Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh TUYN TP CC THI HS GII MN TON LP 6(C P N Y ) Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh S 1 Thi gian lm bi 120 pht Cu 1 : (2 im) Cho biu thc 1 2 21 22 32 3+ + + +=a a aa aAa, Rt gn biu thc b, Chng minh rngnu a l s nguyn th gi tr ca biu thc tm c ca cu a, l mt phn s ti gin. Cu 2: (1 im)Tm tt c cc s t nhin c 3 ch sabc sao cho12 = n abcv 2) 2 ( = n cbaCu 3: (2 im) a. Tm n n2 + 2006 l mt s chnh phng b. Cho n l s nguyn t ln hn 3. Hi n2 + 2006 l s nguyn t hay l hp s. Cu 4: (2 im)a. Cho a, b, n e N* Hy so snh n bn a++v ba b. ChoA = 1 101 101211;B = 1 101 101110++ . So snh A v B. Cu 5: (2 im) Cho 10 s t nhin bt k : a1, a2, ....., a10. Chng minh rng th no cng c mt shoc tng mt s cc s lin tip nhau trong dy trn chia ht cho 10. Cu 6: (1 im)Cho 2006 ng thng trong bt k 2 ngthng no cng ct nhau. Khng c 3 ng thng no ng qui. Tnh s giao im ca chng. P NCu 1: Ta c: 1 2 21 22 32 3+ + + +=a a aa aA= 11) 1 )( 1 () 1 )( 1 (2222+ + +=+ + + + +a aa aa a aa a a iu kin ng a -1 ( 0,25 im). Rt gn ng cho0,75 im. b.Gi d l c chung ln nht caa2 + a 1 v a2+a +1 ( 0,25 im). V a2 + a 1 =a(a+1) 1 l s l nn d l s l Mt khc, 2 =[ a2+a +1 (a2 + a 1) ] . d Nn d = 1 tc l a2 + a + 1v a2 + a 1 nguyn t cng nhau.( 0, 5 im) Vy biu thc A l phn s ti gin. ( 0,25 im) Cu 2:abc =100a + 10 b + c = n2-1(1) cba = 100c + 10 b + c= n2 4n + 4(2) (0,25 im) T (1) v (2) 99(a-c) = 4 n 5 4n 5 . 99 (3) (0,25 im) Mt khc:100 |n2-1 | 999 101 | n2 | 1000 11 |n|31 39 |4n 5 | 119 (4) ( 0, 25 im) T (3) v (4)4n 5=99 n = 26 Vy:abc = 675 ( 0 , 25 im) Cu 3: (2 im) a) Gi s n2 + 2006 l s chnh phng khi ta t n2 + 2006 = a2 ( ae Z) a2 n2 = 2006 (a-n) (a+n) = 2006 (*) (0,25 im). + Thy : Nu a,n khc tnh cht chn l th v tri ca (*) l s l nn khng tha mn (*)( 0,25 im). + Nu a,n cng tnh chn hoc l th (a-n).2 v (a+n).2 nn v tri chia ht cho 4 v v phi khng chia ht cho 4 nn khng tha mn (*)(0,25 im). Vy khng tn ti n n2 + 2006 l s chnh phng. (0,25 im). b) n l s nguyn t > 3 nnkhng chia ht cho 3. Vy n2 chia ht cho 3 d 1 do n2 + 2006 = 3m + 1 + 2006 = 3m+2007= 3( m+669) chia ht cho 3. Vy n2 + 2006 l hp s. ( 1 im). Bi 4: Mi cu ng cho 1 imTa xt 3 trng hp1 =ba 1 >ba 1 ba a>b a+m > b+n.M n bn a++ c phn tha so vi 1 l n bb a+ ba c phn tha so vi 1 l bb a, vn bb a+ < bb a nn n bn a++ < ba (0,25 im). TH3: ba x=0; y-5=12 => y=17hoc 2x+1=3=> x=1; y-5=4=>y=9 (0,25) vy (x,y) = (0,17); (1,9) (0,25) b.(1) Ta c 4n-5 = 2( 2n-1)-3(0,25) 4n-5 chia ht cho2n-1 => 3 chia ht cho2n-1 (0,25) Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh =>* 2n-1=1 => n=1 *2n-1=3=>n=2(0,25) vy n=1;2 (0,25) c. (1)Ta c 99=11.9 B chia ht cho 99 => B chia ht cho 11v B chia ht cho 99(0,25) *B chia ht cho 9 => ( 6+2+4+2+7+x+y) chia ht cho 9(x+y+3) chia ht cho 9=> x+y=6 hoc x+y =15 -B chia ht cho 11=> (7+4+x+6-2-2-y) chia ht cho11=> (13+x-y)chia ht cho 11 x-y=9 (loi) hoc y-x=2 (0,25) y-x=2 v x+y=6 => y=4; x=2 (0,25) y-x=2 v x+y=15 (loi) vy B=6224427(0,25) Cu2: a. Gi dl c chung ca 12n+1v 30n+2 ta c5(12n+1)-2(30n+2)=1 chia ht cho d(0,5) vy d=1 nn 12n+1 v 30n+2 nguyn t cng nhaudo 2 301 12++nnl phn s ti gin (0,5) b. Ta c 221 x=3 Bi 2. Val mt s t nhin vi mi a eZ nn ta< 5 ta=>a= {0,1,2,3,4}. Nghala={0,1,-1,2,-2,3,-3,4,-4}.Biudintrntrcscccsnyulnhn-5vnhhn5do -5 x + 1 + x + 2 + x + 3 + . . . . . . .. . .. . . . + x + 100 =5750 =>(1+ 2 + 3 + . .. + 100) + ( x + x+ x . . . . . . . + x ) =5750 101 x 50+100 x= 5750 100 x + 5050=5750 100 x = 5750 5050100 x =700 x =7 Cu 2. a)eg cd ab abc + + = 100 10000 deg= 9999cd ab 99 + +( ) eg cd ab + + . 11. b). 10 28 + 8 .9.8ta c 10 28 + 8 .8 (v c s tn cng l 008)nn 10 28 + 8 . 9.8 vy 10 28 + 8.72 Cu 3.Gi s giy mi lp thu c l x (Kg) th ( x-26) .11 v ( x-25) . 10. Do (x-15) e BC(10;11) v 200 x 300=> x-15 = 220=> x = 235. S hc sinh lp 6A l:(235 26) : 11 + 1 = 20. hs S hc sinh lp 6B l:(235 25) : 10 + 1 = 22 hs. Cu 4.S th nht bng: 119: 76 = 2221 (s th hai)S th ba bng: 119: 32 = 2227 (s th hai) Tng ca 3 s bng 2227 21 22 + + (s th hai) = 2270(s th hai) S th hai l : 210 : 2270 = 66 ;s th nht l: 2221. 66 = 63 ; s th 3 l:2227.66 = 81 Cu5: ng thng a chia mt phng ra hai na mt phng Xt 3 trng hp a). Nu c 4 im A, B, CD thuc cng mt na mt phng th a khng ct on thng no. b).Nuc 1im(ChnghnimAthucnamtphng)baimB,C,Dthucnamtphngith ng thng a ct ba on thng AB, AC, AD c). Nu c 2 im chng hn (A v B) thuc mt na mt phng hai im kia (C v D) thuc mi mt phng i th a ct bn on thng AC, AD, BC, BD S 5 Thi gian lm bi 120 pht Bi 1 (3): a) So snh: 222333 v 333222 b) Tm cc ch s x v y s2 8 1 y xchia ht cho 36 c) Tm s t nhin a bit 1960 v 2002 chiacho a c cng s d l 28 Bi 2 (2): Cho : S = 30 + 32 + 34 + 36 + ... + 32002 a) Tnh S b) Chng minh S. 7 Bi 3 (2):Tm s t nhin nh nht, bit rng khi chia s ny cho 29 d 5 v chia cho 31 d 28 Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Bi 4 (3): Cho gc AOB = 1350. C l mt im nm trong gc AOB bitgc BOC = 900

