TP GII THI HC SINH GII TON LP 9Thi gian: 120 pht S 01Bi 1: (3 im)Rt gn cc biu thc sau: a) A = 3 5 3 52 3 5 2 3 5+ ++ + b) B = ( ) ( )14 2 13 12 2 11 44 52 + + +b) C =10 24 40 60 7 40 + + + +Bi 2: (2 im)a) Tm cc gi tr ca x biu thc sau c ngha: D = 12 1 x x b) Gii bt phng trnh:3 2 5 x Bi 2: (2 im)1. Tm cc s thc x, y, z tha mn iu kin: ( )11 22x y z x y z + + + +2Gii phng trnh: 2 211 12x x x x + + + Bi 3: (2 im)1. Tm gi tr nh nht ca biu thc: A = 24 3 x x +2. Tm gi tr ln nht ca biu thc:B =( ) ( ) 2 6 x x Bi 4: (2 im) Cho ng trn (O), ng knh AB v tip tuyn d ti B. Trn d ly hai im C v D. (B nm gia C v D). CA v DA ct ng trn (O) ln lt ti M v N. Chng minh AM. AC = AN. ADBi 5: (2 im)T im P nm ngoi ng trn (O) v hai tip tuyn PA v PB ti ng trn (A,B l hai tip im) . Gi H l chn ng vung gc k t A n ng knh BC ca ng trn. Chng minh PC ct AH ti trung im I ca AH.=== HT===TP GII THI HC SINH GII TON LP 9Thi gian: 150 pht S 05Bi 1: (3 im)1. Tnh gi tr biu thc: a) A =4 80 7 2 45 7 5 20 7 b) B = 3 35 2 13 5 2 13 + + 2. Rt gn biu thc P =2 2 3 1 4 3 x x x x + , vi 3 4 x Bi 2: (3 im)1. Gii phng trnh: a) 24 4 2 x x x + b) 2 2 24 5 4 8 4 9 3 5 x x x x x x + + + + + + 2. Cho 0 < x < 2, tm gi tr nh nht ca biu thc: A = 9 22xx x+Bi 3. (2,5im)Cho na ng trn (O) ng knh AB = a. Gi Ax, By l cc tia vung gc vi AB( Ax, By thuc cng mt na mt phng b AB). Qua im M thuc na ng trn(O) (M khc A v B) k tip tuyn vi na ng trn (O); n ct Ax, By ln lt E v F.Gi K l giao im ca AF v BE, 1. Chng minhMK AB . 2. Khi MB =3 .MA, tnh din tch tam gic KAB theo a. Bi 4: (1,5 im) Cho tam gic ABC, trc tm H l trung im ca ng cao AD. Chng minh rng tgB. tgC = 2 ==== HT=====TP GII THI HC SINH GII TON LP 9Thi gian: 150 pht S 06Bi 1: (1,5 im)1. Phn tch a thc sau thnh nhn t: ab(a + b) bc(b + c) + ac(a c)2.Chng minh x4 + x3 + 2x2 + x + 1 lun nhn gi tr dng vi mi xBi 2: (1,5 im) Rt gn biu thc: 1. A = ( )2011 2010 2010 20112010 20112011 2010 2010 2011++2.B = 21 2 1 14( 1)x x xx x + + vi x 1 v x 2Bi 3: (2 im) 1. Gii phng trnh sau: 2 23 2 2 1 x x x x x + + + 2. Tm gi tr nh nht ca biu thc: A = 2 6 10 2 2023 x y x y xy + + + vi x, y l cc s thc khng m.Bi 4: (2,5 im)Cho tam gic ABC nhn, H l trc tm, cc ng thng BH v CH ln lt ct AC v AB ti M v N, 0120 NHM .1. Chng minh AMN ABC , tnh MNBC.2. Tnh AHBCBi 5: (1,5 im)Cho tam gic ABC cn ti A c AB = AC = 13cm; BC = 10cm. Tnh cos ABi 6: (1 im) Cho biu thc M =( ) ( )3 22 1 2 2 1 n n n + + vi n NChng minh M M16====HT====TP GII THI HC SINH GII TON LP 9Thi gian: 150 pht S 07Bi 1: (1,5 im)1.Cho x v y l hai s khc nhau sao cho x2 y = y2 x .Tnh gi tr ca biu thc A = x2 + 2xy + y2 3x 3y 2.Tm 3 ch s tn cng ca tng 9994 + 999.Bi 2: (1,5 im)Khng x dng my tnh, rt gn cc biu thc sau:1. ( ) ( )3 2 2 3 2 2 3 2 2 3 2 2 + + +2. 