Tr ng THCS NguynnhChiu Nm hc 2010-2011 K thi ch hc sinh gii lp 9 THCSMn thi : Ton M s:............Thi gian lm bi 150 pht khng k thi gian giao thi gm1 trangC u 1:(2 im) 1) Tnh:9 17 9 17 2 A + + 2) Tnh:( ) ( )6 2 10 5 3 2 3 B + .3) Cho1 22009 1 2008 1 C v2 22.20092009 1 2008 1D + .Khng dng my tnh hy so snh C v D .C u 2: (2im) 1) Cho a thc( ) ( ) 1.2 2.3 3.4 ... . 1 f x x x + + + + +.Tm x ( ) 2010 f x 2) Gii h phng trnh: 2 2 2x y z 6xy yz zx 1x y z 14+ + + '+ + C u 3: (2im) Trong mt phng ta cho im B c nh c ta ( ) 1;1v im A di ng( ) A m;01) Vit phng trnh h ng thng( )md vung gc vi AB ti A.2) Chng minh rng khng c 3 ng thng no ca h( )mdng qui.3) Tm cc im trn mt phng Oxy sao cho ch c 1 ng thng ca h( )mdi quaC u 4: (3 im) Cho tam gic vung cn ABC (vung A), AD l trung tuyn thuc cnh huyn, M l im thay i trn on AD. Gi N v P theo th t l hnh chiu vung gc ca M xung cc cnh AB, AC; H l hnh chiu ca N xung ng thng PD.a) Tnh s o gc NEB.b) Xc nh v tr ca M tam gic AHB c din tch ln nht.b) Chng minh rng khi M thay i, ng thng HN lun i qua mt im c nh.C u 5: (1im) Cho cc s 1 2 2009, a , . . . ,a ac xc nh theo cng thc sau: + + +n2a(2n 1)( n n 1) vi n = 1, 2, , 2008.Chng minh rng: 2 th phng trnh sau c nghim2ax2 + bx +1 - a = 0THI HS GII Nm hc 07-08 MON TOAN BI:Bi 1: (2) Rt gn biu thc 2 1 2 1 A x x x x + + Bi 2 (3) Cho biu thc2a 1 2a1 .1 1 2 1a a a a a aBa a a a _+ + + ,a/ Rt gn B.b/ Chng minh rng 23B.Bi 3: (3). Vi a, b, c, d l cc s dng tha mn a.b = c.d =1.Chng minh bt ng thc:( ) ( ) ( ) 4 2 a b c d a b c d + + + + + +.Bi 4 (3). Chng minh rng:2 2 2 2 2 2 2 2 2 2 2 21 1 1 1 1 1 1 1 1 1 1 1...1 2 3 1 3 4 1 2006 2007 1 2007 2008C + + + + + + + + + + + +l s hu t.B i 5(3). Cho ba s x, y, z tha mn 2 2 23 3 311x y zx y z + + '+ + Hy tnh tng x y z + +.Bi 6 (3). Cho( ) ABC ABAC . Gi I l tm ng trn ni tipABC . ng thng AI ct ng trn ngoi tipABC ti D.a/ Tm tm ng trn ngoi tipBIC .b/ Gi M, N ln lt l tip im ca ng trn ni