,y)( thi chn hc sinh gii Ton Quc gia nm hc 2009 2010) Li gii: Trc ht ta chng minh rng vi mi s nguyn n2 tn ti cp s nguyn dng l (xn , yn) sao cho sao cho nn ny x 4 152 2= +Tht vy vi n = 2 , chn x2 = 1 , y2= 1Gisvin2tnticpsnguyndngl(xn,yn)saochosaocho nn ny x 4 152 2= + . Ta chng minh rng mi cp(X= 2,215n nn ny xYx y +=),(X=2,215n nn nx yYx y =+)thomn1 2 24 15+= +nY XTht vy 2 2215215||

\| + ||

\|n n n nx y x y m= 4 (2 215n ny x + ) = 4. 4n = 4n+1 V xn , yn l nn xn = 2k + 1, yn = 2l + 1 (k, l Z)12+ + =+ l ky xn n vk lk l x yn n =+ +=2) 1 2 ( ) 1 2 (2.iuchngtrngmttrongccs 2,2n n n nx y y x +ll.Vvyvin+1tnticcstnhinlxn+1vyn+1thomn 1 21214 15++ += +nn ny x Tr li bi ton 6: Vi n = 1, phng tnhny x 4 152 2= + c 1 nghim t nhin l (x,y)= (2,0) Vi n = 2, phng tnh ny x 4 152 2= + c 2 nghim t nhin l (x,y)= (4,0); (1,1) Gi s vi n 2, phng tnh ny x 4 152 2= + c n nghim t nhin l (x1,y1), (x2,y2), , (xn,yn)khi(x,y)=(2xk,2yk)(1 kn)lccnghimtnhincaphngtrnh ny x 4 152 2= ++1.Theochngminhtrnphngtrnh ny x 4 152 2= ++1lic1nghimt nhinl.Vyphngtnh ny x 4 152 2= ++1ctnhtn+1nghimtnhin.Biton6 c gii quyt. ===========================================================71HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Bi ton 7: Tm tt c cc cps nguyn dng(x,y) sao cho y xy x+2 2 l s nguyn v l c ca 1995. ( thi Olympic ton Bungari nm 1995) Li gii : Trc ht ta chng minh B : Cho s nguyn t p = 4q + 3 (q N). Gi s x, y l cc s nguyn sao cho x2 + y2 chia ht cho p, Khi x v y chia ht cho p. Tht vy nu x M p th y M p . Gi s x khng chia ht py khng chia ht cho pTheo nh l nh Phecma ta c x p-11 (modp)x4q+21 (modp). Tng ty4q+2 1 (modp) . Ta c x2 + y2M px2 -y2 (modp)(x2)2q+1 (-y2)2q+1 (modp) x4q+2 - y4q+2 (modp) 1 - 1 ( modp) p = 2 (v l). B c chng minh. p dng b vo bi ton 7: Gi s tn ti cc s nguyn dng x,y sao cho x> y , y xy x+2 2 l s nguyn v y xy x+2 2l c ca 1995 . t k = y xy x+2 2thx2 +y2 = k( x y) v k l c ca 1995= 3.5.7.19.N?u kM 3 th k= 3 k1 (k1N*) (k1 khng chia ht cho 3) x2 + y2M 3xM 3 v yM 3x = 3x1 , y = 3y1 (x1 , y1 N*, x1 > y1) ) (1 1 12121y x k y x = + . N?u k = 1 th x2 + y2 = x y. l iu v l v x2 + y2 x + y> x y (v x,y1 ) Nu k = 5 th x2 + y2 = 5(x y)(2x - 5)2 + (2y +5)2 = 50 x = 3 , y = 1 hoc x = 2 , y = 1Nu k = 7 , tng t nh trn, tn ti k2N* sao cho k = 7 k2 (k2 khng chia ht cho 7)x = 7x2 , y = 7y2(x2, y2 N* , x2 > y2) v) (2 2 22222y x k y x = +Nu k M 19 th tn ti k3 N* sao cho k = 19k3 (k3 khng chia ht cho 19 ),x = 19x3 , y = 19y3(x3, y3 N* , x3 > y3 ) v) (3 3 32323y x k y x = +Vy tt c cc cp s nguyn dng (x,y) cn tm c dng (3c, c), (2c, c), (c, 2c), (c, 3c) trong c {1,3,7,19,21,57,133,399} . Bi ton 8: Tm tt c cc cp s nguyn dng (x,y) sao cho sA = y xy x+2 2 l s nguyn v l c ca 2010. ( thi Olympic Ton khu vc duyn hi