14 6 5 2 25 31, 5 2 +++Bi 3: (1,5 im)Tm gi tr nh nht ca cc biu thc:1. A =2 3 x x 2. B =( ) ( )2 22011 1 x x + 3.C = 2 2 2 22 6 4 11 3 2 6 4 x y x y x y x y + + + + + + + +Bi 4: (1,5 im)1. Chng minh bt ng thc 1 1 x y y x xy + (vi x 1; y 1)2. Gii phng trnh: 32 1 2 12xx x x x++ + Bi 5: (2,0 im)Cho hnh ch nht ABCD c AB = 6; BC = 4. V AHBD. Gi M l trung imca HB; N l trung im ca CD. Tnh MA2 + MN2.Bi 6: (2,0 im)Qua mt im M ngoi ng trn (O; R) k ng thng qua O ct ng trn A v B, k qua M mt ng thng khc ct ng trn C v D1. So snh cc tch MA. MB v MC. MD.2. K tip tuyn MT vi ng trn (T l tip im) . Chng minh MTC MDT ===HT==== TP GII THI HC SINH GII TON LP 9Thi gian: 150 pht S 08Bi 1: (1,5 im) Cho biu thc : A = x(x + 1)(x + 2)(x + 3) + 1 1. Chng t rng vi mi s nguyn x, gi tr ca A l s chnh phng.2. Tm s nguyn x sao cho A = 25Bi 2: (1,5 im)1. Chng minh rng:a) 1 1 1 1 1...... 91 2 2 3 3 4 98 99 99 100+ + + + + + + + + +b) 1 1 1 1....... 282 3 4 225+ + + + 0; b > 0 th a ba bb a+ +Bi 4: (2,0 im)Cho hnh ch nht ABCD, M l trung im AD, N l trung im BC. Trn tia ica tia DC ly im P. Tia PM ct on thng AC Q.Chng minh : QNM MNP Bi 5: (1,5 im)Gi r l bn knh ng trn ni tip ca mt tam gic m di cc ng cao l1 2 3; ; h h h. Chng minh h thc: 1 2 31 1 1 1r h h h + +====HT====TP GII THI HC SINH GII TON LP 9Thi gian: 150 pht S 10Bi 1: (2,0 im) Cho bit a3 + b3 + c3 = 3abc . Tnh gi tr biu thc :A = 1 1 1a b cb c a| `| `| `+ + + . ,. ,. ,Bi 2: ( 4,5 im) Khng x dng my tnh, hy:1.So snh hai s: a =2009 2011 +v b =2 2010 .2.Rt gn cc biu thc sau: A = 3 2 2 3 2 217 12 2 17 12 2 + +3. Tnh gi tr ca biu thc: B = 2 21 1x x x xx x x x ++ + + vi x = 3 2 24Bi 3: ( 4,0im)Gii cc phng trnh sau: 1. 37 1 2 x x + + 2. Cho hm s: y =1 2 2 7 6 2 x x x x + + a) Tm cc gi tr ca x y c ngha ? b) Tm gi tr nh nht ca hm s cho.Bi 4: (4 im) Cho t gic ABCD c AD = BC. Gi M, N ln lt l trung im ca AB v CD. ng thng MN ct cc ng thng AD v BC theo th t ti E v F. Chng minh rng E F Bi 5: (4 im) T im A nm ngoi ng trn (O) v hai tip tuyn AB, AC (B, v C l hai tip im). Gi E, F ln lt l trung im AB, AC. T im T bt k trn