tipABC vi cc cnh AB, BC. K l hnh chiu vung gc ca C xung ng thng AI. Chng minh M, N, K thng hng.Bi 7 (3). ChoABC . Mt ng thng song song vi cnh BC ct AB ti D v ct AC ti E. Chng minh rng vi mi im P trn canh BC, ta lun c din tchD P E khnh ln hn 14 din tchABC .ng thng DE v tr no th din tchD P E t gi tr ln nht.Bi 2: Tnh gi tr ca biu thc: viLi gii:Ta c:(1)9 12Tr ng THCS NguynnhChiu Nm hc 2010-2011 T ng thc (1) suy ra: x3 = 3x2 x = 3(3x 1) x = 8x 3x4 = 3x3 x2 = 3(8x 3) (3x 1) = 21x 8x5 = 3x4 x3 = 3(21x 8) (8x 3) = 55x 21 Vy P = Bi 3: Chng minh rng Li gii:Ta co: Li co Suy ra: (1)Tng t nh vy, ta c:(2)Cng v theo v hai bt ng thc (1) v (2) ta c PCM.Bi 4: Cho a, b, c l di cc cnh v p l na chu vi ca mt tam gic. Chng minh rng: Li gii:t x = p a, y = p b, z = p c. Khi x, y, z l cc s dng v: a = y + z, b = z + x, c = x + yp dng bt ng thc Cauchy, ta c:v Tng t nh vy, ta c v Cng v theo v 3 bt ng thc trn ri rt gn, ta c: Hay l Du ng thc xy ra khi v ch khi a = b = c, tc tam gic cho l tam gic u.Bi tn c chng minh.Bi 5: Cho hnh vung ABCD c cnh bng a. Mt gc 450 quay xung quanh nh A v nm bn trong hnh vung ct cnh BC, CD ln lt ti M v Na) Chng minh rng a(BM + DN) + BM.DN = a2b) ng thng AM ct ng thng CD ti E. Chng minh Li gii:10Tr ng THCS NguynnhChiu Nm hc 2010-2011 a) Trn tia i ca tia DC ly im F sao cho FD = BMD dng nhn thyABM =ADF(cnh, gc, cnh)AF = AMMt khc:NAF =NAD +DAF =NAD +MAB =BAD MAN = 900 450 = 450T suy ra:MAN =FAN(cnh, gc, cnh)MN = FN =BM + DNXt tam gic vung CMN, ta c: MN2 = CM2 + CN2(BM + DN)2 = (a BM)2 + (a DN)2(1)Khai trin (1) ri rt gn, ta c: a(BM + DN) + BM.DN =a2. PCMb)Ta c:EAF =MAN +NAF = 450 + 450 = 900EAF l tam gic vung (H thc lng trong tam gic vung) Hay l:.PCM.Mt s luyn thi vo chuyn Ton 9 13B i 1 (1 ):Cho : M = x2 + y2+xy-3x-3y+2011. Vi gi tr no ca x,y th M t gi tr nh nht. Tm gi tr ?B i 2 (1 ): Chng minh rng 1 1 1... 22 1 3 1 ( 1) n n+ + + 0. Chng minh rng: ( ) ( ) 2 21422 2 2 1 212 2 2+ ,_