a) Tnh gc AOC b) Gi OD l tia i ca tia OC. So snh hai gc AOD v BOD HNG DNBi 1 (3):a) Ta c 222333 = (2.111)3.111 = 8111.(111111)2.111111(0,5) 333222 = (3.111)2.111 = 9111.(111111)2 (0,5) Suy ra: 222333 > 333222 b) s2 8 1 y x.36 ( 0sx, y s 9 , x, y e N ) + + + +4 29 ) 2 8 1 (..yy x(0,5) { } 9 ; 7 ; 5 ; 3 ; 1 4 2 = y y .(x+y+2) . 9 => x+y = 7 hoc x+y = 16 => x ={ } 7 ; 9 ; 0 ; 2 ; 4 ; 6(0,25) Vy ta c cc s: 16812; 14832; 12852; 10872; 19872; 17892(0,25) c) Ta c a > 28 => ( 2002 - 1960 ) . a => 42 .a(0,5)=> a = 42 (0,5) Bi 2 (2): a) Ta c 32S = 32 + 34 + ... + 32002 + 32004 (0,5) Suy ra: 8S = 32004 - 1 => S = 81 32004(0,5) b) S = (30 + 32 + 34 ) + 36(30 + 32 + 34 ) + ... + 31998(30 + 32 + 34 ) = = (30 + 32 + 34 )( 1 + 36 + ... + 31998 )= 91( 1 + 36 + ... + 31998 )(0,75) suy ra: S. 7 (0,25) Bi 3 (2):Gi s cn tm l: aTa c a = 29q + 5 = 31p +28(0,5) 29(q - p) = 2p + 23 V 2p + 23 l nn( q - p) l => q - p> 1.(0,75) V a nh nht hay q - p = 1 => p = 3;=> a = 121 (0,5) Vy s cn tm l 121 (0,25) Bi 4 (3):a)theogithitCnmtronggcAOBnn tia OC nm gia hai tia OB v OA => gc AOC + gc BOC = gc AOB => gc AOC = gc AOB - gc BOC => gc AOC = 1350 - 900 =450 b) v OD l tia i ca tia OC nn C, O,Dthnghng. Do gc DOA + gc AOC = 1800 (hai gc kb) => gc AOD = 1800 - gc AOC = 1800 - 450 =>gcAOD= 1350 gc BOD = 1800 - 900 = 900 Vy gc AOD > gc BOD ------------------------------------------------ S 6 Thi gian lm bi 120 pht Bi 1( 8 im 1. Tm ch s tn cng ca cc s sau:a)571999 b) 931999 2. Cho A= 9999931999 - 5555571997. Chng minh rng A chia ht cho 5. Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh 3 . Cho phn s ba(0 < a < b) cng thm m n v (m > 0) vo t v mu th phn s mi ln hn hay b hn ba? 4. Cho s16 * 4 * 710 * 155 c 12 ch s . chng minh rng nu thay ccdu * bi cc chc s khc nhau trong ba ch s 1,2,3 mt cch tuth s lun chia ht cho 396. 5. chng minh rng: a) 31641321161814121< + + ; b) 1633100399...34333231100 99 4 3 2< + + + Bi 2: (2 im ) Trn tia Ox xc nh cc im A v B sao cho OA= a(cm), OB=b (cm) a) Tnh di on thng AB, bit b< ab) Xc nh im M trn tia Ox sao cho OM = 21(a+b). P N Bi 1:1. Tm ch s tn cng ca cc s sau: ( 1 im ) tm ch s tn cng ca cc s ch cn xt ch s tn cng ca tng s : a) 571999 ta xt71999 Ta c: 71999 = (74)499.73 = 2041499. 343 Suy ra ch s tn cng bng 3 ( 0,25 im ) Vys 571999 c ch s tn cng l : 3 b) 931999 ta xt 31999 Ta c: 31999 = (34)499. 33 = 81499.27 Suy ra ch s tn cng bng 7(0,25 im ) 2. Cho A = 9999931999 - 5555571997 . chng minh rng A chia ht cho 5 chng minh A chia ht cho 5 , ta xt ch s tn cng ca A bng vic xt ch s tn cng ca tng s hng. Theo cu 1b ta c: 9999931999 c ch s tn cng l 7 Tng t cu 1a ta c: (74)499.7 =2041499.7 c ch s tn cng l 7( 0,25 im ) Vy A c ch s tn cng l 0, do A chia ht cho 5.( 0,25 im ) 3 (1 im )Theo bi ton cho a < b nn am < bm ( nhn c hai v vi m)( 0,25 im ) ab +am < ab+bm ( cng hai v vi ab) ( 0,25 im ) a(b+m) < b( a+m) m bm aba++ yOz = 180 a Khi ; tOy = 21a t,Oy = 21( 180 a)=>tOt, =) 180 (2121a a += 900 Cu 2; Gi s trong 20 im, khng c 3 im no thng hng. Khi , s ng thng v c l;19 . 20:2 = 190 Trong a im, giskhng c 3im no thnghng.Sng thngv c l ; (a 1 )a : 2 . Thc t, trong a im ny ta chi v c 1 ng thng. Vy ta c ;190 ( a- 1)a : 2 + 1 = 170 =>a = 7 x t y t z O Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh S 8 Thi gian lm bi : 120 Bi 1 : (3 ) Ngi ta vit ccs t nhin lin tip bt u t 1 n 2006 lin nhau thnh mt s t nhin L . Hi s t nhin L c bao nhiu ch s . Bi 2 : (3) C bao nhiu ch s gm 3 ch s trong c ch s 4 ? Bi 3 : (4)Cho bng gm 2007 nh sau : 17 36 19 Phn u ca bng nh trn . Hy in s vo ch trng sao cho tng 4 s 4 lin nhau bng 100 v tnh : a)Tng cc s trn bng . b)Tng cc ch s trn bng . c)S in th 1964 l s no ? P N Bi 1 : C 9 s c 1 ch s t 1 n 9 ( 0.25) C 90 s c 2 ch s t 10 n 99 (0.5) C 900 s c 3 ch s t 100 n 999 (0.5) Cc s c 4 ch s l t 1000 n 2006 c :2006 - 1000 + 1 = 1007 s (0.5) S ch s ca s t nhin L l :9 + 90.2 + 900.3 + 1007.4 = 6917 (ch s ) (1.25) Bi 2 : C 900 s c 3 ch s t 100 n 999 (0.25) Ta chia 900 s thnh 9 lp , mi lp c 100 s (0.25) c cng ch s hng trm . Lp th nht gm 100 s t 100 n 199Lp th hai gm 100 s t 200 n 299 Lp th 9 gm 100 s t 900 n 999(05) Xt 9 lp th lp th 4 c 100 s u c ch s 4 hng trm . 8 lp cn li hng trm khc 4 nn ch s 4 nu c th hng chc v hng n v (0.25) .Xt lp th nht th cc s c ch s 4 lm hng n vgm : 104, 114194 (c 10 s ) (05) cc s c 4 ch s lm hng chc l140,141,142,..149 (c 10 s) (0.5) Nhng s 144 c mt trong c 2 trng hp vy lp th nht s lng s c ch s 4 l :10 + 10 - 1 = 19 (s)(0.25) By lp cn li cng theo quy lut y . Vy s lng s c 3 ch s c ch s 4 l : 100 + 19.8 = 252 s (0.5) Bi 3 : Ta dng cc s 1; 2; 3 . nh s cho cc phn u bng (0.25) . 12 3 4 56 7 8 910 28 17 19 36 28 17 19 36 28 17 V cc s 4; 5; 6; 7 v 3; 4; 5; 6 nn s s 3 v s 7 bng nhau s 3 l 19 (0.5) 100 - (17 + 19 + 36) = 28 Vy s 1 l s 28 ( 0.25) 100 - (17 + 19 + 36) = 28 . Vy s in th 5 l s 28 ( 0.25) Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh s in s 6 cng l s 17 (0.25) Ta c : 2007 = 501.4 + 3 Vy ta c 501 nhm 4 , d 3 cui l th 2005; 2006; 2007 vi cc s 28; 17; 19 (0.5)a)Tng cc s trn bng l : 100.501 + 28 +17 +19 = 50164 (1) b) Tng cc ch s mi nhm l : 2 + 8 +1 + 7 +1 +9 + 3 + 6 = 37(0.5) Tng cc ch s trn bng l :37.501 + 2 + 8 + 1 + 7 +1 +9 = 18567c) 1964 . 4 . vy s in th 1964 l s 36 .(0.5) -------------------------------------------------------- S 9 Thi gian lm bi: 120 pht Bi 1: (1 im)in du thch hp vo trng: Nu ab v b10 a 10 Vit tp hp M cc s chn a tha mn a 10 C bao nhiu s chn nh hn n (nN) Bi 2: (2 im)Cho A = 3 + 32 + 33 + 34 + 3100 chng minh A chia ht cho 120. Bi 3: (2 im)Cho cc s 0; 1; 3; 5; 7; 9. Hi c th thit lp c bao nhiu s c 4 ch s chia ht cho 5 t su ch s cho. Bi 4: (2 im) Tng s trang ca 8 quyn v loi 1 ; 9 quyn v loi 2 v 5 quyn v loi 3 l 1980 trang. S trang ca mt quyn v loi 2 ch bng s trang ca 1 quyn v loi 1. S trang ca 4 quyn v loi 3 bng s trang ca 3 quyn v loi 2. Tnh s trang ca mi quyn v mi loi. Bi 5: (1,5 im)Cho c s o bng 1250. V tia oz sao cho = 350. Tnh trong tng trng hp. Bi 6: (1,5 im) Cho ba im A, B, C nm ngoi ng thng a. Bit rng c hai on thng BA, BC u ct ng thng a. Hi ng thng a c ct on thng AC khng? V sao? HNG DN Bi 1: (1 im) in du thch hp vo trng l( Nu ab v b10 a 10) 0,25 M = 0; 2; 4; 6; 8; 10 0,25 Ta phi xt hai trng hp: + S n l s chn, lc s chn nh hn n l0,25 + S n l s l, lc s chn nh hn n l0,25 Bi 2: (2 im) Ta nhm lm 25 nhm, mi nhm 4 s hng nh sau:A = (3 + 32 + 33+ 34) ++ (397+398+399+3100) = 3 (1 + 3 + 32+33)+.+ 397(1+3+32+33) 0,5 Ta li thy: 1 + 3 + 32+33 = 40Nn A = 40. (3 + 35 +39 ++397 ) 0,5 = 40.3 (30 + 34 +38 ++396 ) 0,5 = 120. (30 + 34 +38 ++396 )iu ny chng t A120 (pcm) 0,5 Bi 3: (2 im) Mi s c dng: ; 0,25 * Vi- C 5 cch chn ch s hng nghn (v ch s hng nghn phi khc 0). 0,5 - C 6 cch chn ch s hng trm. - C 6 cch chn ch s hng chc 0,25 Vy dng c 5.6.6 = 180 s. 0,5 * ViCch chn tng t v cng c 180 s.S thit lp c l 180+180=360 s 0,5 (c 4 ch s chia ht cho 5 t 6 ch s cho) Bi 4: (2 im) Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Ta k hiu: Loi 1: LI; Loi 2 : LII; Loi 3: LIII V s trang ca mi quyn v LII bng s trang ca 1 quyn LI , nn s trang ca 3 quyn LII bng s trang ca 2 quyn LI 0,5 M s trang ---------------------------------------- S 10 Thi gian lm bi: 150 pht(Nm hc 1998-1999) Bi 1: (4 im) Cho A = 7 + 73 + 75 + ... + 71999Chng minh rng A chia ht cho 35. Bi 2: (4 im) Tm s nguyn t p p + 10 v p + 14 u l cc s nguyn t. Bi 3: (4 im) Cho 19981. ..........31211 + + + + =nm vi m, n l s t nhin. Chng minh rng m chia ht cho 1999. Nu bi ton tng qut. Bi 4: (4 im) Cho phn s 00 200020002099 1999199919= Av phn s 20001999= BSo snh A v B. Bi 5: (4 im) t A i t H Ni v Ph L, t B i t Ph L ln H Ni, chng gp nhau ln th nht ti mt a im cch H Ni 25 Km. Khi xe n Ph L th lp tc quay tr li H Ni, cn xe kia n H Ni lp tc quay tr v Ph L .... C nh vy cho n ln gp nhau ln th 3 th hai xe cch H Ni l 5 Km. Tnh qung ng t Ph L i H Ni. P N Bi 1:A = 7 + 73 + 75 + ... + 71999 = (7 + 73) + (75 + 77) + ..... + (71997 +71999) A = 7(1 + 72) + 75(1 + 72) + ... + 71997(1 + 72) A = 7.50 + 75 .50 + 79.50 + ...+ 71997.50 => A Chia ht cho 5 (1) A = 7 + 73 + 75 + ... + 71999 = 7.( 70 + 72 + 74 + ... + 71998) => A Chia ht cho 7 (2) M CLN(5,7) = 1 => A Chia ht cho 35. Bi 2:-Nu p l s nguyn t chn => p = 2. Khi : p + 10 = 12 khng l s nguyn t. Vy p = 2 loi. -Nu p l s nguyn t l => p =3 hoc p = 3k + 1 hoc p = 3k + 2. +./ p = 3 => p + 10 = 13 l s nguyn t v p + 14 = 17 l s nguyn t. Vy p = 3 l s nguyn t tho mn iu kin u bi. +./ p = 3k + 1 (k e N*) => p + 14 = 3k + 15 = 3(k + 5) Chia ht cho 3 v k + 5 > 5 Nn p + 14 l hp s. Vy p = 3k + 1 loi +./ p = 3k + 2 (k e N*) => p + 10 = 3k + 12 = 3(k + 4) Chia ht cho 3 v k + 4 > 4 Nn p + 10 l hp s. Vy p = 3k + 2 loi Bi 3: 19981. ..........31211 + + + + =nm.T1n1998c1998sNnvphic1998shngtaghp thnh 999 cp nh sau: |.|

\|+ + + |.|

\|+ + |.|

\|+ + |.|

\|+ =100019991. ..........19961311997121199811nm

1000 . 9991999.......1996 . 319991997 . 21999.1998 . 11999+ + + + =Quy ng tt c 999 phaan s ny ta c: Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh 1998 . 19978 . 1996 .... .......... .......... .......... .......... 9 . 8 . 7 . 6 . 5 . 4 . 3 . 2 . 1. 1999 . 1999 . 1999 ........ . 1999 . 1999 . 1999999 998 997 3 2 1a a a a a anm + + + + + +=Vi a1 , a2 , a3 , ........... , a998 , a999 e N 1998 . 1997 . 1996 .......... .......... .......... 3 . 2 . 1) ......... .( 1999999 998 997 3 2 1a a a a a anm + + + + + +=V 1999 l s nguyn t. Nn sau khi rt gn, a v dng phn s ti gin th t s vn cn tha s 1999. Vy m Chia ht cho 1999. Bi 4: 2000 20000000 20000000001999 19990000 199900000000 200020002099 1999199919+ ++ += = AB = = =+ ++ +=20001999100010001 . 2000100010001 . 1999) 1 10000 100000000 ( 2000) 1 10000 100000000 ( 1999 Vy A = B. Bi 5: Hai xe i ngc chiu nhau, gp nhau ln th nht th c 2 xe i c 1 ln qung ng H Ni - Ph L. V c hai xe cch H Ni 25 Km vy xe i t H Ni v i c qung ng 25 Km. V 2 xe li quay li on ng trn nn phi gp nhau ln 2, ln gp ny c 2 xe i c 3 ln qung ng H Ni - Ph L v nh vy ln gp th 3 th 2 xe i c 5 ln qung ng H Ni - Ph L. 1 ln qung ngH Ni -Ph L th xe t t H Niv i c 25 Km. Vy 5ln qung ng H Ni - Ph L th xe i c qung ng l: 25 Km x 5 = 125 Km. Thc t th xe i c 2 ln qung ng H Ni - Ph L v thm 5 Km. Vy qung ng H Ni - Ph L l: (125 - 5) : 2 = 60 (Km). p s: 60 Km. ------------------------------------------------------------------------- S 11 Thi gian lm bi: 120 pht I. TRC NGIM: in du x vo thch hp:( 1 im)

II. T LUN: Cu 1:Thc hin cc php tnh sau: (4 im) a.729 . 723 9 . 162 . 54 . 18 234 . 9 . 327 . 81 . 243 729 . 21812 2+ ++ b. 100 . 99199 . 9814 . 313 . 212 . 11+ + + + + c.110014131212 2 2 2< + + + + d. 6 29 19 99 20 9 1527 . 2 . 7 6 . 2 . 58 . 3 . 4 9 4 . 5 CungSai a. S -551bng 5 + 51 (0.25 im) . S 1173 bng 780

(0.25 im) c) S -1145bng 11- 45 (0.25 im) d) Tng -351+ 232bng -11513 (0.25 im) Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Cu 2: (2 im) Mt qung ng AB i trong 4 gi. Gi u i c 31 qung ng AB. Gi th 2 ikm gi u l 121 qung ng AB, gi th 3 i km gi th 2 121 qung ng AB. Hi gi th t i my qung ng AB? Cu 3: (2 im) a. V tam gic ABC bit BC = 5 cm; AB = 3cm ;AC = 4cm. b. Ly im O trong tam gic ABC ni trn.V tia AO ct BC ti H, tia B0 ct AC ti I,tia C0 ct AB ti K. Trong hnh c c bao nhiu tam gic. Cu 4: (1 im)a. Tm hai ch s tn cng ca cc s sau: 2100; 71991 b.Tm bn ch s tn cng ca s sau: 51992 P N I - T LUN. Cu 1: Thc hin cc php tnh. Cu a. =+ + +729 . 723 162 . 6 . 2 . 9 243 . 9 . 39 . 81 3 . 243 729 . 21813 2 2729 . 723 1944 . 729 243 . 729729 729 . 21812+ ++12910 . 7292910 . 729) 723 1944 243 ( 729) 729 2181 ( 729= =+ ++=Cu b. Ta c: ;21112 . 11 =;31213 . 21 = ;41314 . 31 = ..;;99198199 . 981 =1001991100 . 991 =Vy = + + + + +100 . 99199 . 9814 . 313 . 212 . 11 = + + + + + 1001991991981413131212111

1009910011 = . Cu c. Ta c: ;21112 . 11212 = 621 >> 801(2) Ta c+ +601601.+ 601601+ +801+801 +.+801801+= 127123 4413180206020=+= + = + (3) T (1) , (2), (3) Suy ra: 801791781......431421411+ + + + + +>127 Bi 2: V s trang ca mi quyn v loi 2 bng 32 s trang ca 1 quyn loi 1. Nn s trangca 3 quyn loi 2 bng s trang ca 2 quyn loi 1 M s trang ca 4 quyn loi 3 bng 3 quyn loi 2. N s trang ca 2 quyn loi 1 bng s trang ca 4 quyn loi 3Do s trang ca 8 quyn loi 1 bng :4 .8 : 2 = 16 ( quyn loi 3) S trang ca 9 quyn loi 2 bng9 .4 : 3 = 12 (qun loi 3) Vy 1980 chnh l s trang ca 16 + 12+ 5 = 33(quyn loi 3) Suy ra: S trang1 quyn v loi 3 l 1980 : 33 = 60 ( trang) S trang 1 quyn v loi 2 l 8034 . 60=(trang) S trang 1 quyn v loi1 l;12023 . 80=( trang) Bi 3:Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh T1; 2; ; nc n s hng Suy ra 1 +2 ++ n =2). 1 ( n n + M theo bi ra ta c 1 +2 +3+..+n=aaaSuy ra2). 1 ( n n + =aaa = a . 111 = a . 3.37Suy ra: n (n+1) = 2.3.37.a V tch n(n+1) Chia ht cho s nguyn t 37 nn n hoc n+1 Chia ht cho 37 V s 2). 1 ( n n +c 3 ch s Suy ra n+1 < 74 n = 37 hoc n+1 = 37 +) Vi n= 37 th 703238 . 37= ( loi) +) Vi n+1 = 37 th 666237 . 36= ( tho mn) Vy n =36 v a=6 Ta c: 1+2+3+..+ 36 = 666 Bi 4 :A, 1,5 im V mi tia vi 1 tia cn li to thnh 1 gc. Xt 1 tia, tia cng vi 5 tia cn li to thnh 5 gc. Lm nh vy vi 6 tia ta c 5.6 gc. Nhng mi gc c tnh 2 ln do c tt c l1526 . 5=gc B, 1 im . T cu a suy ra tng qut. Vi n tia chung gc cn( 21 n)(gc). S 14 Thi gian lm bi 120 pht Bi 1(3 im). a.Tnh nhanh: A = 1.5.6 2.10.12 4.20.24 9.45.541.3.5 2.6.10 4.12.20 9.27.45+ + ++ + + b.Chng minh : Vi keN* ta lun c : ( )( ) ( ) ( ) ( ) 1 2 1 1 3. 1 k k k k k k k k + + + = + . p dng tnh tng : S =( ) 1.2 2.3 3.4 ... . 1 n n + + + + + . Bi 2: (3 im).a.Chng minh rng : nu ( )11 ab cd eg + + .th :deg 11 abc . . b.Cho A = 2 3 602 2 2 ... 2 . + + + +Chng minh : A.3 ; 7 ; 15. Bi 3(2 im). Chng minh :