cung nh BC v tip tuyn th ba ct on thng EF ti M. Chng minh MA = MTBi 6: (1,5 im) Chng minh rng: 1.a = 1021 1 200 M2.b =20 1339 39 40 + M3.c =2007 20052005 2007 2006 + M======HT=====TP GII THI HC SINH GII TON LP 9Thi gian: 150 pht S 11Bi 1: (5,0im)1. Rt gn biu thc: A = ( ) ( ) ( )( ) ( ) ( )3 3 32 2 2b c c a a ba b c b c a c a b + + + + Khng x dng my tnh, rt gn cc biu thc sau:2.. P =4 7 4 7 ++23.Q = 2 2 2 2 2 21 1 1 1 1 11 1 ... 12 3 3 4 2005 2006+ + + + + + + + +Bi 2: (3,0im)1.Chng minh 3 3( ) a b ab a b + +vi mi , 0 a b . 2.p dng kt qu trn, chng minh bt ng thc : 3 3 3 3 3 31 1 111 1 1 a b b c c a+ + + + + + + + vi mi a, b, c l cc s dng tha mn1 abc .Bi 3: (2 im) Gii phng trnh:( ) ( ) ( )13 1 4 3 3 03xx x xx + + + Bi 4: (3 im) Cho tam gic ABC vung ti A. Chng minh rng tan2ABC ACAB BC+Bi 5: (4 im) Cho on thng AB. Trn cng na mt phng b AB v hai tia Ax v By cng vung gc vi AB. Gi O l trung im AB. Trn tia Ax ly im C, trn tia By ly im D sao cho 090 COD .1. Chng minh AC + BD = CD.2. Chng minh CD l tip tuyn ca ng trn ng knh AB.Bi 6: (3 im)1. Chng minh rng 16n 1 chia ht cho 15 nhng khng chia ht cho 17 vi n l2. Chng minh rng:( ) ( )3 2 3 1 21 1m nx x x x+ ++ + + + M vi mi s t nhin m v n. ======HT=====TP GII THI HC SINH GII TON LP 9Thi gian: 150 pht S 12Cu 1 (3,0 im)1. Phn tch x5 + x 1 thnh nhn t.2. Phn tch s 10 000 000 099 thnh tch ca hai s t nhin khc 1.Cu 2 (3,0 im) Rt gn biu thc: A = 11 6 2 3 5 7 3 5 22 3 14 5 3 + + + + + Cu 3: (4 im) 1.Tm gi tr nh nht ca biu thc: 2007 2008 2009 2010 M x x x x + + + 2. Gii phng trnh: 1 122 4x x x + + + + Cu 4: (4 im)1. Chng minh rng 2 16 19 2n n n++ chia ht cho 17 vi mi s t nhin n.2. Chng minh rng : Vi mi s nguyn dng n, gi tr biu thc 224 13 2 1 1nMn n n ++ + khng th l mt s t nhin.Cu 5: (6 im) Cho hnh thang vung ABCD ( 090 A B ), tia phn gic ca gc C i qua trung im O ca AB.1. Chng minh rng CD l tip tuyn ca ng trn (O; OA)2. Gi H l tip im ca CD vi ng trn (O; OA). Gi K l giao im ca ACV BD. Chng minh rng KH song song vi AD. 3. Cho bit AB = 2a. Tnh tch ca hai on thng AD v BC theo a==== HT======TP GII THI HC SINH GII TON LP 9Thi gian: 150 pht S 13Cu 1 ( 3im ):1. Rt gn cc biu thc sau: a)A = 3 512 6 3 20 10 3 +.b)B = 2 2 a b c ac bc a b c ac bc + + + + + + + +(a >0, b > 0, c > 0 , a + b c)Cu 2: (3im)1. Chng minh rng vi mi s t nhin n ta c: A(n) = 27.5 12.6n n+chia ht cho 192. Tm cc s nguyn x v y sao cho xy x y = 6Cu 3: (3im)1. Chng