+ + + + + ,_

+ x xaaaa x x x xaaCu 5: Cho t gic ABCD c hai ng cho AC v BD ct nhau O. Gi din tch ca t gic ABCD l S, gi din tch ca cc tam gic AOB v COD ln lt l S1 v S2.Chng minh rng iu kin cn v hai cnh AB v CD song song vi nhau l: 2 1S S S + .( 19) kho st cht lng HSG lp 9 Mn: TonBi 1: (2 im):a) Chng minh rng:( ) ( ) ( )4 421 1 1 6 2 + + + + x x x xvi mi x khng m;b) Gii h phng trnh:( ) ( ) ' + + +0 1 11 54 42x xx x xc) Gii phng trnh:. 0 3 10 3 5 102 2 + x x x xBi 2 (2 im):a) Tm ba s nguyn t lin tip a, b, c sao cho 2 2 2c b a + +cng l s nguyn t?14Tr ng THCS NguynnhChiu Nm hc 2010-2011 b) Cho hai s x, y tho mn: 3 2 4 92 2 + x xy xy y x. Hy tnh gi tr ca biu thc 22:16 8162 2 32 + y yyx x xxA?c) Tm cc s nguyn a phng trnh ( ) 0 40 2 32 + + a x a x c nghim nguyn?Bi 3 (2,5 im):a) Gii phng trnh nghim nguyn: x2 + y2 xy x y + 1 =0;b) Gii phng trnh:. 2 1 12 2 2+ + + + + x x x x x xBi 4 (1 im): Cho x>0; y>0 tho mn 11 +yx.Hy tm gi tr nh nht ca biu thc xyyxP 2006 16 + ?Bi 5 (2,5 im): Cho ng trn (O) ng knh BC=2R v im A thay i trn (O) (A khng trng vi B, C). ng phn gic trong gc A ca tam gic ABC ct (O) ti K (K khc A). H AH vung gc vi BC.a) t AH=x. Tnh din tch S ca tam gic AHK theo R v x. Tm x sao cho S t gi tr ln nht.b) Chng minh rng khi A thay i, tng AH2+HK2 lun l mt i lng khng i. Tnh s o gc B ca tam gic ABC bit 53HKAH. 20 kho st cht lng HSG lp 9 Mn: TonCu1: Chng minh rng1 12 2 22+ + n Al s chnh phng khiN AvN n .C u 2:ChoathcP(x)nguynvP(x)chiahtcho3khi{ } 2 ; 1 ; + + k k k xvi Z k . Chng minh rng: P(m) chia ht cho 3 viZ m .C u 3:a) Gii phng trnh( )( ) 2 10 3 1 1 1 + + x x x xb) Gii h phng trnh '+ + + a y x y xy x y x6362) ( ) ( (a l tham s v a>0)C u 4:Cho hnh vung ABCD ngoi tip ng trn (O;R) v M l mt im trn ng trn . Gi di MA, MB, MC, MD ln lt l a, b, c, d.Chng minh rng: a2b2 +b2d2 =10R4C u 5:Cho a, b, c l cc s thc dng. Tm gi tr nh nht ca:S=

,_

+ba321.

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+cb321.

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+dc321.

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+ad321 21 kho st cht lng HSG lp 9 Mn: TonCu 1: a) Gii phng trnh3 3 3 3 2 y x trn tp hp cc s hu t.15Tr ng THCS NguynnhChiu Nm hc 2010-2011 b) Gii h phng trinh ( ) ( )' + + + + +4 142xy y xy x xy xCu 2: Vit phng trnh ng thng i qua im A(4;3), ct trc tung ti im c tung l mt s nguyn dng, ct trc honh ti im c honh l s nguyn t.Cu 3:a) Cho xy = 1 v x > y. Chng minh 2 22 2+y xy xb) Cho a, b, c l di ba cnh ca mt tam gic tho mn a+b+c=2. Chng minh: 2 22 2 2< + + + abc c b aCu 4: (trng 2 thay bng bi hnh thoi) Cho ng trn (O) ng knh BC=2R v im A thay i trn (O) (A khng trng vi B, C). ng phn gic gc A ca tam gic ABC ct (O) ti K (K khc A). Hai AH vung gc vi BC.a) t AH=x. Tnh din tch S ca tam gic AHK theo R v x.Tm x sao cho S t gi tr ln nht.b) Chng minh rng khi A thay i, tng AH2 +HK2 lun l mt i lng khng i. Tnh s o gc B ca tam gic ABC bit 53HKAH. 22 kho st cht lng HSG lp 9 Mn: Ton1-a, Tm tt c cc s t nhin c 3 ch sabcsao cho