2 3 41 1 1 1...2 2 2 2n+ + + +< 1. Bi 4(2 im). a.Cho on thng AB =8cm. im C thuc ng thngAB sao cho BC =4cm. Tnh di on thng AC. b.Cho101ngthngtrongbtchaingthngnocngctnhauvkhngcba ng thng no cng i qua mt im. Tnh s giao im ca chng. P NBi 1. a. 1.5.6 2.10.12 4.20.24 9.45.541.3.5 2.6.10 4.12.20 9.27.45+ + ++ + + = ( )( )1.5.6 1 2.2.2 4.4.4 9.9.91.5.621.3.5 1 2.2.2 4.4.4 9.9.9 1.3.5+ + += =+ + +. b.Bin i : ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 1 1 2 1 3 1 k k k k k k k k k k k k( + + + = + + = + p dng tnh :Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh ( )( )( )( ) ( )( ) ( ) ( )3. 1.2 1.2.3 0.1.2.3. 2.3 2.3.4 1.2.3.3. 3.4 3.4.5 2.3.4....................................3. 1 1 2 1 1 n n n n n n n n= = = + = + + + Cng li ta c :( )( )( )( ) 1 23. 1 23n n nS n n n S+ += + + = . Bi 2. a.Tch nh sau :

( ) ( )deg 10000 100 9999 99 abc ab cd eg ab cd ab cd eg = + + = + + + + . Do9999 11;99 11 . .( )9999 99 11 ab cd + .M : ( )11 ab cd eg + + .(theo bi ra) nn :deg 11. abc .b.Bin i : *A =( ) ( ) ( ) ( ) ( ) ( ) ( )2 3 4 3 4 59 60 3 592 2 2 2 2 2 ... 2 2 2 1 2 2 1 2 ... 2 1 2 + + + + + + + + = + + + + + +=( )3 593 2 2 ... 2 3. + + + .*A = ( ) ( ) ( )2 3 4 5 6 58 59 602 2 2 2 2 2 ... 2 2 2 + + + + + + + + += =( ) ( ) ( )2 4 2 58 22. 1 2 2 2 . 1 2 2 ... 2 . 1 2 2 + + + + + + + + + = ( )4 587 2 2 ... 2 7 + + + . . *A = ( ) ( ) ( )2 3 4 5 6 7 8 57 58 59 602 2 2 2 2 2 2 2 ... 2 2 2 2 + + + + + + + + + + + + = =( ) ( ) ( )2 3 5 2 3 57 2 32 1 2 2 2 2 1 2 2 2 ... 2 1 2 2 2 + + + + + + + + + + + + = =( )5 5715. 2 2 ... 2 15. + + + .Bi 3. Ta c :( )21 1 1 1.1 1 n n n n n< =

p dng : 2 2 21 1 1 1 1 1 1 11 ; ;...; .2 2 3 2 3 1 n n n< < < 2 3 41 1 1 1...2 2 2 2n+ + + +< 11 1.n BC) AC + BC = AB AC = AB - BC = 4 cm. b. - Mi ng thng ct 100 ng thng cn li nn to ra 100 giao im. - C 101 ng thng nn c : 101.100 = 10100 giao dim. -Do mi giao im c tnh hai ln nn s giao im l : 10100 : 2 = 5050 giao im. Lu : Hc sinh gii cch khc ng vn cho im ti a. Bi hnh khng v hnh khng chm im. ------------------------------------------------------------- S 15 Thi gian lm bi 120 phtC BACBAGenerated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Cu 1: Cho S = 5 + 52 + 53 + + 52006 a, Tnh S b, Chng minh S 126 Cu 2. Tm s t nhin nh nht sao cho s chia cho 3 d 1; chia cho 4 d 2 ; chia cho 5 d 3; chia cho 6 d 4 v chia ht cho 11. Cu 3.Tm cc gi tr nguyn ca n phn s A = 3 21nn + c gi tr l s nguyn. Cu 4. Cho 3 s 18, 24, 72. a, Tm tp hp tt c cc c chung ca 3 s . b, Tm BCNN ca 3 s Cu 5. Trn tia cho 4 im A, B, C, D. bit rng A nm gia B v C; B nm gia C v D ; OA = 5cm; OD = 2 cm ; BC = 4 cm v di AC gp i di BD. Tm di cc on BD; AC. P NCu 1. (2). a, Ta c5S = 52 + 53 +54 ++52007 5S S = (52 + 53 +54 ++52007) (5 + 52 + 53 + + 52006) 4S = 52007-5 Vy S = 20075 54 b, S = (5 + 54) + (52 + 55)+(53 + 56) +.. + (52003 +52006) Bin i cS = 126.(5 + 52 + 53 ++ 52003) V 126 126 S 126 Cu 2. (3) Gi s phi tm l x. Theo bi ra ta c x + 2 chia ht cho 3, 4, 5, 6. x + 2 l bi chung ca 3, 4, 5, 6 BCNN(3;4;5;6) = 60 . nen x + 2 = 60.n Do x = 60.n 2(n = 1;2;3..) Mt khc x 11 ln lt cho n = 1;2;3. Ta thy n = 7 th x = 418 11 Vy s nh nht phi tm l 418. Cu 3. (1). Ta c 3 2 3 3 5 3( 1) 5 531 1 1 1n n nn n n n+ + += = = + A c gi tr nguyn 51 n nguyn. M 51 n nguyn 5 (n-1) hay n-1 l c ca 5 Do 5 = {1;5} Ta tm c n =2 n =0 n =6 n = -4 Cu 4 (2)A, Tm c cc (18); (24) ; (72) ng cho 0,5 C (18;24;72)= {1; 2; 3; 6} b, Ta c 72 e B(18) 72 e B(24) BCNN (18;24;72) = 72. Cu 5. (2) ODBACx

V A nm gia B v C nnBA +AC = BC BA +AC =4(1) Lp. lun B nm gia A v D. Theo gt OD < OA D nm gia O v A. (0,5) M OD + DA = OA 2 + DA =5 DA =3 cm Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Ta c DB + BA = DA DB +BA =3 (2) (0,25) (1) (2) AC DB = 1 (3) (0,25) theo ra : AC = 2BD thay v (3) Ta c 2BD BD = 1 BD = 1 (0,25) AC = 2BD AC = 2 cm(0,25) ------------------------------------------------------------- S 16 Thi gian lm bi: 120 pht Cu 1: (2 im) Cho 2 tp hpA = {n e N /n (n + 1) 12}. B = {x e Z / x< 3}. a. Tm giao ca 2 tp hp. b. c bao nhiu tch ab (vi a eA;b e B) c to thnh, cho bit nhng tch l c ca 6. Cu 2: ( 3 im). a. Cho C = 3 + 32 + 33 + 34 + 3100chng t C chia ht cho 40. b. Cho cc s 0; 1; 3; 5; 7;9. Hi c th thit lp c bao nhiu s c 4ch s chia ht cho 5t su ch s cho. Cu 3: (3 im). Tnh tui ca anh v em bit rng 5/8 tui anh hn 3/4 tui em l 2 nm v 1/2 tui anh hn3/8 tui em l 7 nm. Cu 4: (2 im). a. Cho gc xoy c s o 1000. V tia oz sao chogc zoy = 350. Tnh gc xoz trong tng trng hp. b. Din t trung im M ca on thng AB bng cc cch khc nhau. P N Cu 1: Lit k cc phn t ca 2 tp hp a. A = { 0, 1, 2, 3}B ={ - 2, -1, 0, 1, 2, }0,5 im A B = { 0, 1, 2,}0,5 im. b. C 20 tch c to thnh -2-1012 000000 1-2-1012 2-4-2024 3-6-3036 Nhng tch l c ca 6: +1; + 2+ 3+ 60,5 im Cu 2:a. B = (3 + 32 + 33+ 34) ++ (397+398+399+3100) = 3 (1 + 3 + 32+33)+.+ 397(1+3+32+33)0,5 im=40.(3+35+39 ++397 ) : 400,5 im b. Mi s c dng abc0, abc5. Viabc0 - C 5 cch chn ch s hng nghn (v ch s hng nghn khng phi l s 0). - C 6 cch chn ch s hng trm. - C cch chn ch s hng chc. Vy 5 . 6 . 6 = 180 s. Viabc5Cch chn tng tv cng c 180 s. Vy ta thit lp c 360 s c 4 ch s chia ht cho 5 t 6 ch s cho0,5 im. Cu 3: 1/2 tui anh th hn 3/8 tui em l 7 nm. Vy tui anh hn 6/8 tui em l 14 nm0,5 im. M 5/8 tui anh ln hn 3/4 tui em l 2 nm,nn 1-5/8 = 3/8 tui anh = 14-2 = 12 nm. 1 im Vy tui anh l: 12:3/8 = 32 tui.0,5 im Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh 3/4 tui em = 32 14 = 18 tui 0,5 im Tui em l: 18:3/4 = 24 tui0,5 im Cu 4:a, C 2 cch v tia OZ (c hnh v) Gc XOZ = 650 hoc 13501 im b, C th din t trung im M ca on thng AB bng 3 cch khc nhau M l trung imMA+MB=AB MA=MB=AB/2 Ca on thng AB MA=MB ---------------------------------------------------------------- S 17 Thi gian lm bi: 120 pht A/. BI Cu 1: (2,5 im) C bao nhiu s c 3 ch s trong c ng mt ch s 5? Cu 2:Tm 20 ch s tn cng ca 100! . Cu 3: Ngi ta th mt s Bo vo ao th sau 6 ngy bo ph kn y mt ao. Bit rng c sau mt ngy th din tch bo tng ln gp i. Hi : a/. Sau my ngy bo ph c na ao? b/. Sau ngy th nht bo ph c my phn ao? Cu 4: Tm hai s a v b( a < b ), bit: CLN( a , b ) = 10 v BCNN( a , b ) = 900. Cu 5: Ngi ta trng 12 cy thnh 6 hng, mi hng c 4 cy. Hy v s v tr ca 12 cy . P N Cu 1: (2,5 im) Chia ra 3 loi s: *5ab . Trong s a c 9 cch chn ( t 0 n 9, tr s 5 ). S b cng vy.Nn cc s thuc loi ny c : 9.9 = 81 ( s )(1 im) * 5 a b. Trong s a c 8 cch chn ( t 1 n 8, tr s 5 ).S b c 9 cch chn. Nn cc s thuc loi ny c: 9.8 = 72 ( s ) (0,5 im) * 5 ab. Trong s a c 8 cch chn , s b c 9 cch chn.Nn cc s thuc loi ny c : 8.9 = 72 ( s )(0,5 im)V3dngtrnbaogmttcccdngs phI mv 3 dng l phn bit.Nn s lng ccs t nhin c 3ch s trong c ng mt ch s 5 l: 81 + 72 + 72 = 225 ( s ) p s: 225 ( s )(0,5 im) Cu 2: ( 2,5 im) *Ccthas5trong100!(khiphntchccthaschiahtcho5)l: 100 100245 25+ = (thas)(1 im) * Cc tha s 2 c trong 100! l: 100 100 100 100 100 1002 4 8 16 32 64 ((((+ + + + + (((( = 50 + 25 + 12 + 6 + 3 + 1 = 97( s )(1 im) Tch ca mi cp tha s 2 v 5 tn cng bng mt ch s 0. Do : 100! C tn cng bng 24 ch s 0. Vy 20 ch s tn cng ca 100! l20 ch s 0. Cu 3: (1,5 im) Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh a/. V 6 ngy bo phkn ao v c sau 1 ngy din tch bo tng ln gp i nn ph kn na ao th phi sau ngy th 5.(0,5 im) b/. Sau ngy th x s phn ao b che ph l: Vi x = 5, ta c: 1: 2 = 12(ao) Vi x = 4, ta c: 12 : 2 = 14(ao) Vi x = 3, ta c: 14 : 2 = 18(ao) Vi x = 2, ta c: 18 : 2 = 116(ao) Vi x = 1, ta c: 116 : 2 = 132 (ao)(0,5 im) Vy sau ngy th nht th bo ph c: 132 (ao)(0,5 im) Cu 4: (1,5 im) V CLN( a, b)= 10, suy ra : a = 10x ;b = 10y (vi x < y v CLN(x, y)= 1 )(0,5 im) Ta c : a.b = 10x . 10y = 100xy (1) Mt khc: a.b = CLN(a, b) . BCNN(a, b) a.b = 10 . 900 = 9000(2)(0,5 im) T (1) v (2), suy ra: xy = 90 Ta c cc trng hp sau: X12359 y9045301810 T suy ra a v b c cc trng hp sau: a1020305090 y900450300180100 Cu 5: (1 im) Ta c s : --------------------------------------------------------------- S 18 Thi gian lm bi: 120 pht Cu 1: (2)Vi q, p l s nguyn t ln hn 5 chng minh rng: P4 q4 . 240 Cu 2: (2) Tm s t nhin n phn b3 4193 8++=nnAa. C gi tr l s t nhin b. L phn s ti gin c. Vi gi tr no ca n trong khong t 150 n 170 th phn s A rt gn c. Cu 3: (2) Tm cc nguyn t x, y tha mn : (x-2)2 .(y-3)2 = - 4 Cu 4: (3) Cho tam gic ABC v BC = 5cm. im M thuc tia i ca tia CB sao cho CM= 3 cm. a. Tnh di BM b. Cho bit gc BAM = 800 , gc BAC = 600 . Tnh gc CAM. c. V cc tia Ax, Ay ln lt l tia phn gic ca gc BAC v CAM . Tnh gc xAy. d. Ly K thuc on thng BMv CK = 1 cm.Tnh di BK. Cu 5: (1) Tnh tng: B = 100 . 972....10 . 727 . 424 . 12+ + + +P N Cu 1: (2) Ta c: p4- q4 = (p4 1 ) (q4- 1); 240 = 8 .2.3.5 Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Chng minh p4 1 .240 - Do p >5 nn p l s l (0,25) + Mt khc: p4 1= (p-1) (p+1) (p2 +1)(0,25) --> (p-1 v (p+1) l hai s chn lin tip=> (p-1) (p+1) .8(0,25) + Do p l s l nn p2l s l ->p2 +1 . 2(0,25) - p > 5 nn p c dng: + p = 3k +1 --> p 1 = 3k + 1 1= 3k .3--> p4 1 . 3 + p = 3k + 2 -->p + 1= 3k + 2 + 1= 3k +3 . 3-->p4-1 .3 (0,25) - Mt khc, p c th l dng: + P =5k +1 --> p 1= 5k + 1- 1= 5k.5 --> p4 - 1.5 + p = 5 k+ 2 --> p2 + 1 = (5k +2)2+1= 25k2+ 20k +5 . 5 --> p4 - 1. 5(0,25 ) + p = 5k +3 --> p2 +1 = 25k2 + 30k +10. --> p4 1 . 5 + p = 5k +4 --> p + 1 = 5k +5 . 5 --> p4 1 . 5(0,25) Vy p4 1 . 8 . 2. 3 . 5 hay p4 1 . 240 Tng t ta cng c q4- 1 . 240 (0,25) Vy: (p4 - 1) (q4 1)= p4 q4.240 Cu 2: (2) a. 3 418723 4187 ) 3 4 ( 23 4193 8++ =+ + +=++=n nnnnA Ae N th 187 . 4n + 3 => 4n +3e { } 187 ; 11 ; 17 (0,5) + 4n + 3 = 11-> n = 2+ 4n +3 = 187 --> n = 46 + 4n + 3 = 17 -> 4n = 14 -> khng c ne N (0,5) Vy n = 2; 46 b.A l ti gin khi 187 v 4n + 3 c UCLN bng 1 -> n=11k + 2 (k e N) -> n=17m + 12 (m e N)(0,5) c) n = 156 ->;1977= An = 165 -> 3989= An = 167 -> 61139= A (0,5) Cu 3: (2) Do 4 = 12 . (- 4)= 22.(-1) n c cc trng hp sau: a. == == = = 1311 24 31 ) 2 (2yxyxyx(0,5) hoc == = = 1111 2yxyx(0,5) b. ==== = = 2422 21 32 ) 2 (2 2yxyxyx (0,5) hoc === = 2022 2yxyx (0,5) Cu 4: (3) a. M, B thuc 2 tia i nhau CB v CM -> C nm gia B v M. ->BM = BC + CM = 8 (cm)(0,5) b. C nm gia B,M -> Tia AC nm gia tia AB, AM ->ZCAM =Z BAM -Z BAC = 200 (0,75) B A M x KC y Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh c. CZ xAy =Z x AC+ZCAy= 21Z BAC + 21Z CAM =21 ( Z BAC +Z CAM) = 21ZBAM = 21.80 = 400 (0,75) d. + Nu K e tia CM -> C nm gia B v K1 -> BK1 = BC + CK1 = 6 (cm)(0,5) + Nu Ke tia CB -> K2 nm gia B v C -> BK2= BC = CK2=4 (cm)(0,5 ) Cu 5: (1) Ta c)4111(324 . 12)4111(314 . 11 = = );....10171(3210 . 72);7141(327 . 42 = = ......;)1001991(32100 . 972 = (0,5) B=)1001991.....1017171414111(32 + + + + B= 503310099.32)100111(32= = (0,5) ---------------------------------------------------------------- S 19(Vng trng 09 10) (Thi gian lm bi 150 pht) Cu 1:a, cho A = 4 + 22 + 23+ 24 + + 220