minh bt ng thc:( ) ( )3 5 2 0 x x + > vi 0 x 2. Gii phng trnh:1 x x 2 x 3 x 32 2 + +Cu 4: (3 im)Cho a, b, c l cc s thc tha mn: 0; 02 4 2 02 7 11 0a ba b ca b c + + ' + .Tm gi tr ln nht v nh nht ca biu thc Q = 6a + 7b + 2006c.Cu 5: (4 im)Cho ng trn (O; R), ng knh AB. Gi dl tip tuyn ca ng trn, C l tip im. D v E ln lt l cc hnh chiu ca A v B trn ng thng d.1. Chng minhhai im D v E ngoi ng trn (O)2. K CHAB (H AB). Chng minh AD. BE = CH2.Cu 6: (4 im)Cho hnh thang cn ABCD ( BC // AD). Gi M v N ln lt l trung im ca hai yBC v AD. Trn tia i ca tia AB ly im P bt k, PN ct BD ti Q. Chng minh MN l tia phn gic ca gc PMQ.&&&&&&& HT&&&&&&&TP GII THI HC SINH GII TON LP 9Thi gian: 150 pht S 14Bi 1: (3 im) Cho phn thc ( ) ( ) ( )( ) ( )2 22 2 22a b c a b c ab bc caMa b c ab bc ca+ + + + + + ++ + + +a) Tm cc gi tr ca a, b, c M c ngha ?b) Rt gn phn thc M.Bi 2: (4 im) Rt gn cc biu thc sau:a)2 4 5 21 8010 2A + +b) 2 1 2 12 1 2 1x x x xBx x x x+ + + vi x 2 Bi 3: (3 im)a) Chng minh rng 2 2 111 12n n + ++chia ht cho 133 vi mi s t nhin n.b) Tm nghim t nhin ca phng trnh xy 4x = 35 5y Bi 4: (3 im) a) Gii phng trnh 0 5 2 6 1412 + + x x b) Cho a, b, c 0. Chng minh rng a + b + cab bc ca + +Bi 5: (3,5 im) Cho na ng trn (O; R) , ng knh AB, M l im di chuyn trn na ngtrn. V tiptuyn ti M ct cc tip tuyn vi na ng trn ti A v ti B theo th t C v D . im M v tr no trn na ng trn th t gic ACDB c din tch nh nht ? Tnh theo R din tch nh nht . Bi 6: (3,5 im)Cho na ng trn (O; R) , ng knh AB. AC v BD l hai dy cung ca na ng trn ct nhau ti im M bn trong na ng trn (O). Chng minh AM. AC + BM. BD = AB2 . ********************** HT*************************** THI CHN HC SINH GII TON LP 9Thi gian: 150 pht S 15Bi 1: (3 im) a) Tm s d ca php chia S : 101 trong :S = 1n + 2n + 3n + ..+ 97n + 98n + 99n vi n l s t nhin lb) Chng minh rng : Nu n l s t nhin l th 2 3 4 13 2n n + ++chia ht cho 25Bi 2: (4 im)Khng x dng my tnh, rt gn cc biu thc sau: a)A =2 12 140 2 12 1407 512 140 32 5 28 12 140+ + + + + +b) B =4 4 4 4 x x x x + + Bi 3: (3 im) Gii phng trnh : a) 2 3 x x + b)( )2 23 10 12 x x x x + Bi 4: (4 im)Cho ng trn tm O, ng knh AB, k dy CD ty khng vung gc vi AB v ct AB ti I. Gi M v N ln lt l hnh chiu ca A v B trn CD.Chng minh CM = DN.Bi 5: (4 im) Cho hnh bnh hnh ABCD c AC > BD. K CEAB, CFAD. Chng minh : AB. AE + AD. AF = AC2Bi 6: (2 im) Chng minh bt ng thc: 1 1 1 3a b b c c a a b c+ + >+ + + + +( vi a, b, c > 0)*******HT****** THI CHN HC SINH GII TON LP 9Thi gian: 150 pht S 16Bi 1: (3 im)a) Tm cc ch s x, y 1234 72 xyMb) Vi n l s t nhin, chng minh rng 9.10n + 18 chia ht cho 27Bi 2: (4 im)Khng x dng my tnh, rt gn cc biu thc sau: a) A =6 3 3 6 3 3 + b) B = 4 7 4 73 2 4 7 3 2 4 7+ ++ + Bi 3: (4 im)a) Gii phng trnh: 1 4 4 9 9 ..... 