( ) ' 2221n cban abc vi n l s nguyn ln hn 2.b, Tm cc s x, y, z nguyn dng tho mn ng thc:2(y+z)=x(yz -1)c, Cho s nguyn t p>3. Bit rng c s t nhin n sao cho trong cch vit thp phn ca s pn c ng 20 ch s . Chng minh rng : Trong 20 ch s ny c t nht 3 ch s ging nhau.1. Cho x l s thc tho mn 3 0 xTm gi tr ln nht v gi tr nh nht ca: P= x x 5 +(3-x) 2 + x2. a, Gii phng trnh : xx xxx xx ++ ++33332222 b, Gii h phng trnh :' + + + + +1 4162 2 2z y xz x y z x yz y x3. Cho tam gic u ABC cnh a ngoi tip ng trn (0). Mt tip tuyn ca ng trn ct cnh AB, AC th t D, E. t AD=x; AE=y; DE=z.16Tr ng THCS NguynnhChiu Nm hc 2010-2011 Chng minh rng : a, 2 2 2z xy y x + b, ECAEDBAD+ khng i khi tip tuyn DE thay i. 23 kho st cht lng HSG lp 9 Mn: TonCu1: 1) Cho h phng trnh ' +b y a xa y x 3 (a, b l tham s)Xc nh b h lun c nghim (x; y) vi mi a.2) Vit phng trnh ng thng i qua im M(2; 3), ct trc honh v trc tung ti cc im A(a; 0) v im B(0; b) sao cho a, b l cc s nguyn t.Cu 2: Cho ng trn tm O ni tip tam gic ABC, tip xc vi cc cnh BC, CA, AB ln lt ti D, E, F. V DH vung gc vi EF (HEF). Chng minh rngH C A H B A .Cu 3: 1) Chng minh rng F(n) = 4n + 15n 1 lun chia ht cho 9 vi mi n nguyn dng.2) Tm tt c cc cp s nguyn (x; y) tho mn x3 = y3 +2y2+1.Cu 4: Cho a, b, c 1 v a3+b3+c3 = 6.Tm gi tr ln nht ca biu thc A = a2+ b2 + c2 . 24 kho st cht lng HSG lp 9 Mn: TonCu 1: a) Cho a, b > 0, c 0. Chng minh rng:c b c a b ac b a+ + + + + + 01 1 1b) Gii h phng trnh: '+ + +2 1 111 12 22 2x y y xy xCu 2: a) Cho p 5 l s nguyn t sao cho 2p+1 cng l s nguyn t. Chng minh rng p+1 chia ht cho 6 v 2p2+1 khng phi l s nguyn t.b) Gii phng trnh nguyn: x3 = y3 + xy +8Cu 3: a) Tm tt c cc s nguyn n bt ng thc: (n2 1)x < -3n3 4n2 +n + 2 ng vi mi s nguyn dng x.b) Cho x, y, z l cc s thc khng m v x+y+z=1. Tm gi tr ln nht ca P = xy+yz+zx.Cu 4: Cho hnh vung ABCD tm O. Gi K, N ln lt l trung im ca AB, BC v F l trung im ca NC. T A k ng thng song song vi KF ct CD ti G. Chng minh FG l tip tuyn ca ng trn tm O ni tip trong hnh vung. 25 kho st cht lng HSG lp 9 Mn: TonBi 1 (2 im): a) Chng minh rng nu b l s nguyn t ln hn 3 v 10b+1 cng l s nguyn t th 5b+1 chia ht cho 6.b) Tm nghim nguyn ca phng trnh x2+x+13=y2.17Tr ng THCS NguynnhChiu Nm hc 2010-2011 Bi 2 (1 im): Cho xyz = 1, chng minh rng : 1111111+ +++ +++ + xz z yz y xy x.Bi 3 (3 im): Rt gn biu thc:xxxx xxx21.12 111 2222