Hi A c chia ht cho 128 khng? b, Tnh gi tr biu thc 104 . 265 . 2 13 . 21012 12++4 910 102 . 35 . 3 11 . 3 + Bi 2 : a, Cho A = 3 + 32 + 33 + + 32009 Tm s t nhin n bit rng 2A + 3 = 3n b, Tm s t nhin c ba ch s chia ht cho 5 v 9 bit rng ch s hng chc bng trung bnh cng ca hai ch s kia Bi 3 :Cho p v p + 4 l cc s nguyn t( p > 3) . Chng minh rng p + 8 l hp sBi 4 :Tm hai s t nhin bit tng ca chng bng 84 ,CLN ca chng bng 6. Bi 5: Gi A v B l hai im trn tia Ox sao cho OA = 4 cm ; OB = 6 cm . Trn tia BA ly imCsao cho BC = 3 cm .So snh AB viAC HNG DN CHMBiHng dn chmim 1 a,2A A= 221 . 27 A. 128 b, = 104 . 278 . 21012 + 16 . 316 . 3910 = 3 + 3= 6 0.5 0.5 0.5 0.5 2 a, Tm cn = 2010 b, Gi s phi tm labc theo bi ra ta ca + b + c . 9v 2b = a + c nn 3b . 9 b . 3vy b{ } 9 ; 6 ; 3 ; 0 eabc . 5ce{ } 5 ; 01 0.5 Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Xt sabo ta c s 630 Xt s5 abta c s 135 ; 765 0.5 3 P c dng 3k + 1; 3k + 2keN Dng p = 3k + 2 th p + 4 l hp s tri vibip = 3k + 1 p + 8 = 3k + 9 . 3 p + 8 l hp s0.5 0.5 0.5 0.5 4 Gi 2 s phi tm l a v b ( asb) ta c (a,b) = 1 nn a = 6a

b= 6btrong (a,b) = 1 ( a,b,a,beN) a + b = 14 a135 a13119 A61830 B786654

0.5 0.5 1 5 x OB C A Hai im A v B trn tia Oxm OA< OB (41) 0.5 0.5 0.5 0.5 ----------------------------------------------------------------- S 20 Thi gian lm bi: 120 pht Cu 1:(2) Thay (*) bng cc s thch hp : a) 510* ; 61*16 chia ht cho 3. b) 261* chia ht cho 2 v chia 3 d 1 Cu 2: (1,5) Tnh tngS = 1.2 + 2.3 + 3.4 + ... + 99.100 Cu 3: (3,5 ) Trn con ng i qua 3 a im A; B; C (B nm gia A v C)c hai ngi i xe my Hng v Dng. Hngxut pht tA, Dng xut pht t B. Hcngkhihnhlc 8 gi cng n Cvo lc 11 gi cng ngy. Ninh i xe p t C v pha A, gp Dng luc 9 gi v gp Hng lc 9 gi 24 pht. Bit qung ng AB di 30 km, vn tc ca ninh bng 1/4 vn tc ca Hng. Tnh qung ng BC Cu 4: (2) Trn on thng AB ly 2006 im khc nhau t tn theo th t t A n B l A1; A2; A3; ...; A2004. T im M khng nm trn on thng AB ta ni M vi cc im A; A1; A2; A3; ...; A2004 ; B. Tnh s tam gic to thnh Cu 5: (1) Tch ca hai phn s l 158. Thm 4 n v vo phn s th nht th tch mi l 1556. Tm hai phn s . P N

Cu 1 a) 510* ; 61*16 chia ht cho 3 th: 5 + 1 + 0 + * chia ht cho 3; t tm c * = 0; 3; 6; 9(1) b) 261* chia ht cho 2 v chia 3 d 1 th: Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh * chn v 2 + 6 + 1 + * chia 3 d 1; t tm c * =4(1) Cu 2 S =1.2 + 2.3 + 3.4 + ... + 99.100 3.S =(1.2 + 2.3 + 3.4 + ... + 99.100).3 (0,5) =1.2.3 + 2.3.3 + 3.4.3 + ... + 99.100.3 =1.2.3 +2.3.(4 - 1) + 3.4.(5 - 2) + ... + 99.100.(101 - 98) (0,5) =1.2.3 - 1.2.3 + 2.3.4 - 2.3.4 + 3.4.5 - ... - 98.99.100 + 99.100.101 S=99.100.101: 3 = 33. 100 . 101 = 333300 (0,5) Cu 3 Thi gian i t A n C ca Hng l:11 - 8 = 3 (gi) Thi gian i t B n C ca Dng l: 11 - 8 = 3 (gi) Qung ng AB l 30 km do c 1 gikhong cch ca Hng v Dng bt i 10 km. V vy lc 9 gi Hng cn cch Dng l 20 km, lc Ninh gp Dng nn Ninh cng cch Hng 20 km. n 9 gi 24 pht, Ninh gp Hng do tng vn tc ca Ninh v Hng l: 20 :) / ( 502460 . 206024h km = =Do vn tc ca Ninh bng 1/4 vn tc ca Hng nn vn tc ca Hng l: [50 : (1 + 4)] . 4 = 40 (km/h) T suy ra qung ng BC l: 40 . 3 - 30 = 90 (km) p s:BC = 90 km Cu 4: (2) Trn on thng AB c cc im A; A1; A2; A3; ...; A2004 ; B do , tng s im trn AB l 2006 im suy ra c 2006 on thng ni t M n cc im . Mi on thng (v dMA)c thkthp vi 2005 on thngcn livcc on thng tngng trn AB to thnh 2005 tam gic.Do 2006 on thng s to thnh2005 . 2006 = 4022030 tam gic (nhng lu l MA kt hp vi MA1 c 1 tam gic th MA1 cng kt hp vi MA c 1 tam gic v hai tam gic ny ch l 1) Do s tam gic thc c l: 4022030 : 2 = 2011015 Cu 5: (1) Tch ca hai phn s l 158. Thm 4 n v vo phn s th nht th tch mi l 1556 suy ra tch mi hn tch c l 1556 - 158 = 1548 y chnh l 4 ln phn s th hai. Suy ra phn s th hai l 1548 : 4 = 1512= 54 . T suy ra phn s th nht l: 158 : 54 = 32 ------------------------------------------------------------------- S 21 Thi gian lm bi: 120 pht Cu 1: (1.5) Chng minh cc phn s sau y bng nhau: 5325;53532525 ;535353252525 Cu 2: (1,5) Khng quy ng mu hyo snh hai phn s sau: 6737v 677377 Cu 3: (2) Tm s t nhin x, bit: 51002010030) 5 ( + = xxCu 4: (3) Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Tui trung bnh ca mt i vn ngh l 11 tui. Ngi ch huy l 17 tui. Tui trung bnh ca i ang tp (tr ngi ch huy) l 10 tui. Hi i c my ngi. Cu 5: (2) Cho gc xOy v gc yOz l hai gc k b nhau. Gc yOz bng 300 . a.V tia phn gic Om ca gc xOy v tia phn gic On ca gc yOz. b.Tnh s o ca gc mOn. P N Cu 1: 5325101 . 53101 . 2553532525= = (0.5) 532510101 . 5310101 . 25535353252525= = (0.5) Vy 535353252525535325255325= = (0.5) Cu 2: 677300670300>m 67730067306730670300> = (1)(0.5) Ta c : 673067371 = v6773006773771 = (2)(0.5) T (1) v (2)6737677377> (0.5) Cu 4: Gi s i vn ngh c n ngi. Tng s tui i vn ngh tr ngi ch huy l m. Ta c:1117=+nm (1)v 101 = nm (2)(1) T (1) m = 11n 17(3) (2) m = 10n 10 (4)(1) T (3) v (4)11n 17 = 10n 10 n =7(1) p s: S ngi trong i vn ngh l: 7 Cu 5: a.Tnh c ZyOn = 150 ; ZmOy = 750 (1) Ch ra cch v v v ng.(0.5) b.Tnh c ZmOn = 900 (0.5) ---------------------------------------------------------------------- S 22 Thi gian lm bi: 120 pht. Cu I : 3 Thc hin php tnh bng cch hp l : 1)A = 2006 .... 3 2 163 . 373737 37 . 636363+ + + + 2)B= 237373735124242423.200651951755200641941744:53337331353123712191212.4161||||.|

\|+ + ++ + + + + Cu II : 2 Tm cc cp s (a,b) sao cho :45 5 4 . b aCu III : 2 O m y n xz Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh ChoA =31 +32+33 + .....+ 32006

a, Thu gn A b, Tm x 2A+3 = 3x Cu IV : 1 So snh: A = 1 20051 200520062005++ vB =1 20051 200520052004++ Cu V: 2 Mt hc sinh c quyn sch trong 3 ngy. Ngy th nht c c 52 s trang sch; ngy th 2 c c 53 s trang sch cn li; ngy th 3 c c 80% s trang sch cn li v 3 trang cui cng. Hi cun sch c bao nhiu trang? P N CU I : 1) 1,5 A = 2006 .... 3 2 163 . 373737 37 . 636363+ + + + = 2006 .... 3 2 1) 63 . 10101 .( 37 ) 37 . 10101 .( 63+ + + + ==+ + + +2006 .... 3 2 1) 10101 10101 .( 63 . 37 0 2) B = 237373735124242423.200651951755200641941744:53337331353123712191212.4161||||.|

\|+ + ++ + + + + =1010101 . 5 . 471010101 . 3 . 41.200611911711 5200611911711 4:5313711911 35313711911 . 12.4147||||.|

\||.|

\|+ |.|

\|+ + +|.|

\| +|.|

\| + = 5 . 473 . 41).45. 4 .(4147 = 3(1,5) CU 2: 2 - b=0 => 9+a. 9 => a = 0 - B =5 => 14+a .9 => a = 4 CU III: 2 a)A=31+32+33+.....+320063A=32+33+34+.....+320073AA=32007-3A= 23 32007