100 100 165 x x x x + + + + + + + + b) Gii phng trnh nghim nguyn:

5 3 2 11 x y xy Bi 4: (2 im) Chng minh bt ng thc:

1 2a b c da b d a b c b c d a c d< + + + 0)Bi 5: (3 im)Cho hnh thang ABCD y AB = 3 cm, y CD = 9 cm, AD = 4 cm , BC = 6 cmTnh din tch hnh thang.Bi 6: (4 im)Cho na ng trn tm O, ng knh AB = 2R v E l mt im di chuyn trn nang trn ( E khc A v B). Tip tuyn ti E ca na ng trn ct hai tip tuyn kt A v B ca na ng trn ln lt C v D. Gi H l hnh chiu ca E trn AB.a) Chng minh HE l phn gic ca CHD.b) Chng minh CD. EH khng i khi E di chuyn trn na ng trn.----HT ---- THI CHN HC SINH GII TON LP 9Thi gian: 150 pht S 17Cu 1: (4 im)a) Rt gn biu thc:

2 2 36 2A++12 185 2 6++b) Chng minh bt ng thc: 2010 20112010 20112011 2010+ > +Cu 2: (3 im)a) Tm gi tr ca x biu thc 212 2 5 x x + c gi tr ln nht .b) Tm gi tr nh nht ca biu thc : P =( ) ( )2 21995 1996 x x + + + Cu 3: (3 im) Gii phng trnh sau: 29 20 2 3 10 x x x + + +Cu 4: (4 im)Cho na ng trn tm O ng knh AB = 2R, CD l dy cung song song vi ABv khong cch t O ti CD cng di bng CD.a) Tnh CD theo R.b) Gi H l hnh chiu ca C n AB v I l giao im ca AB vi tip tuyn v t CVi na ng trn O. Chng minh HI = 4 HO, t tnh HI v CI theo R.Cu 5: (4 im)Trn cnh AB, BC, CA ca tam gic ABC ly cc im C1, A1, B1 tng ng sao cho AA1 , BB1 , CC1 ng qui ti O. ng thng qua O songsong vi AC ct A1B1 v C1B1 ti K v M. Chng minh OK = OMCu 6: (2 im)Chng minh rng: Nu n l s t nhin chia ht cho 4 th 2n 1 chia ht cho 15. === HT=== THI CHN HC SINH GII TON LP 9Thi gian: 150 pht S 18Cu 1: (3 im)a) Rt gn biu thc sau: A = ( ) ( ) ( )3 3 32 2 23 x y z xyzx y y z z x + ++ + + + b) Chng minh rng: Nu 2 2(1 ) (1 )x yz y xzx yz y xz vi x y ; 1 yz ; 1 xz ; 0 x ; 0 y ; 0 z th 1 1 1x y zx y z+ + + +Cu 2: (4 im)a) 2 29 2 14 9 2 147 2 7 2| ` | ` ++ +. , . ,b)14 7 15 5 1 5 2:1 2 1 3 7 55 2 ] ++ + ] ]Cu 3: (3 im)Gii cc phng trnh sau: a)( ) ( )3 4 9 x x x b) 1 14 6 0 x xx x| ` | `+ + + . , . ,Cu 4: (3 im)a) Tm s d khi chia 31993 cho 7b) Gii phng trnh nghim nguyn xy+ 3x 2y 9 = 0Cu 5: (3 im)Cho hnh thoi ABCD c 0120 A . Tia Ax to vi tia AB gc BAx