,_

+++ Bi 4 (2 im): Tm gi tr ln nht ca biu thc 4 2 4 29 13 x x x x A + + vi1 0 x .Bi 5 (2 im): Cho hnh vung ABCD c cnh 1 n v. Trn cnh BC v CD ln lt ly M v N sao cho MC+CN+MN=2. Cc on thng AM, AN ln lt ct ng cho DB ti I, K. Chng minh cc on thng BI, LK, KD lp thnh ba cnh ca mt tam gic vung. 26 kho st cht lng HSG lp 9 Mn: TonCu 1: a) Chng minh rng vi mi s t nhin1 nth s 1 23+n chia ht cho 3n+1 nhng khng chia ht cho 3n+2. b) Tm tt c cc s nguyn x,y bit x>y>0 tho mn x3+7y=y3+7xCu 2: Gii phng trnha x x x x + 4 4 4 2 3(a l tham s).Cu 3: T im P nm ngoi ng trn tm O bn knh R k hai tip tuyn PA, PB vi A, B l cc tip im. Gi H l chn ng vung gc h t im A n ng knh BC.a) Chng minh rng PC ct AH ti trung im ca AH;b) Tnh AH theo R v PO=d.Cu 4: Cho a, b, c l cc s thc khng m tho mn a+b+c=1. Tm gi tr ln nht ca biu thc ( ) 1 + + c b c ab P. 27 kho st cht lng HSG lp 9 Mn: TonCu 1: a) Chng minh rng vi mi s t nhin1 nth s 1 23+n chia ht cho 3n+1 nhng kkhng chia ht cho 3n+2. b) Tm tt c cc s nguyn x,y bit x>y>0 tho mn x3+7y=y3+7xCu 2: Gii phng trnha x x x x + 4 4 4 2 3(a l tham s).Cu 3: T im P nm ngoi ng trn tm O bn knh R k hai tip tuyn PA, PB vi A, B l cc tip im. Gi H l chn ng vung gc h t im A n ng knh BC.c) Chng minh rng PC ct AH ti trung im ca AH;d) Tnh AH theo R v PO=d.Cu 4: Cho a, b, c l cc s thc khng m tho mn a+b+c=1. Tm gi tr ln nht ca biu thc ( ) 1 + + c b c ab P. 28 kho st cht lng HSG lp 9 Mn: TonCu 1: a) Gii phng trnh: ( )( )1014 1 . 1 4 1323+ ++yy yxx.b) Gii h phng trnh '+ + 28 1 6 822 2 2x x yy x y y xCu 2: a) Trn ng thng 8x 13y + 6 = 0, hy tm cc im c to nguyn nm gia hai ng thng x =10 v x = 50. 18Tr ng THCS NguynnhChiu Nm hc 2010-2011 b) Tm s t nhin n, bit rng khi b i ba ch s tn cng bn phi ca n th c s mi c gi tr bng 3n .Cu 3:Cho cc s thc x>0,0 ytho mn x3+y3=x-y. Tm gi tr ln nht ca biu thc P=x2+y2. Cu 4: Cho tam gic IDC ngoi tip ng trn tm O. K ng thng AB tip xc vi (O) ti M v song song vi CD (A thuc ID, B thuc IC). K ng knh MN ca ng trn (O). Gi giao im ca IN vi AB l M/. Chng minh rng:a) AM/.CN=BM/.DN;b) AM/=BM. 29 kho st cht lng HSG lp 9 Mn: TonCu 1: a) Rt gn biu thc: ( ) ( )b b a ab b a a b ab ab abA++ + +2 33 vi a>0, b>0 vb a .b) Tnh gi tr ca biu thc: ( )13 32+ x xxy vi3 2 + x .Cu 2:a) Cho biu thc:( ) ( ) ( )2 2 22 3 x z z y y x B + + + . Tm cc s nguyn x, y, z 1 0 B.b) Cho1 1 xv n l s nguyn dng. Chng minh rng: ( ) ( )n n nx x 2 1 1 + + .Cu 3:a) Gii phng trnh:( ) 12 10 32 2 + x x x x .b) Gii bt phng trnh:2 1 30 16 9 1 4 3 4 + + > + x x x x .Cu 4:a) Cho tam gic ABC vung ti A, M l trung im ca AC. ng thng qua A vung gc vi BM ct BC ti D tnh t s DBDC.b) Cho tam gic ABC cn ti A. Trn cnh BC ko di v pha C ly mt im M. Mt ng thng d qua im M ct AC v AB theo th t ti N v P. Chng minh rng: CNCMBPBM khng i khi M v d thay i.Cu 5:Chng minh rng vi mi x, y khc nhau v khc 0 th:( )12 22 2 211]1