(1) b) Ta c : 2. 23 32007+3 = 3x =>32007 -3 +3 = 3x => 32007 = 3x => x = 2007 (1) CU IV: 1 A = 1 20051 200520062005++ < 2004 1 20052004 1 200520062005+ ++ += ) 1 2005 ( 2005) 1 2005 ( 200520052004++= 1 20051 200520052004++ = B VyA < BCU V : 2 Gi x l s trang sch, x eN Ngy 1 c c lx52 trang S trang cn li l x-x52 =x53 trangNgy 2 c c l 53.53x= x259 trang S trang cn li lx53- x259 = x256trangGenerated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Ngy th 3 c c l :x256.80% +30 = 12524x + 30 Hay :x52+x259+12524x + 30 =x=> x =625 trang S 625 trang ------------------------------------ S 23 Thi gian lm bi: 120 pht Bi 1 (1,5): Dng 3 ch s 3; 0; 8 ghp thnh nhng s c 3 ch s: a. Chia ht cho 2 b. Chia ht cho 5 c. Khng chia ht cho c 2 v 5 Bi 2 (2): a. Tm kt qu ca php nhn A =33 ... 3 x 99...9 50 ch s50 ch s b. Cho B = 3 + 32 + 33 + ... + 3100 Tm s t nhin n, bit rng 2B + 3 = 3n Bi 3 (1,5 ): Tnh a.C = 101 100 99 98 ... 3 2 1101 100 99 98 ... 3 2 1+ + + + + + + + + + + b.D = 3737.43 4343.372 4 6 ... 100+ + + + Bi 4 (1,5): Tm hai ch s tn cng ca 2100. Bi 5 (1,5): Cho ba con ng a1, a2, a3 i t A n B, hai con ng b1, b2 i t B n C v ba con ng c1, c2, c3, i t C n D (hnh v). Vit tp hp M cc con ng i t A dn D ln lt qua B v C Bi 6 (2): Cho 100 im trong khng c ba im no thng hng. C qua 2 im ta v mt ng thng. c tt c bao nhiu ng thng. P N Bi 1 (1,5): a. 308; 380;830(0,5) b. 380830(0,5) c. 803 Bi 2 (2): a) (1) A =50 chu so333...3_ x 50 chu so1 00..0- 1| | |\ .= 50 chu so 50 chu so 50 chu so33...300...0 -33...3 (0,5)= 49 chu so 49 chu so33 ... 33 00 ... 0033 ... 33 33 ...3266 ... 67_ (0,25). Vy A =49 chu so49 chu so33 ...3266 ... 67_ (0,25) ABCD a1 a2 a3 b1 b2 c1 c2 c3 Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh b) (1 )B= 3 + 32 + 33 + ... + 399 + 3100(1) 3B=32 + 33 + ... + 3100 + 3101(2)(0,25) Ly (2) tr (1) ta c: 2B = 3101 - 3(0,25) Do : 2B + 3 = 3101 (0,25) Theo bi 3B + 3 = 3n . Vy n = 101 (0,25) Bi 3 (1,5): a) (0,75) C = 101 100 99 98 ... 3 2 1101 100 99 98 ... 3 2 1+ + + + + + + + + + + Ta c: *, 101 + (100 + 99 + ... + 3 + 2 + 1) =101 + 101.100 : 2 = 101 + 5050 = 5151(0,25) *,101 - 100 + 99 - 98 + ... + 3 - + 1 =50 cap(101 - 100) + (99 - 98) + ... + (3 - 2)+ 1_= 50 + 1 = 51(0,25) Vy C = 515110151= (0,25) b) (0,75) B = 3737.43 4343.372 4 6 ... 100+ + + + Ta c:3737.43 - 4343.37 = 34.43.101 - 43.101.37 = 0(0,5) Vy B = 0 ( v 2 = 4 + 6 + ...+ 100 = 0)(0,25) Bi 4 ( 1,5): Ta c: 210 = 1024(0,25) 2100 = ( )10102= 102410 = ( )521024 (0,75) =(......76)5 = ....76(0,5) Vy hai ch s tn cng ca 2100 l 76 Bi 5 (1,5): Nu i t A n D bng con ng a1: a1 b1 c1; a1 b1 c2; a1 b1 c3; a1 b2 c1; a1 b2 c2; a1 b2 c3;(0,5) i t A n D bng con ng a2: a2 b1 c1; a2 b1 c2; a2 b1 c3; a2 b2 c1; a2 b2 c2; a2 b2 c3;(0,5) i t A n D bng con ng a3: a3 b1 c1; a3 b1 c2; a3 b1 c3; a3 b2 c1; a3 b2 c2; a3 b2 c3;(0,5) Vy tp hp M: M = { a1 b1 c1; a1 b1 c2; a1 b1 c3; a1 b2 c1; a1 b2 c2; a1 b2 c3; a2 b1 c1; a2 b1 c2; a2 b1 c3; a2 b2 c1; a2 b2 c2; a2 b2 c3; a3 b1 c1; a3 b1 c2; a3 b1 c3; a3 b2 c1; a3 b2 c2; a3 b2 c3;} Bi 6 ( 2):Chn mt im. Qua im v tng im trong 99 im cn li, ta v c 99 ng thng(0,5) Lm nh vy vi 100 im ta c 99.100 ng thng (0,5) Nhng mi ng thng c tnh 2 ln, do tt c c 99.100 : 2 = 4950 ng thng (1) -------------------------------------------------------- S 24 Thi gian lm bi: 120 pht Bi 1(2) a. Tnh tng S = 18 16 14 .... 6 4 22 . 550 135 4500 27+ + + + ++ + + b. So snh:A = 1 20071 200620072006++ vB = 1 20061 200620062005++ Bi 2 (2) a. Chng minh rng: C = 2 + 22 + 2 + 3 + +299 + 2100chia ht cho 31 b. Tnh tng C. Tm x 22x -1- 2 = C Bi 3 (2) Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Mt s chia ht cho 4 d 3, chia cho 17 d 9, chia cho 19 d 13. Hi s chia cho1292 d bao nhiu Bi 4 (2) Trong t thi ua, lp 6A c42 bn c t 1 im 10 tr ln, 39 bn c 2 im 10 tr ln, 14 bn c t 3 im 10 tr ln, 5 bn c 4 im 10, khng c ai c trn 4 im 10. Tnh xem trong t thi ua lp 6A c bao nhiu im 10 Cu 5 (2) Cho 25 im trong khng c 3 im thng hng. C qua 2 im ta v mt ng thng. Hi c tt c bao nhiu ng thng? Nu thay 25 im bng n im th s ng thng l bao nhiu. P NBi 1 a. S = 270.450 270.550 270(450 550) 2700003000(2 18).990 902+ += = =+ b. Ta c nu1ab 8100 Nn 3200 > 2300

b)A = 101 : 1717101 : 40417210101 : 17171710101 : 1212121717404172171717121212 + + +174 2 121741721712 += + = AVy A = 1710 hay A =B = 1710 3). s c 4 ch s*26*, 4ch s khc nhau m 4 ch s *26* chia ht cho c 4 s 2; 5;3;9.Ta cn tho mn : S m bo chia ht cho 2 nn s l s chn. S chia ht cho 5 nn s phi c ch s tn cng l s 0 hoc 5.S va chia ht cho 3 v9 .Nn s phi c tng cc ch s chia ht cho 9. Vy : Ch s tn cng ca s l 0*260. Ch s u l s 1 Do s cho l 1260 4 ) Bi 4. Tm s t nhin n. M 1! +2!+3! +...+n! l bnh phng ca mt s t nhin.Xt : n = 1 1! = 12 n = 2 1! +2! = 3 n=3 1! + 2! + 3! = 9 =32 n = 4 1!+ 2! +3! + 4! =33 Vi n >4 thn! = 1.2.3.........n l mi s chn .Nn 1!+2!+......+n! =33 cng vi mt s chn bngsc ch s tn cng ca tng l ch s 3 .Nn n khng phi l s chnh phng. Vy chc hai gi tr n=1 hoc n=3 th 1! +2! + 3! +4! +.......+n!l s chnh phng. 5 )Gii 1 gi xe th nht i c21qung ng AB. 1 gi xe th 2 i c 31 qung ng AB . Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh 1 gi c 2 xe i c 21+31= 65 qung ng AB. Sau 10 pht = 61gi: Xe th nht i c61. 21=121qung ng AB. Qung ng cn li l:1 -=121 1211 (ca AB) Thi gian hai xe cng i qung ng cn li l: 1211:65= 1011 gi = 1 gi 6 pht. Hai xe gp nhau lc 7 gi 10 pht + 1 gi 6 pht = 8 gi 16 pht . p n: 8 gi 16 pht.(0,25) 6) Hnh hc. (t v hnh) (2) V :

xOy= 1200 ,

AOy = 750, im A nm trong gc xOy nn tia OA nm gia hai tia Ox v Oy. Ta c :0 0 0 xOA =xOy - AOy =120- 75= 45im B c th hai v tr : B v B. (0,75)+, Ti B th tia OB nm ngoi hai tia Ox, OA nn 0 0 0 BOx + xOA = 135+ 45= 180 . Do

0BOA = BOx + xOA =180. Nn 3 im A,O,B thng hng. (0,75)+, Cn ti B th :

xOB' = 1350< 1800, 0 0 0AOB'= xOB' - xOA = 135- 45= 90 . Nn 3 im A,O, B khng thng hng.(0,5) ----------------------------------------------------------- S 26 Thi gian lm bi: 120 pht Cu 1:Tnh tng 2 3 1001 1 1 1...3 3 3 3A = + + + +Cu 2:Tm s t nhin a, b, c, d nh nht sao cho: 53ab = ; 1221bc = ; 611cd =Cu 3:Cho 2 dy s t nhin 1, 2, 3, ..., 50 a-Tm hai s thuc dy trn sao cho CLN ca chng t gi tr ln nht. b-Tm hai s thuc dy trn sao cho BCNN ca chng t gi tr ln nht. Cu 4:Cho bn tia OA, OB, OC, OD,to thnh cc gc AOB, BOC, COD, DOA khng c im chung. Tnh s o ca mi gc y bit rng: BOC = 3 AOB; COD = 5 AOB; DOA = 6 AOB HNG DN Cu 1: Ta c3A = 1 + 1/3 + 1/32 + ... + 1/399 vy: 3A-A = (1 + 1/3 + 1/32 + ... + 1/399)-(1/3 + 1/32 + ... + 1/3100) 2A= 1-1/3100 = (3100-1)/ 3100 suy raA= (3100-1) )/ 2.3100 Cu 2: Ta c 12/21= 4/7, cc phn s 3/5, 4/5,6/11 ti ginnn tn ti cc s t nhin k, l, m sao cho a=3k, b=5k,b=4n,c=7n,c=6m,d=11m.Tccngthc5k=4n,v7k=6mtac4n. 5v7n. 6m(4,5)=1; (7,6)=1 nn n. 5, n.6 mt khc (5,6) =1 do n.30 cc s t nhin a, b, c, d nh nht v phi khc 0 , ta chn n nh nht bng 30 suy ra: k =24, m=35 vy a=72, b=120, c=210, d=385. cu 3:Gi a v b l hai s bt k thuc dy 1, 2, 3, ..., 50. Gi s a>b. a.Gi d thuc C(a,b) th a-b.d ta s chng minh d 25 tht vy gi s d>25 th b>25 ta c a 50 m b>25 nn0< a-b < 25, khng th xy raa-b.d ; d=25 xy ra khi a=50; b=25 vy hai s c CLN t gi tr ln nht l 50 v 25 b. BCNN(a,b) a.b 50.49=2450 vy hai s c BCNN t gi tr ln nht l 50 v 49 Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh cu 4: (Hc sinh t v hnh)Ta thy : 0AOB+BOC + AOD >180v nu tri li th gc AOD c im trong chung vi ba gc kia. t

AOB= ta c: 0AOB+BOC + AOD + COD = 360 +3+5+6=3600 = 240. Vy:0 0 0 0AOB = 24 ; BOC =72 ; COD = 120 ;DOA = 144 ---------------------------------------------------------- S 27 Thi gian lm bi: 120 pht Cu 1: (3). a. Kt qu iu tra mt lp hc cho thy: C 20 hc sinh thch bng , 17 hc sinh thch bi, 36 hc sinh thch bng chuyn, 14 hc sinh thch bngv bi, 13 hc sinh thchbi v bng chuyn,15hc sinh thch bng v bng chuyn, 10 hc sinh thch c ba mn, 12 hcsinhkhng thch mn no. Tnhxem lp hc c bao nhiu hc sinh? b. Cho s: A = 1 2 3 4 5 6 7 8 9 10 11 12 .58 59 60. - S A c bao nhiu ch s? - Hy xa i 100 ch s trong s A sao cho s cn li l: + Nh nht + Ln nht Cu 2: (2). a. Cho A = 5 + 52 + + 596. Tm ch s tn cngca A. b.Tm s t nhin n : 6n + 3 chia ht cho 3n + 6 Cu 3: (3). a. Tm mt s t nhin nh nht bit rng khi chia s cho 3 d 2, cho 4 d 3, cho 5 d 4 v cho 10 d 9. b.Chng minh rng: 11n + 2 + 122n + 1 Chia ht cho 133. Cu 4: (2).Cho n im trong khng c 3 im no thng hng . C qua hai im ta v 1 ng thng. Bit rng c tt c 105 ng thng. Tnh n? P NCu 1: (3). a. V c s cho (1,5). - S hc sinh thch ng 2 mn bng v bi: 14 10 = 4 (hs) - S hc sinh thch ng hai mn bi v bng chuyn: 13 10 = 3 (hs). - S hc sinh thch ng hai mn bng v bng chuyn: 15 10 = 5 (hs) - S hc sinh ch thch bng : 20 (4 + 10 + 5) = 1 (hs) - S hc sinh ch thch bi: 17 (4 + 10 + 3) = 0 (hs). - S hc sinh ch thch bng chuyn: 36 (5 + 10 + 3) = 18 (hs). Vy: S hc sinh ca lp l: 1 + 0 + 18 + 4 + 10 + 5 + 3 + 12 + = 53 (hs). b. (1,5 ) A = 1 2 3 4 5 6 7 8 9 10 11 12 58 59 60. * T 1 n 9 c : 9 ch s T 10 n 60 c: 51 . 2 = 102 ch s. Vy: S A c 9 + 102 = 111 ch s.(0,5) * Nu xa 100 ch s trong s A th s A cn 11 ch s. Trong s A c 6 ch s 0 nhng c 5 ch s 0 ng trc cc ch s 51 52 53 . 58 59 60. Trong s nh nht c 5 ch s 0 ng trc s nh nht l s c 6 ch s. S nh nht l 00000123450 = 123450 (0,5). * Trong s A c 6 ch s 9. Nu s ln nht c 6 ch s 9 ng lin nhau th s l: 99999960 S ny ch c 8 ch skhng tha mn. S ln nht ch c 5 ch s 9 lin nhau s c dng 99999. Cc ch s cn li 78 59 60. Vy s ln nht: 99999785860. Cu 2: (2,5). a.(1,5). Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh A = 5 + 52 + + 596 5A =52 + 53 + + 596 + 597