bng 150 v ct cnh BC ti M, ct ng thng DC ti N.Chng minh 2 2 21 1 43 AM AN AB+ Cu 6: (4 im)Cho na ng trn ng knh AB. Hai dy cung AC v BD ct nhau ti H.Chng minh AH. AC + BH. BD c gi tr khng i. ---- HT ---- THI CHN HC SINH GII TON LP 9Thi gian: 150 pht S 19Cu 1: (3 im)a) Tm s d khi chia 31998 + 51998 cho 13.b) Tm tt c cc s t nhin n n4 + 4 l s nguyn t.Cu 2: (4 im) Khng x dng my tnh, rt gn cc biu thc sau: a) A = 14 6 5 2 25 31, 5 2 +++b) B = ( )( )28 2 15 8 2 15 : 3 1 2007 . 2008 2 2007 + + +Cu 3: (3 im)a) Tm gi tr nh nht ca biu thc2010 x x .b) Gii phng trnh: 2 25 15 3 3 1x x + +Cu 4: (3 im)a) Chng minhng thc: ( )1 1 11 1 1 n n n n n n + + + + vi n nguyn dngb) Chng minh bt ng thc:1 1 1 1..... 12 1 1 2 3 2 2 3 4 3 3 4 100 99 99 100+ + + + 0; b > 0. Chng minh a ba bb a+ +Bi 3: (4,0 im) Gii cc phng trnh: a)5 14 20 3 9 45 49 3xx x + b)2 222 2 2 2x xx x+ + + + Bi 4: (3,5 im) Cho tam gic ABC. Gi M, N , P l cc im ln lt nm trn cc cnh BC, CA, AB ca tam gic ABC sao cho cc ng thng AM, BN, CP ng qui ti O. Chng minh : . . 1AP BM CNPB MC NA Bi 5: (3,,5 im) T mt im A ngoi ng trn tm O, k hai tip tuyn AB v AC vi B, C l cc tip im. Trn on OB ly im N sao cho BN = 2ON. ng trung trc ca on thng CN ct OA ti M. Hy tnh t s AMAO.Bi 6: (2 im) Tm nghim nguyn dng ca phng trnh: 2(x + y) + 16 = 3xy--------------HT----------- THI CHN HC SINH GII TON LP 9Thi gian: 150 pht S 23Bi 1: (4im)Rt gn cc biu thc sau: a) 2 1 2 2 12 2 2 2 2 + ++ b) 2 5 2 52 3 5 2 3 5+ ++ + Bi 2: (6im) a) Gii phng trnh:

2 22 21 5 1 25 1259 45 16 80 3 912 16 4 9x xx x+ ++ + + b) Chng minh1 1 1y x y xx y x y++ >+ + + + c)Tm gi tr nh nht v ln nht ca biu thc: Q = 24 5 x x + +Bi 3: (3im)Cho mt im P nm trong ng trn (O; R). Hai dy cung di ng ABv CD cng i qua P v vung gc vi nhau ti P.Chng minh AB2 + CD2 c gi tr khng i.Bi 4: (4im)Cho hnh ch nht ABCD c AB = 2BC. Trn cnh BC ly im E . Tia AE ctng thng CD ti F. Chng minh 2 2 21 1 14 AB AE AF- =Bi 5: (3im) Gii phng trnh nghim nguyn: x2y + 2x2 y2 + 1 = 0 -----------------HT----------------------- THI CHN HC SINH GII TON LP 9Thi gian: 150 pht S 24Bi 1: (2 im)Bit a + b + c = 0 v abc 0. Chng minh rng: 2 2 2 2 2 2 2 2 21 1 10b c a c a b a b c+ + + + + Bi 2: (4 im)a) Khng x dng my tnh, hy rt gn cc biu thc sau: A =( ) ( ) ( ) ( )2 1 3 1 6 1 5 2 2 3 + + + b) Cho x, y tha mn: ( ) ( )2 22010 2010 2010 x x y y + + + + Tnh gi tr biu thc B = x + y Bi 3: (4 im)a) Gii phng trnh sau: 