+yyxxy xy xxyy x. 30 kho st cht lng HSG lp 9 Mn: TonCu 1: (2,0 im)a) Rt gn biu thc:,_

+

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+++a aaa aP11112 212 2122 vi a>0 v a 1.b) Gii phng trnh: 22 222 22 ++ ++xxxx.Bi 2: (3,0 im)19Tr ng THCS NguynnhChiu Nm hc 2010-2011 a) S o hai cnh gc vung ca mt tam gic vung l nghim ca phng trnh bc hai (m 2)x2 2(m 1)x + m = 0. Hy xc nh gi tr ca m s o ng cao ng vi cnh huyn ca tam gic l 52.b) V cc ng thng x = 6; x = 42; y = 2; y = 17 trn cng mt h trc to . Chng minh rng trong hnh ch nht gii hn bi cc ng thng trn khng c im nguyn no thuc ng thng 3x + 5y = 7. (im M(x,y) c gi l im nguyn nu x, y cng l s nguyn).Bi 3: (2,0 im)Cho t gic ABCD c cc cnh i din AD ct BC ti E v AB ct CD ti F. Chng minh rng iu kin cn v t gic ABCD ni tip c ng trn l: EA.ED + FA.FB = EF2.Bi 4: (2,5 im)Cho tam gic ABC cn A, AB = 32BC, ng cao AE. ng trn tm O ni tip tam gic ABC tip xc vi AC ti F.a) Chng minh rng BF l tip tuyn ca ng trn ngoi tip tam gic ECF.b) Gi M l giao im ca BF vi ng trn (O). Chng minh rng BMOC l t gic ni tip.Bi 5: (0,5 im)Mi im ca mt phng c gn vi mt trong hai mu: xanh hoc trng. Chng minh rng tn ti mt tam gic u, vi cnh bng 1 hoc3(n v di), c ba nh cng mu. 31 kho st cht lng HSG lp 9 Mn: TonPhn 1 (Mi cu 5 im): HS ch cn trnh by vn tt cch gii v ghi p s.1. Tm s ln nht trong hai s 1 20071 2007;1 20071 20073333222222221111++++2. Cho hnh thang ABCD (BC//DA), hai ng cho ct nhau ti O, Ly P trn AB sao cho PO//BC. Bit BC=3, DA=7. Hy tnh OP.3. Gii h phng trnh '1 038 21 01 0yxx y4. Xt s nguyn dng a v cc s thc x, y, z tho mni. 2x+a=yii. a+y=xiii. x+y=zTm gi tr ln nht c th c ca tng x+y+z.Phn 2 (Mi cu 10 im): Hc sinh phi trnh by li gii.1. Cc im A, B, C, D ly trn cc cnh PQ, QR, RS, SP ca t gic PQRS sao cho ABCD l hnh bnh hnh v AC, BD, PR, QS ng quy. Chng minh rng PQRS l hnh bnh hnh.2. a) Cho f(x)=x2+6x+c,Z c . Chng minh rng f(0)+f(-1) l s nguyn l.b) Cho g(x)=x3+px2+qx+r ; Z r q p , ,. Chng minh rng nu c g(0), g(-1) u l th phng trnh g(x)=0 khng th c ba nghim nguyn.20Tr ng THCS NguynnhChiu Nm hc 2010-2011 3. Cho hnh ch nht ABCD. Ly XAB, YBC; SDAX=5, SXBY=4, SYCD=3. Tnh din tch tam gic DXY.4. K hiu ]x l s nguyn ln nht khng vt qu x. Tnh tng ] ] ] ] ] ]50 49 48 ... 3 2 1 + + + + + +5. Cc im D, E, F ly trn cc cnh ca tam gic ABC sao cho AEF CED CDE BDF BFD AFE ; ;a. Chng minhBAC BDF b. Bit AB=5, BC=8, CA=7. Tnh BD.6. Chng minh rng vi mi s nguyn dng n s( )n n n n4 1900 25 121 + lun chia ht cho 2000.7. Cho cc s thc a, b, c, d tho mn ' + + + +11 1 11c a b c a bc b aHy tnh gi tr ca biu thc ca c bc b ab aM+ +++ +++ +111111 8. T gic li ABCD c tnh cht AB=CD, . Ly cnh BC lm y, dng ra ngoi tam gic vung cn EBC. Chng minh rng tam gic EAD cng vung cn. 32 kho st cht lng HSG lp 9 Mn: TonCu 1: (1,5 im) Cho biu thc 1 4 