ng bng Bc B nm hc 2009 2010) Li gii: Trn c s li gii ca bi ton 7 ta ch cn tm cc nghim nguyn dng ca cc phng trnh :) (2 2y x k y x = +vi k { 2,5, 10}. Phng tnhx2 + y2 = 2 (x- y) khng c nghimnguyn dng . Tht vy gi s x,yN* , x > yv x2 + y2 = 2 (x- y) ===========================================================72HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 x2+y2 2x+y2>2(xy).liuvl.Phngtrnhx2+y2=5(x-y)ccc nghimnguyn dng l (x,y) = (3,1), (2,1). Phng trnh x2 + y2 = 10 (x- y) (x-5)2 + (y+5)2 = 50 c cc nghim nguyn dng l (x,y) = (6,2) ; (4,2) . Vy tt c cc cp s nguyn dng (x, y) tho mn bi l(3c, c), (2c, c), (c, 2c) , (c, 3c) , (6c, 2c) , (4c, 2c) , ( 2c, 6c), (2c, 4c) trong c {1,3,6,7,201}Cui cng l mt s bi ton dnh luyn tpBi ton 9: Chng minh rng vi mi s nguyn dng n, phng tnh7x2 + y2 = 2n+2 lun c nghim nguyn dng. Bi ton 10: Chng minh rng vi mi s nguyn dng n, phng trnh x2 + 15y2 = 4n c ng n nghim t nhin . Bi ton 11: Cho s nguyn dng n. Gi Sn l tng cc bnh phng ca cc h s ca a thc f(x) = (1+x)n.Chng minh rng S2n+1 khng chia ht cho 3 ( thi chn i tuyn Vit Nam d thi Olympic Ton Quc t nm 2010) Bi ton 12: Chng minh rng tn ti v s s nguyn dng n sao cho 2n +2 chia ht cho n . Bi ton 13: Chng minh rng tn ti v s s nguyn dng n sao cho tt c cc c nguyn t ca n2 + n + 1 khng ln hnn . ( thi chn i tuynUkraina d thi Olympic ton quc t nm 2007) Bi ton 14: Vi mi s nguyn dng n > 1, k hiu p(n) l c nguyn t ln nht ca n. Chng minh rng tn ti v s s nguyn n > 1 sao cho: p(n) < p(n+1) < p(n+2) . Bi ton 15: Cho cc s nguyn a,btho mn a>b > 0 . Chng minh rng tn ti v s s nguyn dng n sao cho an + bn chia ht cho n . Bi ton 16: Chng minh rng tn ti v s s nguyn t p c tnh cht sau: Tn ti v s nguyn dng n sao cho p 1 khng chia ht cho n v n! +1 chia ht cho p. ( thi chn i tuyn ca Mnva d thi Olympic ton Quc t nm 2007). Biton17:Chngminhrngtntivssnguyndngnsaocho5n-21chia ht cho n. ( thi Olympic ton ca Braxin nm 2008) ===========================================================73HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 NH L LAGRANGE V NG DNG ng nh SnChuyn Lng Vn Ty Ninh Bnh 1.NH L LAGRANGE 1.1. NH L ROLLE nh l: Nu( ) f x l hm lin tc trn on[ ; ] a b , c o hm trn khong( ; ) a bv( ) ( ) f a f b =th tn ti( ; ) c a b sao cho'( ) 0 f c = . Chng minh: V( ) f x lin tc trn [a; b] nn theo nh l Weierstrass( ) f x nhn gi tr ln nht Mv gi tr nh nht m trn [a; b]. - KhiM = m ta c( ) f x l hmhng trn [a; b], do vi mi( ; ) c a b lun c '( ) 0 f c = . -KhiM>m,v( ) ( ) f a f b = nntntic (a; b) saocho( ) f c m = hoc ( ) f c M = , theo b Fermat suy ra'( ) 0 f c = . H qu 1: Nu hm s( ) f x c o hm trn (a; b) v( ) f x c n nghim (n l s nguyn dng ln hn 1) trn (a; b) th'( ) f x c t nht n - 1 nghim trn (a; b). H qu 2: Nu hm s( ) f x c o hm trn (a; b) v'( ) f x v nghim trn (a; b) th( ) f x c nhiu nht 1 nghim trn (a; b). H qu 3: Nu( ) f x c o hm trn (a; b) v'( ) f x c nhiu nht n nghim (n l s nguyn dng) trn (a; b) th( ) f x c nhiu nht n + 1 nghim trn (a; b). Cc h qu trn c suy ra trc tip t nh l Rolle v n vn ng nu cc nghim l nghim bi (khi( ) f x l a thc).Cchqutrnchotatngvvicchngminhtntinghimcngnhxc nh s nghim ca phng trnh, v nu nh bng mt cch no ta tm c tt c cc nghim ca phng trnh (c th do m mm) th ngha l khi phng trnh c gii. T nh l Rolle cho php ta chng minh nh l Lagrange, tng qut hn, ch cn ta n ti ngha ca o hm (trung bnh gi tr bin thin ca hm s). 1.2. NH L LAGRANGE (Lagrange's Mean Value Theorem) ===========================================================74HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 nhl:Nu( ) f x lhmlintctrnon [ ; ] a b ,cohmtrnkhong( ; ) a b thtnti ( ; ) c a b sao cho ( ) ( )'( )f b f af cb a=. Chng minh:Xt hm s: ( ) ( )( ) ( )f b f aF x f x xb a= . Tac:F(x)lhmlintctrnon[ ; ] a b ,c o hm trn khong( ; ) a b v( ) ( ) F a F b = .Theo nh l Rolle tn ti( ; ) c a b sao cho'( ) 0 F c = . M ( ) ( )'( ) '( )f b f aF x f xb a= , suy ra ( ) ( )'( )f b f af cb a=. nh l Rolle l mt h qu ca nh l Lagrange (trong trng hp( ) ( ) f a f b = ) ngha hnh hc:

nhlLagrangecho php ta c lng t s ( ) ( ) f b f ab a do n cn c gi l nh l Gi tr trung bnh (Mean Value Theorem). T cho ta tng chng minh cc nh l v s bin thin ca hm s, t nn mng cho nhng ng dng ca o hm. nh l: Cho hm s( ) f x c o hm trn khong( ; ) a b . - Nu'( ) 0, ( ; ) f x x a b > th( ) f x ng bin trn( ; ) a b . - Nu'( ) 0, ( ; ) f x x a b < th( ) f x nghch bin trn( ; ) a b . - Nu'( ) 0, ( ; ) f x x a b = th( ) f x l hm hng trn( ; ) a b . Chng minh: Gi s'( ) 0, ( ; ) f x x a b > v 1 2 1 2, ( ; ), x x a b x x < , theo nh l Lagrange, tn ti 1 2c (x ; x ) sao cho 2 12 1( ) ( )'( )f x f xf cx x=. M 1 2'( ) 0 ( ) ( ) ( ) f c f x f x f x > < ng bin trn (a; b). Chohms( ) f x thamnccgithitca nh l Lagrange,th (C), A(a;f(a)), B(b;f(b)). Khitrn(C)tntiimC(c;f(c)), c (a; b) m tip tuyn ca (C) ti C song song vi ng thng AB. Joseph Louis Lagrange (1736 - 1813) ===========================================================75HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Nu trong gi thit ca nh l Lagrange ta thm vo gi thit'( ) f xng bin hoc nghch bin trn [a; b] th ta c th so snh ( ) ( ) f b f ab a vi'( ), '( ) f a f b . C th:'( ) f xng bin trn [a;b] ( ) ( )'( ) '( )f b f af a f bb a < < '( ) f xnghch bin trn [a;b] ( ) ( )'( ) '( )f b f af a f bb a > > T y cho ta tng ng dng nh l Lagrange chng minh bt ng thc v nh gi cc tng hu hn. Cng tng t nu trong gi thit ca nh l Lagrange ta thm vo gi thit'( ) f xngbinhocnghchbintrn[a;b]thtacthsosnh ( ) ( ) f c f ac avi ( ) ( ) f b f