5A A = 597 - 5 A = 975 - 54 Tac: 597 c ch s tn cng l 5 597 5 c ch s tn cng l 0. Vy: Ch s tn cng ca A l 0. b. (1). C: 6n + 3 = 2(3n + 6) 9 6n + 3chia ht 3n + 6 2(3n + 6) 9 chia ht 3n + 6 9 chia ht 3n + 6 3n + 6 = 1 ; 3 ;9 3n + 6- 9- 3- 1139 n- 5- 3- 7/3- 5/3- 11 Vy; Vi n = 1 th 6n + 3 chia ht cho 3n + 6. Cu 3: (2,5). a. (1). Gi s t nhin cn tm l a (a > 0, a e N) Theo bi ra ta c: - a chia cho 3 d 2 a 2 chia ht cho 3 - a chia cho 4 d 3 a 3 chia ht cho 4 - a chia cho 5 d 4 a 4 chia ht cho 5 - a chia cho 10 d 9 a 9 chia ht cho 10 a = BCNN(3, 4, 5, 10) = 60. b.(1,5). 11n + 2 + 122n + 1 = 121 . 11n + 12 . 144n =(133 12) . 11n + 12 . 144n = 133 . 11n + (144n 11n) . 12 Tac: 133 . 11n chia ht 133;144n 11n chia ht (144 11) 144n 11n chia ht 133 11n + 1 + 122n + 1 Cu 4: (2). S ng thng v c qua n im: ( ) 11052n n = n .(n 1) = 210 = 2 . 5 . 3 . 7 = 10 . 14 n. (n 1) = 6 . 35 = 15 . 14. V n v n 1 l 2 s t nhin lin tip nn: n = 14 Vy n = 14. ----------------------------------------------------------- S 28 Thi gian lm bi: 120 phtBi 1:(2,25 im)Tm x bit a) x+1 75 25= b) x-4 59 11=c)(x-32).45=0 Bi 2:(2,25 im)Tnh tng sau bng cch hp l nht: a)A = 11 + 12 + 13 + 14 + ..+ 20. b)B = 11 + 13 + 15 + 17 + ..+ 25. c)C = 12 + 14 + 16 + 18 + ..+ 26. Bi 3:(2,25 im)Tnh: a)A= 5 5 5 5...11.16 16.21 21.26 61.66+ + + +b)B= 1 1 1 1 1 12 6 12 20 30 42+ + + + +c)C = 1 1 1 1... ...1.2 2.3 1989.1990 2006.2007+ + + + + Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Bi 4:(1 im)Cho:A= 2001 20022002 200310 1 10 1; B = 10 1 10 1+ ++ +. Hy so snh A v B. Bi 5:(2,25 im) Cho on thng AB di 7cm. Trn tia AB ly im I sao cho AI = 4 cm. Trn tia BA ly im K sao cho BK = 2 cm. a)Hy chng t rng I nm gia A v K. b)Tnh IK. P NBi 1:(2,25 im) a) x=7 1 225 5 25 =;b) x=5 4 45 44 8911 9 99 99+ = =; c) x = 32 Bi 2:(2,25 im)Tnh tng sau bng cch hp l nht: a) A = (11 + 20) + (12 + 19) + (13 + 18) + (14 + 17) + (15+ 16)=31 + 31 + 31+31+ 31 = 31.5= 155 b)B = (11+25)+(13+23)+(15 + 21)+(17 +19) = 36.4 = 144. c)C = (12 +26)+(14+24)+(16 +22)+(18 +20) = 38.4 = 152. Bi 3:(2,25 im)Tnh: a)A= 1 1 1 1 1 1 1 1 1 1 5...11 16 16 21 21 26 61 66 11 66 66 + + + + = =b)B= 1 1 1 1 1 1 1 1 1 1 1 1 61 12 2 3 3 4 4 5 5 6 6 7 7 7 + + + + + = =c)C = 1 1 1 1 1 1 1 1 20061 ... ... 12 2 3 1989 1990 2006 2007 2007 2007 + + + + + = =Bi 4:(1 im)Ta c:10A = 20022002 200210 10 9 = 1 +10 1 10 1++ +(1) Tng t: 10B = 20032003 200310 10 9 = 1 +10 1 10 1++ +(2) T (1) v (2) ta thy : 2002 20039 910 1 10 1>+ + 10A > 10BA > B Bi 5:(2,25 im) a) Trn tia BA ta c BK = 2 cm.BA = 7cm nn BK< BA do im K nm gia A v B. Suy ra AK + KB = ABhay AK + 2 = 7 AK = 5 cm. Trn tia AB c im I v K m AI < AK (v 4 n+2 e ( 18) ={ } 18 ; 9 ; 6 ; 3 ; 2 ; 1 (0,25 im) +, n + 2= 1 n= - 1 (loi) +,n + 2= 2 n= 0+, n + 2= 3 n= 1+, n + 2= 6 n= 4+, n + 2= 9 n= 7+,n + 2= 18 n= 16 Vy n e{ } 16 ; 7 ; 4 ; 1 ; 0th B e N(0,25im ) c. (1 im) Ta c 55 =5.11 m (5 ;1) = 1(0,25 im) Do C = . y x1995 55 115..CC ( )( ) 12 (0.25 im)(1) => y = 0 hoc y = 5 +, y= 0 : (2)=> x+ 9+5 ( 1+9 +0).11 => x = 7(0,25 im) +, y =5 : (2)= > x+9 +5 (1+9+5 ). 11 => x = 1(0,25 im) Ba 2 (2 im) a( 1im) M = 140010...26010140105610+ + + += 28 . 255...13 . 10510 . 757 . 45+ + + + (0,25 im) = |.|

\| + + + + 281251...131101101717141.35( 0, 25 im) Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh = 145286.3528141.35= = |.|

\|( 0,5 im) b. (1 im)S = 153153153153153143133123113103+ + + + > + + + + => S >11515=(1) ( 0,5im) S=103103103103103143133123113103+ + + + < + + + +=> S 1 < S < 2 Bi 3:Gigi go np l a (ng/kg) ; khi lng go np mua l b (kg)(0,25 im) Suy ra gi go t la .1080; khi lng go t mua lb .100120 ( 0,25 im) S tin ngi th nht phi tr l a.b (ng)(0,25 im) S ting ngi thhai phi tr l 10096. .100120. .10080= b aa.b (0.75im) Vy ngi th hai tr t tin hn ngi th nht . T l % t hn l: % 4 . : . .10096. = |.|

\| b a b a b a(0,5 im) BI 4V hnh chnh xc(0,5 im) a.Bn im A,B, M, N thng hng v chng cng nm trn ng thng MN (0,5 im) b. (1 im) BM = AB AM = 2 (cm) (0,25im) M,N e tia AB m BM > BN ( 2 > 1) => N nm gia B v M.( 0,25 im) MN = BM BN = 1 cm = BN.=> N l ng trung im ca BM .(0,5 im). c. ng trn tm N i qua B nn CN = NB = 1 cm (0,25 im) ng trn tm A i qua N nnAC = AN = AM + MN = 4 cm(0.25 im) Chu viA CAN = AC + CN = NA = 4 + 4+1= 9 (cm) (0,5 im) S 30 Bi 1 :Tm x bita ) x + (x+1) +(x+2) +...... +(x +30) = 620b)2 +4 +6 +8 +..............+2x = 210 Bi 2 : a) chng t rng trong 3 s t nhin lin tip lun c 1 s chia ht cho 3 b) cho A =( 17n +1 )(17n +2 )3 . vi mi n N e Bi 3:ChoS = 1+3+32 +33+.........+348 +349 a )chng tS chia ht cho 4 b) Tm ch s tn cng ca S c)Chng t S =21 350 Bi 4 : Tm 2 sa ,bN etho mn : 12a + 36b = 3211 Bi 5 :Cho(2a + 7b)3 . ( a,bN e )Chng t : (4a + 2b ) 3 .

Bi 6 : Ly 1 tgiy ct ra thnh 6 mnh .Ly 1 mnh bt k ct ra thnh 6 mnh khc . C nh th tip tc nhiu lna) Hi sau khi ct mt s mnh no ,c th c tt c 75 mnh giy nh khng ? b)Gi s cui cng m c 121 mnh giy nh .Hi ct tt c bao nhiu mnh giy ? Bi 7 : Cho on thng AB .Hy xc nh v tr ca im C trn on thng AB sao cho CAsCBBi 8 :V on thng AB =5 cm .Ly 2 im C ,D nm gia A v B sao cho :AC +BD=6 cma)chng t im C nm gia B v Db)Tnh di on thng CD Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh P N Bi 1 : a) 31x + 620230 ) 30 1 (=+155 15 . 31 620 31 = = xx=155 :31= 5 b) 2102) 2 2 (=+ x x 210 ) 1 ( = + x x 210=2.3.5.7 =(2.7)(3.5)=14.15 Vy x= 14 Bi 2 : a) gi 3 s t nhin lin tip l x ,x+1, x+2( x) N e - Nu x = 3k ( tho mn ) .Nu x= 3k +1 th x+2 =3k+1+2 =(3k +3 ) 3 . -Nu x = 3k +2th x +1 = 3k+1 +2 = (3k +3 )3 . Vytrong 3 s t nhin lin tip c 1 s chia ht cho 3 b)Nhn thy17n, 17n +1 , 17n + 2l 3 s t nhin lin tip m 17n khng chia ht cho 3 ,Nn trong 2 s cn li 1 s phi3 .Do :A =( 17n +1 )(17n +2 )3 . Bi 3: a )Ta c :S = (1+3)+(32+33)+.......+(348+349) = 4+32(1+3)+......+ 348(1+4) 4 . b )S = (1+3+32 +33)+(34+35+36+37)+........+(344+345+346+347) +348 +349 Cc tng 4 s hng u chia ht cho 10 ,do tn cng bng 0 Mt khc 338 + 349 = 34.12 + 348 .3 = .....1 + ....1 .3 = .............4VyS c tn cng bng 4 c )S = 1+3+32 +33+.........+348 +349 3S= 3 +3+32 +33+.........+348 +349+ 350s s 3 =350 1 2S= 350 1Suy ra S =21 350 Bi 4 : Nhn thy 12 a4 . v 36 b 4 . m 3211 khng chia ht cho 4 , Vy khng c 2 s t nhin no tho mn Bi5 :Ta c ( 6a + 9b ) 3 . hay ( 2a + 7b +4a + 2b )3 . .M (2a +7b )3 .Nn(4a + 2b )3 . Bi 6 : a)Khi ta ct 1 t giy thnh 6 mnh th s mnh giy tng thm 5 .Ct nhiu ln nh th th tng s mnh giy tng thm 5k (k l t giy em ct ) .Ban u ch c 1t giy ,Vy tng s cc mnh giy l 5k + 1 S ny chia 5 d 1 : vy khng th c c tt c 75 mnh giy nh ( v 75 5 . ) b) Ta c 5k +1 = 121 k=24 .Vy ta ct c tt c 24 mnh giyBi 7 : -Gi M l trung im ca AB suy ra MA = MBv M eABXt 3 trng hp a ) CMta c MA = MBsuy ra CA = CB b ) C nm gia A v M CA < MA CA < MB(1) M nm gia C v BnnMB < CB (2) T (1) & (2) CA < CBc ) C nm gia M v BCB < MB CB < MA ( 3) M nm giaA v Cnn MA < CA(4)T (3) v (4) CA < CBABMC Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Tm li C eMA th ta lun c CAs CB

Bi 8 : C nm gia A v B nn : AC + CB = AB= 5V AC + BD= 6 AC + CB < AC + BD CB < BDCnm gia D v B b ) BD = BC + CD v AC + BD = 6nn AC + BC + CD = 6(BC + AC) + CD = 6 CD = 6 AB = 6 -5=1 VyCD= 1 S 31 Thi gian lm bi: 150 pht Nm hc 2009 - 2010 Cu 1 (2 im) Tnh a/ A = 1 2 3 . . . 98 99 100 1011 2 3 . . . 98 99 100 101+ + + + + + + + + + + b/ B = 423134 846267 . 423133423133 846267 . 423134+ Cu 2 (2 im) a/ Chng minh rng: 1028 + 8 chia ht cho 72 b/ Cho A = 1 + 2 + 22 + 23 + . . . + 22001 + 22002B = 22003 So snh A v B c/ Tm s nguyn t p p + 6; p + 8; p + 12; p + 14 u l cc s nguyn t. Cu 3 (2 im)Ngi ta chia s hc sinh lp 6A thnh cc t, nu mi t 9 em th tha 1 em, cn nu mi t 10 em th thiu 3 em. Hi c bao nhiu t, bao nhiu hc sinh ? Cu 4 (3 im) Cho +ABC c BC = 5,5 cm. im M thuc tia i ca tia CB sao cho CM = 3 cm. a/ Tnh di BM b/ Bit BAM = 800; BAC = 600. Tnh CAM Bit

BAM= 800;