31 2) (22+ x xx xx Pa) Tm tt c cc gi tr ca x P(x) xc nh. Rt rn P(x).b) Chng minh rng: Nu x>1 thP(x).P(-x) < 0.Cu 2: (1,5 im) Tm gi tr nguyn ca x v y trong ng thc:2x3 + xy = 7.Cu 3: (2 im) Gii h phng trnh ' + + 11 3 62 22y xy x xy xCu 4: (3 im) ng trn tm O ni tip tam gic ABC tip xc vi cc cnh BC, CA, AB tng ng ti cc im D, E, F. ng trn tm O/ nm trong gc A ca tam gic ABC tip xc vi cnh BC v phn ko di ca cc cnh AB, AC tng ng ti cc im P, M, N.a) Chng minh rng: BP=CD.b) Trn ng thng MN ta ly cc im I v K sao cho CK//AB, BI//AC. Chng minh rng BICE l hnh bnh hnh.c) Gi (S) l ng trn i qua ba im I, K, P. Chng minh rng (S) tip xc vi cc ng thng BC, BI, CK.Cu 5: (2 im) Cho a, b, c > 0. Chng minh rng:b a c a c b c b a a c c b b a + +++ +++ ++++++ 212121313131 33 kho st cht lng HSG lp 9 Mn: Ton21Tr ng THCS NguynnhChiu Nm hc 2010-2011 Cu 1: (2,5 m)a) Gii h phng trnh: ( )( )' x x y zx xy x z212b) Gii phng trnh: 44 222++xx xxxCu 2: (2,5 im)a) Gi s 1 22 2;1 4 412 2+ + +x xxBx xA. Xc nhZ x 32 B AC+ nhn gi tr nguyn.b) Gi a v b l hai nghim ca phng trnh bc hai0 12 x x . Chng minh rng 2 2) (+ ++ + + n n n nb a b a n P l nhng s nguyn v chia ht cho 5 vi mi s nguyn dng n.Cu 3: (2,0 im)Cho tam gic ABC, ng cao CH ( AB H ). Gi CM, CN (AB N M ,) ln lt l phn gic cc gc ACH, gc BCH. Tm ng trn ngoi tip tam gic CMN trng tm ng trn ni tip tam gic ABC. Chng minh rng 2.BM ANSABC. ( ABCSl k hiu din tch tam gic ABC).Cu 4: (2,0 im)Cho tam gic nhn ABC, cc ng cao BD, CE; trung tuyn AM. Qua A k cc ng thng song song vi CE v BD ct BD v CE th t ti P v Q. Gi K l giao im ca AM v PQ. Chng minh APDK t gic ni tip.Cu 5: (1 im)Bit rng hai s thc cng du v tho mn ng thc:( ) ( ) 0 4 4 32 2 3 3 + + + + + + y x y x y x .Hy tm gi tr ln nht ca biu thc y xQ1 1+ . 34 kho st cht lng HSG lp 9 Mn: TonCu 1. (2 im) Cho a a aa a aa a aa a aP44442222 + +a) Tm iu kin ca a biu thc P c ngha v rt gn P.b) Tm a 5 < P .Cu 2. (1,5 im) Cho a thc f(n) = n5 5n3 + 4n vi n nguyn dng. Chng minh rng f(n) chia ht cho 120 vi mi gi tr nguyn dng ca n.Cu 3. (1,5 im) Gii phng trnh:13 6 1 72+ + + x x x x .Cu 4. (1,5 im) Cho a, b, c l ba s thc dng. Chng minh rng:c b aa cac ac bcb cb aba b+ + +++++23 323 323 3353535.Cu 5. (1,5 im) Cho hnh thoi ABCD c oA 120 . Tia Ax to vi tia AB gc BAx bng 15o v ct cnh BC ti M, ct ng thng DC ti N. Chng minh: 2 2 234 1 1AB AN AM +.Cu 6. (1,5 im) Gi s t gic ABCD c ng trn ng knh AB tip xc vi ng thng CD. Chng minh rng nu AD//CB th ng trn ng knh CD tip xc vi AB. 35 kho st cht lng HSG lp 9 Mn: TonBi 1: (2 im)22Tr ng THCS NguynnhChiu Nm hc 2010-2011 a) Chng minh rng nu p l s nguyn t ln hn 3 th ( p 1)( p + 1) chia ht cho 24.b) Tm nghim nguyn dng ca phng trnh: xy 2x 3y + 1 = 0.Bi 2: (2 im) Cho cc s a, b, c khc khng v i mt khc nhau, tho mn iu kin a3+b3+c3 = 3abc. Tnh:,_