2 25 545 5 x x x x + b) Vi x, y khng m . Tm gi tr nh nht ca biu thc: P = 2 3 2 2011, 5 x xy y x + +Bi 4: (3 im)Cho na ng trn (O) ng knh AB. ng trung trc ca AB ct na ng trn ti I. Trn tia i ca tia IO ly im C sao cho IO = IC. T C v hai tip tuyn CD v CE vi na ng trn (D v E l hai tip im). Trn cung DE ly im S (S khc I) , tip tuyn ti S ca na ng trn ct CD v CE lnlt ti H v K. Tnh s o gc HOK.Bi 5 :(3 im)Cho tam gic ABC vung ti A, phn gic trong AD. Chng minh rng: 1 1 2AB AC AD+ Bi 6: (4 im)a) chng minh rng : Vi mi s t nhin n ta c: 2 1 2 121 17 15n n + ++ +khng chia ht cho 19.b) Gii phng trnh nghim nguyn: 5x + 25 = 3xy + 8y2---------------------ht------------------------

THI CHN HC SINH GII TON LP 9Thi gian: 150 pht (Khng k thi gian pht ) S 25

Bi 1: (3 im) a) Rt gn phn thc sau: ( ) ( )( ) ( )22 22 24 121 2 12x x x xx x x x+ + + + + + + b) Tnh gi tr ca biu thc: 22 22 x yzPx y+ bit 0; 0 x y v 3x y = 3z ; 2x + y = 7zBi 2: (6 im) a) Rt gn biu thc: Q = 3 4 56 8 10 27 36 45+ ++ + + + + b) Gii phng trnh: 22 1 5 0 x x x + + c) Ba s dng a, b, c tha mn b c ;a b c + v ( )2a b a b c + + Chng minh ng thc: ( )( )22a a ca cb cb b c+ + Bi 3: (3 im)Cho tam gic ABC cn ti A. Mt ng trn c tm O trn BC v tip xc vi AB v AC. K tip tuyn d ca ng trn (O) ct hai cnh AB v AC ln lt ti P v Q.Chng minh : BC2 = 4BP. CQBi 4: (4 im)Cho tam gic ABC . K ng thng bt k i qua trng tm G ca tam gicct cc cnh AB v AC ln lt P v Q.Chng minh rng: 3AB ACAP AQ+ Bi 5: (4 im)a) Chng minh rng: 2 2 2 2 2 2 2 23 5 7 19....... 11 .2 2 .3 3 .4 9 .10+ + + + 0; b > 0 th a ba bb a+ +Cu 3: (4 im)Cho hnh ch nht ABCD, M l trung im AD, N l trung im BC. Trn tia ica tia DC ly im P. Tia PM ct on thng AC Q.Chng minh : QNM MNP Cu 4: (4 im) T mt im A ngoi ng trn tm O, k hai tip tuyn AB v AC vi B, C l cc tip im. Trn on OB ly im N sao cho BN = 2ON. ng trung trc ca on thng CN ct OA ti M. Hy tnh t s AMAO.Cu 5: (4 im) a)Chng minh rng vi n l s t nhin th: 2 2 15 26.5 8 59n n n + ++ + M b) Chng minh rng vi mi s t nhin n 3:B = 3 3 3 31 1 1 1 1...........3 4 5 12 n+ + + + < ======== ht ==========UBND HUYN THNG BNH THI HC SINH GII NM HC 2010-2011PHNG GIO DC- O TOMN TON 9 (Thi gian 150 pht) Cu 1: (4 im) Tnh: 1) A = ( )23 3 3 3 21 33 1 3 1 5 3+ + ++ (1,5 im)2) B =4 2 3 4 2 3 2 20 + + +(1,5 im)3) C =3 5 3 5 3 40 + + +(1 im)Cu 2: (3 im) Giiphng trnh:1) 14 4 2 9 9 42x x + + + +(1,5 im)2) 24 4 2 10 x x x + + (1,5 im)Cu 3: (1 im) Cho 