BAC= 600. Tnh

CAM c/ Tnh di BK thuc on BM bit CK = 1 cm. Cu 5 (1 im)Chng minh rng:121001. . .241231221< + + + + P N Cu 1:a/ A =1015151 . 101= (1 im) b/ B =1423134 846267 . 423133423133 846267 846267 . 423133=+ +(1 im) Cu 2: a/ V 1028 + 8 c tng cc ch s chia ht cho 9 nn tng chia ht cho 9 Li c 1028+8 c3 ch s tn cng l 008 nn chia ht cho 8 Vy 1028 + 8 chia ht cho 72(1/2 im) b/ C 2A = 2 + 22 + 23 + . . . + 22002 + 22003=> 2A A = 22003 1 => A = B 1. Vy A < B.(1/2 im) c/ Xt php chia ca p cho 5 ta they p c 1 trong 5 dng sau: p = 5k; p = 5k + 1; p = 5k + 2; p = 5k + 3; p = 5k + 4 (k eN; k > 0) + Nu p = 5k th do p nguyn t nn k = 1 => p = 5 + Nu p = 5k + 1 => p + 14 = 5(k + 3). 5 v ln hn 5 nn l hp s (loi) + Nu p = 5k + 2 => p + 8 = 5(k + 2) . 5 v ln hn 5 nn l hp s (loi) + Nu p = 5k + 3 => p + 12 = 5(k + 3). 5 v ln hn 5 nn l hp s (loi) BDAC Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh + Nu p = 5k + 4 => p + 6 = 5(k + 2) . 5 v ln hn 5 nn l hp s (loi) Th li vi p = 5 tho mn(1 im) Cu 3:Gi s c thm 4 hc sinh na th khi chia mi t 10 em th cng cn tha 1 em nh khi chia mi t 9 em. Vy cch chia sau hn cch chia trc 4 hc sinh. Mi t 10 hc sinh hn mi t 9 hc sinh l: 10 - 9 = 1 (hc sinh) (1 im) Do s t l:4 : 1= 4 (t)(1/2 im) S hc sinh l:4 . 10 3 = 37 (hc sinh)(1/2 im) Cu 4: V hnh, ghi gi thit + kt lun (1/2 im) a/ C nm gia B v M=> BC + CM = BM(1/2 im) => BM = 3 + 5,5 = 8,5(1/2 im) b/ C nm gia B v M =>AC l tianm gia 2 tia AB v AM(1/2 im) => BAC + CAM = BAM =>

CAM=

BAM

BAC =>

CAM= 800 600 = 200(1/2 im) c/ Xt 2 trng hp: + Nu K nm gia C v M tnh c BK = BC + CK = 5,5 + 1 = 6,5 (cm) + Nu K nm gia C v B tnh c BK = 4,5 (cm)(1/2 im) Cu 5:Ta c: 1100991001121001. . .241231221100199121001...... ..........413124131212312111221< = < + + + + < < < <

S 32 th chn hc sinh gii lp 6 chuyn ton ( Qun Ba nh - Nm hc 1991-1992) Bi 1:( 5 im ) Bi 2: ( 5 im )Tm hai s t nhin a,b tho mn iu kin: a + 2b = 48 v(a,b) + 3 [a,b] = 114 Bi 3:Hnh hc( 6 im ) 1. Cho 3 im A,B,C thng hng v AB + BC =AC. im no nm gia hai im cn li? Ti sao?2. Cho gc aOb v tia 0c nm gia hai tia Oa v Ob. Od l tia i ca tia Oc .Chng minh rng: a.Tia Od khng nm gia hai tia Oa v Ob. b.Tia Ob khng nm gia hai tia Oa v Od. Bi 4: ( 4 im )Tnh t s BAbit (1/2 im) (1/2 im) 57 . 231143 . 23343 . 19531 . 19757 . 10741 . 10941 . 7631 . 74+ + + = + + + = B A1021716361513 25 , 18 7921:60033415 , 06539:75354212 + ||.|

\| ++Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh P N Bi 1Bi 2: A6121824303642 B21181512963 (a,b)36312363 [a,b]42369024903642 (a,b) + [a,b]12911427384114114129 Vy a = 12;b = 18 hoc a = 36 ;b = 6 Bi 4: S 33 thi hc sinh gii lp 6 chuyn ton ( Qun Ba nh- Nm hc 1993-1994) Cu 1: (6 im) Thc hin tnh dy ) 47 , 0 : 29 ( 100 : 2972 651865 445422 53:4521135621176733 3|.|

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\| Cu 2:(5 im) Tm 2 s t nhin tho mn: - Tng ca BSCNN v SCLN ca 2 s y l 174. - Tng ca snh v trung bnh cng ca 2 s y l 57 Cu 3 : (4 im)Cho 5 im A, B, C, D, E trong khng c 3 im nothng hng. - C bao nhiu on thng m mi on thng ni 2 trong 5 im cho.K tn cc on thng y. -Cth dng c mt ng thngkhng i qua im no trong 5im chom ct ng 5 on thng trong cc on thng ni trn khng? Gii thch v sao: Cu 4 : (5 im)Lc 8 gi, mt ngi i xe p t A n B vi vn tc 12km /h. Lt sau ngi th 2 cng i t A n B vi vn tc 20km /h. Tnh ra hai ngi s gp nhau ti B. Ngi th 2 i c na qung ng AB th tng vn tc ln thnh 24km /h. V vy 2 ngi gp nhau cch B4 km.Hi 2 ngi gp nhau lc my gi? P N Bi 1: =367 Bi 2: (a,b) + [a,b] = 174 ;3a + b = 114 b . 3 ;[a,b] . 3 v 174 . 3 (a,b) . 3 a . 3M 3a + b = 114 3a < 114 a < 38 a..369121518212427303336 b..105968778696051423324156 (a,b)363636363636 [a,b]1059626115634518035716829712016536 | | ( ){ } 42 ; 36 ; 30 ; 24 ; 18 ; 12 ; 648 48 2 ; 6 3 3 , 3 , 3 ; 3 144 ; 2 48 2e < = + = +aa b a a a b a b a a b a . . . . . .25215157131157 . 231143 . 23343 . 19531 . 1972157131157 . 50741 . 50941 . 35631 . 7451= = )` = + + + = = + + + =BAB ABA982824 , 2:75354216116125131293037. 96 , 0:7535421= + = ++Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Tng10811226416234818636017430012616842 Bi 4:Hiu vn tc trn na qung ng u l : 20 - 12 = 8 (km/h) Hiu vn tc trn na qung ng sau l : 24 - 12 = 12 (km/h) Hiu vn tc ca na qung ng u theo d nh bng 2/3hiu vn tc trn na qung ng sau. Ch xt na qung ng sau thi gian xe II ui kp xe Itrn thc t bng 2/3thi gian xe hai ui kp xe I theo d nh Thi gian hai xe ui kp nhau sm hn l : 4: 12 = 31h = 20 ' Thi gian hai xe ui kp nhau theo d nh: 20 . 3 = 60 ' = 1h Tho gian xe hai cn ui kp xe mt trn c qung ng : 1 . 2 = 2h Qung ng xe I i trc l: 16 : 2 = 34 h = 1h 20' Thi gian hai xe gp nhau theo d nh: 8 h + 1h 20' +2h = 11h 20' Do hai xe trn thc t gp nhau sm hn d nh 20'Hai xe gp nhau lc 11h 20' - 20' = 11h S 34 thi chn hc sinh gii lp 6 chuyn ton ( Qun Ba nh - Nm hc 1993-1994) Bi1: ( 4 im ) Cho Tnh t s BA Bi 2: ( 4 im )Tm cc ch s a,b sao cho sb a4 7chia ht cho 4 v chia ht cho 7. Bi 3 : ( 4 im ) Lc 8 gi mt ngi i t A dn B vi vn tc 25 km/h. Khi cn cch B20km ngi y tng vn tc ln 30 km/h. Sau khi lm vic B trong 30 pht, ri quay tr v A vi vn tc khng i 30 km/h v n Alc 12 gi 2 pht. Tnh chiu di qung ng AB. Bi 4: ( 4 im )Trn tia Ax ta ly cc im B, C, Dsao cho AB = 5cm; AC = 1cm; AD = 3 cm. Chng minh rngim Dnmgiahai im C v B. Trn on thngABly im M sao choCM = 3 cm. Chng minh rng im C nm gia hai im A v M Bi5: ( 4 im ) Tm phn s ba tho mn iu kin:3274< > < < < > > b',a bb'= 1 Nu d= 72 a' + b' =7 c bng a'54 b'23 A360144 B288216 Nu d= 84a' + b' =6 khng c gi tr ca a' v b' Nu d= 126a' + b' =4 khng c gi tr ca a' v b' Bi 5. Cminh21995 < 5 863 C : 210 =1024, 55 =3025 210 . 3 (211)24 > (211). 26 = 2270 21720.2270 < 21720 . 3172 < 5860 Vy21990 1 29(8q+1) < 23(9q+5) 25q < 86 q < 4 q e {2; 3} q = 2 1723=baq = 3 2532=ba S 41 ( Qun Ba nh -Nm hc 1990-1991) Cu 1: (6 im) Thc hin dy tnh 1021216361513 25 , 18 74921:60033415 , 06539:75354212 + |.|

\| ++Cu 2:(5 im) Tm 2 s t nhina, b,tho mn:a + 2b = 48v(a, b) + 3[ a, b] = 114Cu 3 : (4 im)a,Cho 3 im A, B, C, thng hngv AB + BC = AC. im no nm gia 2 im cn li? Ti sao? b, Cho gc aOb v tia Oc nm gia 2 tia Oa v Ob. Od l tia i ca tia Oc. Chng minh rng:- Tia Od khng nm gia 2 tia Oa v Ob. - Tia Ob khng nm gia 2 tia Oa v Od. Cu4: (6 im) ChoBAs t nh57 . 231143 . 23343 . 19531 . 19757 . 10741 . 10941 . 767 . 314T BA+ + + =+ + + = P N Bi 1:28611022556:25118717216125134118 4973:200112008353:251187+ = + |.|

\| ++= 17 . 8 . 7 . 3 . 215247102 . 56286118756 . 2861 . 25102 . 25 . 187102 . 252861 . 56:251872= + = + = + Bi 2: a+2b = 48 v (a,b) + 3 [a,b] = 114 114 . 3 ;3[a,b] . 3 (a,b) . 3 v a + 2b = 48 a . 2 a . 6 a e { 0, 6, 12, 18, 24, 30, 36, 42} a6121824303642 Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh b2181512963 (a,b)316312363 [a,b]42369024903642 3[a,b]12610827072270108126 (a,b)+3[a,b]12911436084360114168 Bi 4:A = 57 . 3113057 . 418041 . 31505774191014163147157 . 10741 . 10941 . 767 . 314= + = |.|

\|+ + |.|

\|+ = + + +B=57 . 315257 . 432843 . 3124571143323143531719157 . 231143 . 23343 . 19531 . 197= + = |.|

\|+ + |.|

\|+ = + + + 52130=BA S 42 ( Trng THCS L Ngc Hn-Nm hc 1997-1998) Cu 1:a, Chodeg + abcchia ht cho 37. Chng minh rngdeg abcchia ht cho 11. b,Tm x bitx x x x 20 20 20 20chia ht cho 7 Cu 2 : Tm x: 96232923134912151212031125111051476161153: 6= -(((((

+- |.|

\| ++ -- xCu 3 : So snh: 1 19991 1999v 1 19991 19992009198920001999++=++= N MCu 4 :Tnh tng: 308 . 305414 . 11411 . 848 . 5430 . 29 . 28 . 2716 . 5 . 4 . 315 . 4 . 3 . 214 . 3 . 2 . 11+ - - - + + + =+ - - - + + + =BA Cu 5 :Mt ca hng bn trng trong mt s ngy. Ngy th nht bn 100 qu v 101 s cn li. Ngy th hai bn 20 qu v 101 s cn li. Ngy th nht bn 300 qu v 101 s cn li. C bn nh vy th va ht s trng v s trng bn mingy u bng nhau. Tnh tng s trng bn v s ngy ca hng bn. P N Bi 1: Khng chng minh c iu ny v:Xt :37 592 465 127 deg . = + = + abc 127465 deg = abc. 11 b)1001001 . 20 20 20 20 20 x x x x x =7 20 . x (200 + x ) . 7 (4 + x ) . 7 x = 3 Bi 2: 329.71115711425651109534912.603511571110.52176.161735: 6+= + =201619552249504509329.719911.56509= = 391121239190319552016.962159621520161955= = = = x xBi 3:Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh 19991999 + 1 > 19991989 + 1 19992000+ 1 < 19992009 + 1 1 19991 19991 19991 19992009198920001999++>=+ Bi 4: ((

+ + ++ +=((

+ + ++ + ++=+ + + +=+ + +=+ + +) 3 )( 2 )( 1 (1) 2 )( 1 (131) 3 )( 2 )( 1 ( ) 3 )( 2 )( 1 (331) 3 )( 2 )( 1 ( 33) 3 )( 2 )( 1 ( 33) 3 )( 2 )( 1 (1n n n n n n n n n nnn n n nnn n n nn nn n n n n n n n A = 30 . 29 . 28 . 271..... ..........6 . 5 . 4 . 315 . 4 . 3 . 214 . 3 . 2 . 11+ + + += 812045130 . 29 . 284059.3130 . 29 . 2813 . 2 . 1131= =((

B = 485303308 . 5 . 3303 . 4308151343081305134.......1118134815134= = |.|

\| = |.|

\|+ + + |.|

\| + |.|

\|Bi 5: Ngy th nht bn 100 qu v 101 s trng cn li . Ngy th hai bn 200 qu v 101 s trng cn li m s trng hai ngy bn nh nhau 101 s trng cn li sau khi ly 100 qu nhiu hn 101 s trng cn li sau khi ly 200 qu l 100 qu . C nh vy s trng chnh lch trc khi ly 101 s trng cn li sau mi ln ly l 1000 qu. Ln cui cng cn 109 s trng cn li l 900 qu ngy th nht ly 900 qu trng S trng l (900 - 100) : 101 + 100 = 8100 (qu) S ln ly trng l 8100 : 900 = 9 (ln) S 43 Cu 1: (3 im) Tm cc ch sa, b sao chob a96 12chia ht cho 63. Cu 2 : (6 im) Thc hin dy tnh|.|

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\|+ +241283: 25 , 07375 , 13221132333 :15334454514141572 Cu 3 : (4 im)Tm s t nhin c 4 ch s m khi ta em s y nhn vi 5 ri cng thm 6 ta c kt qu l s c4ch s vit bi cc ch s nh s ban u nhng vittheo th t ngc li Cu 4 : (4 im) Trn tia Ox ly cc im A, B, C, D sao cho OA=1cm, OB = 5 cm,AC= 3 cm, BD=6cm. a, Chng minh rng im C nm gia 2 im A v B. b, Tnh di on thng CD. Cu 5 : (3 im) Cho 7 s t nhin tu . Chng minh rng bao gi ta cng c th chn c 4 s m tng ca chngchia ht cho 4. ( Hng dn: Trc ht nhn xt rng trong 3 s t nhin tu bao gi cng c t nht 2 s cng chn hoc cng l) P N Bi 1: 63 : 96 12 b aging bi 1 s 5 Bi 2: Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh ( )( )( )( ) 214:3215 , 25 75 , 28 : 84 , 64 81 , 33 06 , 342 , 1 8 , 0 . 5 , 21 , 0 2 , 0 : 3: 26 +((