cb c vi[ ; ] c a b chotatngchngminhrtnhiubtngthc,nhbtngthc Jensen Ngoi ra nh l Lagrange cn c pht biu di dng tch phn nh sau: nh l: Nu( ) f xl hm lin tc trn on [a; b] th tn ti im( ; ) c a b tha mn:( ) ( )( )baf x dx f c b a = nhlLagrangedngtchphncpdngchngminhmtsbitonlin quan n tch phn v gii hn hm s. 2.MT S NG DNG 2.1.CHNG MINH S TN TI NGHIM CA PHNG TRNH Bi ton 1. Chng minh rng phng trnh acosx + bcos2x + ccos3x lun c nghim vi mi b cc s thc a, b, c. Li gii:Xt bsin2x sin3x( ) asinx+ '( ) osx+bcos2x+ccos3x, x .2 3cf x f x ac R = + = M0 0(0) ( ) 0 (0; ), '( ) 0 f f x f x = = = , suy ra iu phi chng minh. Nhn xt: Bi ton trn c dng tng qut: Cho hm s f(x) lin tc trn [a; b], chng minh rng phng trnh f(x) = 0 c t nht mt nghim trn (a; b). Phng php gii: Xt hm F(x) tha mn F(x) lin tc trn [a; b], F(x) = f(x).g(x) vi mi x thuc (a; b), g(x) v nghim trn (a;b) v F(a) = F(b). Theo nh l Rolle suy ra iu phi chng minh. ===========================================================76HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Bi ton 2. Cho s thc dng m v cc s thc a, b, c tha mn: 02 1a b cm m m+ + =+ +. Chng minh rngax2 + bx + c = 0 c nghim thuc (0; 1). Hng dn: Xt hm s2 1. . .( )2 1m m ma x b x c xf xm m m+ += + ++ +. Tng t ta c bi ton tng qut hn. Biton3.Chosthcdngm,snguyndngnvccsthc 0 1, ,...,na a a tha mn: 1 0... 01n na a am n m n m+ + + =+ + . Chng minh rng11 1 0... 0n nn na x a x a x a+ + + + = c nghim thuc (0; 1). Hng dn: Xt hm s 1 1 0( ) ...1m n m n m n na a af x x x xm n m n m+ + = + + ++ +

Bi ton 4.(nh l Cauchy)Nu cc hm s( ), ( ) f x g xl cc hm s lin tc trn on[ ; ] a b , c o hm trn khong( ; ) a b v'( ) g x khckhngtrnkhong( ; ) a b thtnti( ; ) c a b saocho ( ) ( )'( )( ) ( )f b f af cg b g a=. Ligii:TheonhLagrangeluntnti 0( ; ) x a b saocho 0( ) ( )'( )g b g ag xb a=( ) ( ) g a g b . Xthms ( ) ( )( ) ( ) ( )( ) ( )f b f aF x f x g xg b g a= ,tac:F(x)lhmlintctrnon [ ; ] a b , c o hm trn khong( ; ) a b v ( ) ( ) ( ) ( )( ) ( )( ) ( )f a g b f b g aF a F bg b g a= =.Theo nh l Rolle tn ti( ; ) c a b sao cho'( ) 0 F c = . M ( ) ( )'( ) '( )( ) ( )f b f aF x f xg b g a= , suy ra ( ) ( )'( )( ) ( )f b f af cg b g a=. Nhnxt:nhlLagrangelhqucanhlCauchy(trongtrng hp ( ) g x x = ) Bi ton 5: Cho a + b c = 0. Chng minh rng: asinx+9bsin3x+25csin5x = 0 c t nht 4 nghim thuc [0; ]. ===========================================================77HI CC TRNG THTP CHUYN KHU VC DUYN HI V NG BNG BC B Hi tho khoa hc mn Ton hc ln th III - 2010 Nhn xt: Bi ton ny cng tng t cc bi ton trn. chng minh( ) f xc t nht n nghim ta chng minh F(x) c t nht n + 1 nghim vi F(x) l mt nguyn hm ca ( ) f xtrn (a;b) (c th phi p dng nhiu ln) Li gii:Xt hm s: ( ) 3 5 f x asinx bsin x csin x = , ta c: '( ) os 3 os3 5 os5 f x ac x