++

,_

++b a ca c bc b ac b ab a ca c b.Bi 3: (2 im)a) Tm a phng trnh 1 3 2 3 + a ax x c nghim duy nht.b) Cho tam thc bc hai f(x) = ax2 +bx + c tho mn iu kin 1 ) ( x f vi mi [ ] 1 ; 1 x. Tm gi tr ln nht ca biu thc: 4a2 + 3b2.Bi 4: (1,5 im) Cho gc xOy v hai im A, B ln lt trn hai tia Ox, Oy, tho mn OA OB = m (m l di cho trc). Chng minh rng ng thng i qua trng tm G ca tam gic ABO v vung gc vi AB lun i qua mt im c nh.Bi 5: (2,5 im) Cho tam gic nhn ABC. Gi ha, hb, hc ln lt l cc ng cao v ma, mb, mc ln lt l cc ng trung tuynca cc cnh BC, CA, AB; R v r ln lt l bn knh ca cc ng trn ngoi tip v ni tip ca tam gic ABC. Chng minh rng: rr Rhmhmhmccbbaa+ + +. 36k thi chon hc sinh gii lp 9 thi mn: TonBi 1.1) Gii h phng trnh ' + +3 33 322x yy x2) Gi x1, x2 l nghim ca phng trnh x2 + 2006x + 1 = 0 v x3, x4 l nghim ca phng trnh x2 + 2007x + 1 = 0. Tnh gi tr ca biu thc A = (x1 + x3)(x2 + x3)(x1 x4)(x2 x4) 2006.Bi 2. Cho ng trn (I) ni tip tam gic ABC tip xc vi cc cnh BC, CA, AB ln lt ti M, N, P. Gi Q l chn ng vung gc k t M xung NP. Chng minh rng:1) QM l tia phn gic ca gc BQC.2) Bn im B, C, E, F cng thuc mt ng trn. (E, F th t l giao im ca BI, CI vi NP).Bi 3. Cho a, b, c l 3 s nguyn dng, nguyn t cng nhau v tho mn c b a1 1 1 +. Chng minh rng tng a + b l s chnh phng.Bi 4. Cho x, y, z, t l cc s thc tho mn x2 + y2 < 1. Chng minh rng ( ) ( )( ) 1 1 12 2 2 2 2 + + + t z y x yt xz .23