2x2 + 2y2 = 5xy Tnh D = x yx y+Cu 4: (4 im)1)_Tm s d php chia 2 888 885100 chia cho 13 (1,5 im)2) Gii phng trnh nghim nguyn:a) xy + 2y = 3x + 11 (1,5 im)b) x3 x2y + 3x 2y = 5 (1 im) Cu 5:Cho tam gic ABC c gc B = 760 v AB < AC < BC. Trn cnh BC ly E sao cho AB = CE. Gi M l trung im AC, I l trung im BE. MI ct AB ti H.Tnh s o gc BHI(3 im) Cu 6: Cho hai ng trn (O; R) v (O1; R1) tip xc ngoi ti A. Tip tuyn chung ngoi CD ( C thuc (O); D thuc (O1)). Chng minh CD l tip tuynca ng trn ng knh OO1(5 im) ======== ht=====UBND HUYN THNG BNH THI HC SINH GII NM HC 2009-2010PHNG GIO DC- O TOMN TON 9 (Thi gian 150 pht)Cu 1: (4im)Thc hin php tnh: a) 5 5 5 5 1 2 263 5 1 5 1 2 1+ + + ++ (1im) b) 6 2 5 6 2 5 8 2 15 + + + + (1im) c) 2 6 5 5 3 33 2 5 5 1 3 1 ++ + + + (1im) d) 22 3 2 33+ + +(1im)Cu 2: ( 3im)Gii phng trnh: a)5 1 4 20 1 x x + (1,5im) b) 22 1 2 8 x x x + + (1,5im)Cu 3: (1im) Rt gn biu thc:

2 2 2 2 2 2 2 2 21 1 1Pb c a c a b a b c + ++ + + bit a +b + c = 0(1im)Cu 4: (4im)a) Tm s d ca php chia 3444444444028 chia cho 31 (1,5im)b) Gii phng trnh nghim nguyn: 3x 2y = 11 xy (1,5im) c) Chng minh rng : x5y xy5 chia ht cho 30 ; vi x ; y Z( 1im)Cu 5: (4im)Cho hnh ch nht ABCD, k AH vung gc vi BD ti H. Gi E; K ln ltl trung im ca DH v BC.Chng minh AEEK Cu 6: (4im)T im M ngoi ng trn (O) k hai tip tuyn MA, MB vi ng trn (O)(A; B l hai tip im). Trn cung nh AB ly im N, tip tuyn ti im N cang trn (O) ct MA ti E, MB ti K; ng vung gc vi MO ti O ct tia MA ti C, tia MB ti D. Chng minh : Tam gic CEO v tam gic DOK ng dng.*** HT***1 1 2a b b c c a+ + + +PGD & T THNG BNH THI HC SINH GII LP 9 S GD QUNG NAM NM HC :2008-2009 Mn : TON CHNH THCThi gian : 150 pht Cu 1:( 4 im )a) Tm s d php chia (2im)(20082008 -1 ) : 7b)Chng minh rng : 42n+2 -1 chia ht cho 15 ( n N ) (2im)Cu 2:( 5 im ) Tnh a) A = 2 3 2 32 2 3 2 2 3+ ++ + (2 im) b) 5 2 10207 2 5 2 7B + ++ + +(2 im) c) Chng minh rng a, b, c l cc s khng m v b l s trung bnh cng ca a v c th ta c : (1 im)Cu 3: ( 3 im ) a) Chng minh rng : nu a + b + c = 0th a3 + b3 + c3 = 3abcb) Cho a > 0 ;b > 0 ;c > 0;d > 0Chng minh rng : 1a b c dEb c d a c d a b d a b c + + + + + + + + + + +Cu 4: ( 4 im )Cho tam gic ABC vung cn ti A . Trn cnh BC ly im M . Chng minh rng: BM2 + CM2 =2AM2 Cu 5 :(4 im ) Gi M , N ln lt l trung im ca cc cnh AD v BC ca hnhch nht ABCD. Trn tia i ca tia DC ly im P bt k . Giao im ca AC v ng thng PM l Q.Chng minh rng : QNM MNP Ht