++50015515014511009211310291923- - - + + + - - - 540776 . 2511 . 28.725116 ..2825417223241283:417347381132372:9245141572= ==|.|

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\| |.|

\|+ + Bi 3:Gi s labcdabcd.5 + 6 =abcd a < 2 a = 1 d > 5 bcd 1 .5 +6 =1 dcb d l s l d {5,7,9} d = 5 1 5 6 5 . 5 1 cb bc = + 5000 + 500b + 50c + 31 = 5000 + 100c + 10b + 1 c =Nbbbe+ =5196 495196 49 { } 9 , 4 4 55196 4e + = eb q b Nb b = 4 c = 0 b = 9 c = 51 Loi Nu d = 9 c = 9b +{ } e 5 ; 05395 4bb loi S l 1407 Bi 5:Gi 7 s l a1; a2; ........a7 Trong 3 s t nhin tu bao gi cng c 2 s cng chn hoc cng l Tng ca chng l mt s chn. Xt a1, a2, a3 : Khng mt tnh tng qut gi s a1,2 = a1+ a2 l s chn Xt a4, a5, a6 a4,5 l s chn Xt a3, a6, a7 a3,6 l s chn Xt a1,2;a4,5 ;a3,6l s chn ta chia s ny cho 2 b1,2 ; b4,5 ; b3,6 b1,2,4,5 = b1,2 + b4,5l s chn a1,2 +a4,5 = 2( b1,2 + b4,5 )v (b1,2 + b4,5) . 2 (a1,2 + a4,5 ) . 4 (a1 + a2 + a4 + a5 ) . 4 Vy trong 7 s t nhin tu bao gi c th chn c s m tng ca chng . 4 S 44 ( Trng THCS L Ngc Hn-Nm hc 1997-1998) Bi 1Tnh a,b, Bi 2Tm s t nhin nh nht chia cho 5 th d 1, chia cho 7 th d 5. Bi 3Hait i t hai im A v B v pha nhau. Xe 1 khi hnh lc 7 gi, xe 2 khi hnh lc 7gi 10pht. Bit rng i c qung ng AB xe 1 cn i 2 gi, xe 2 cn i 3 gi. Hai xe s gp nhau lc my gi?Bi 4V tam gic ABC trn cnh BC ly im D ( D khng trng B, C), trn on thng DC ly im E (E khng trng D, C). a, Nhng im no gi l im nm gia hai im no?Nhng tia nonm gia hai tia no? b, Nu BD=3cm, DE=2cm, EC=4cm. Tnh BC c, Gi s gc BAD=m0, gc DAE = n0, gc EAC= t0. Tnh s o gc BAC Bi 5 Tng kt nm hc ca 100 hc sinh gii v 3 mn Vn, Ton , Ngoi ng c 70 hc sinh gii Ton, 50 gii Vn. Trong 40 hc sinh gii Ton+ Ngoi ng, 35 hc sinh gii Ton+ Vn, 20 Hc sinh gii Vn+ Ngoi ng. Hi : a, C bao nhiu hc sinh gii c 3 mn. b, C bao nhiu hc sinh gii Ngoi ng. Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh c, C bao nhiu hc sinh ch gii 1 mn P N Bi 1 a. 26: 2335652331221233233 988123338. 26 1190256 : 26 19 , 125 , 0530= =+= + = |.|

\|+ + = + |.|

\|+ b.4051: 81001.....10191511008108985001.......501451100921 .....1021911= =|.|

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

\| + + |.|

\| + |.|

\| Bi 2: Gi s l n n = 5q + 1 ; n = 7r + 5 q= 54 7 + r (2r + 4). 5 r = 3k + 3 Tm s nh nht r = 3 q = 5 n = 26 Bi 3: Chn qung ng AB lm n v qui cTrong 1 h xe 1 i c 21qung ng AB Trong 1hxe 2 i c 31 qung ng AB Trong 1h c 2 xe i c 65 qung ng AB Trong 10 pht i trc xe 1 i 121 qung ng AB Thi gian xe 2 i gp nhauh101165:1211== 16 pht Hai xe gp nhau lc 7h 10ph + 1h 6 ph = 8h 16ph S 45 Qun Hai B Trng 1996 - 1997 Cu 1:( 5 im) Chng minh rng cc s cdng abcabcchia ht t nht cho 3 s nguyn t. Cu 2 : ( 5 im) Cho dy phn s c vit theo qui lut:- - - ;26 . 212;21 . 162;16 . 112 a,Tm phn s th 45 ca dy s ny. b, Tnh tng ca 45 phn s ny. Cu 3 : ( 5 im) Hai trng A v B c 1500 hc sinh. S hc sinh gii trng A chim 20%; S hc sinh gii trng B chim 15%. Tng cng hai trng c 255 hc sinh gii. Tnh s hc sinh mi trng? Cu 4 : Mt ngi i t A n B vi vn tc 12km /h. Mt lt sau mt ngi khc cng i t A n B vi vn tc 20km /h.Tnh ra 2 ngis gp nhau ti B. Ngi th 2 i c na qung ng AB th tng vn tc ln thnh 24km /h. Hi hai ngi s gp nhau ti a im cch B bao nhiu km? Bit rng qung ng AB di 80km. P N Cu 1: Vy s chia ht cho t nht 3 s nguyn t l 7 , 11, 13 13 ; 11 ; 7 13 . 11 . 7 1001 1000 . . abcabc abc abc abc abc abcabc = = + =Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Cu 2: Cu 3: 20% s hc sinh c hai trng l: 1500 . 20% = 300(hc sinh) 5% s hc sinh trng B l: 300 - 255 = 45 (hc sinh) S hc sinh trng B l: 45 : 5% = 900 (hc sinh) S hc sinh trng A l : 1500 - 900 = 600 (hc sinh) Cu 4: Hiu vn tc ca hai ngi l: 20 - 12 = 8 (km/h) Thi gian ngi th nht i ht qung ng AB l: 80: 12 =320h = 6h40' Thi gian ngi th hai i ht qung ng AB l: 80: 20 = 4 (h) Thi gian ngi th hai i trc ngi th nht l: 6h40' - 4h = 2h40'=38 h Qung ng ngi th nht i trc l: 38 . 12 = 32 (km) Khong cch gia hai ngi khi ngi th hai tng vn tc l: 32 - 8. 2 = 16 (km) Thi gian t khi ngi th hai tng vn tc n lc gp nhau l: 16: (24 -12)= 34h n lc gp ngi th hai i qung ng l: 40 + 24 . 34 = 72 (km) Ch gp cch B l: 80 - 72 = 8 (km) S 46 Qun Hai B Trng 1997 - 1998 Cu 1 ( 6 im)Tsu ch s: 0, 1, 2, 3, 4, 5. Vit tt c cc s c ba ch s khc nhau chia ht cho 3 vcho 5. Cu 2 : ( 6 im)Mt php chia c thng bng 5 v s d l 12. Nu ly s b chia chia cho tng s chia v s d ta c thng l 3 v s d l 18. Tm s b chia. Cu 3 : ( 4 im) Tnh cc tng sau bng cch hp l nht:a,3061272124012101+ + + b, 30695272129240161210191+ + +Cu 4 : ( 4 im) Lp 6A c s hc sinh Gii v Kh chim 127 s hc sinh c lp. S hc sinh Gii v Trung bnh chim 85s hc sinh c lp. S hc sinhKh v Trung bnh c 34 bn, s hc sinh gii hn s hc sinh Yu l 10 bn, lp khng c hc sinh km. Hi lp 6A c bao nhiu bn hc sinh Gii? bao nhiu hc sinh kh? bao nhiu hc sinh Trung bnh? P N Cu 1: 120; 150; 210; 510; 450; 540; 345; 105; 435; 405; 315; 135 Cu 2: Gi s b chia l a; s chia l b (b = 0) Php chia c thng bng 5 s d l 12 S b chia bt 12 bng 5 ln s chia a = 5b+12 ))12984523612311.......21116116111152236 . 2315.......21 . 16516 . 11552236 . 2312= |.|

\| + + + =||.|

\|+ + + = baGenerated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh S b chia chia cho tng s chia v s dc thng l 3 v s d l 18 S b chia bt 18bng 3 ln tng s chia v s d a = (b +12). 3 + 18 = 3b + 54 5b + 12 = 3b + 54 b = 21 a = 117 Vy s b chia l 117. Cu 3: b ) Nhn xt cc phn s u c tng ca t v mu l 401 Cu 4: Cch 1 Phn s ch s hc sinh gii hn yu l: 245185127= |.|

\|+(hc sinh c lp) S hc sinh c lp l:48524. 10 =(hc sinh) S hc sinh gii v yu l: 48 - 34 = 14 (hc sinh) S hc sinhgii l: ( 14 + 10 ) : 2 = 12 (hc sinh) S hc sinh yu l: 12 - 10 = 2(hc sinh) S hc sinh gii v trung bnh l:3085. 48 =(hc sinh) S hc sinh trung bnh l: 30 - 12 = 18 (hc sinh) S hc sinh kh l: 48 - (18 + 2 + 14) = 16 (hc sinh) Cch 2 Lp chia 24 phn mt phn c: 10 : 5 = 2 (hc sinh) S hc sinh trung bnh hn kh l: 24112785= (hc sinh lp) = 2 (hc sinh) S hc sinh trung bnh l: (34 + 2): 2 = 18 (hc sinh) S hc sinh kh l: 18 - 2 = 16 (hc sinh) S hc sinh gii v yu l: 48 - (18 + 16) = 14 (hc sinh) S hc sinh gii l: (14 + 10): 2 = 12 (hc sinh) S hc sinh yu l: 12 - 10 = 2 (hc sinh) S 47 Qun Hai B Trng 1998 - 1999 Cu 1 : Mtngi em 6000000 gi tin titkim " Khng k hn" vi li xut 0,8% mt thng. Hi sau 3 thng ngi thu c bao nhiu tin li ( sau 3 thng mi rt ht c vn ln li) Cu 2 :Mt x nghip lm mt s dng c, giao cho 3 phn xng thc hin. S dng c phn xng I lm bng 28% tng s. S dng c phn xng II lm gp ri s dng c phn xng I. Phn xng III lm t hn phn xng II l 72 chic. Tnh s dng c mi phn xng lm. Cu 3 :Hy vit phn s 1511 di dng tng ca 3 phn s c t s u bng 1 v c mu s khc nhau. Cu 4 :a, Tm mt s c 3 ch s bit rng tch ca s v tng cc ch s ca n l 1360. b, Chng t rng c th tm c nhiu s t nhin ch gm ch s 1 v ch s 0 chia ht cho 1999 P N )63118114118 . 17117 . 16116 . 15115 . 1413061272124012101= = + + + = + + + a63232 4631. 4013061272124012101. 401 4306951306401;2721291272401;2402611240401;210911210401= = |.|

\|+ + + = + + = + = + = + =B BGenerated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Cu 1: S tin ngi c sau thng 1 l: 6000000 . 100,8% = 6048000 (ng)S tin ngi c sau thng 2 l: 6048000 . 100,8% = 6096384 (ng)S tin ngi c sau thng 3 l: 6096384 . 100,8% = 6145155 (ng)Cu 2: So vi tng s, s dng c phn xng 2 lm chim s phn l: % 4223. % 28 =So vi tng s, s dng c phn xng 3 lm chim s phn l: 100% - (42 %+ 28%)= 30%_ So vi tng s, 72 chic chim s phn l: 42% - 30 % = 12% Tng s sn phm c ba phn xng lm l: 72 : 12% = 600 (dng c) S sn phm phn xng 1 lm l: 600 . 28% = 168 (dng c) S sn phm phn xng 2 lm l: 600 . 42% = 252 (dng c) S sn phm phn xng 3 lm l: 600 . 30% = 180 (dng c) Cu 3: Cu 4:a.Gi s l( ) 1360 . = + + abc c b a abc1360 = 5 . 16. 17 = 2 . 2 . 2 . 2 .5 . 17. Ta c 24. 5 < 10017 khng phi l tng cc ch s5 . 17 17 > = x x abc abc. a + b + c < 16 a + b + c24810 abc680340170136 Tch1360136013601360 Vy s l: 680 ; 340; 170; 136 b.Xt dy s: Dy s trn c 1999 ch s ch chai trng hp xy ra -C t nht mt s chia ht cho 1999.G s s l:11...11 (n ch s) 11....10 (n+1 ch s) cng chia ht cho 1999 Khng nh bi cho l ng. -Trong khng c s no chia ht cho 1999 phi tn ti t nht hai s c cng s d trong php chia cho 1999 Hiu hai s ny l mt s ch gm ton ch s 0 v ch s 1 chia ht cho 1999. L lun tng t nh trn ta c khng nh bi cho l ng. S 48 Hai B Trng 1999 - 2000 Cu 1 : Hy so snh hai phn s sau bng tt c cc cchc th c: a, 2000200019992000;20001999b,23214131> + - - - + +Cu 2 :Kt thc hc k I lp 7A c s hc sinh xp loi vn ho bng 83 s hc sinh c xp loi kh. n cui nm c 7 hc sinh vn ln t loi gii v 1 hc sinh loi gii b chuyn loi xung kh nn s hc sinh gii ch bng139 s hc sinh kh. Tnh s hc sinh lp 7A bit c hai hc k lp 7A ch c hc sinh xp loi vn ho Kh v Gii. Cu 3 :Mt thng y nc c khi lng 5,7 kg. Nu trong thng ch cn 25% nc th thng nc c khi lng 2,4 kg. Tnh khi lng thng khng. { }1512161151160460306010604444 4 10 3060 ; 30 ; 20 ; 15 12 ; 10 ; 6 ; 5 ; 4 ; 3 ; 2 ; 1 ) 60 (60441511+ + = + + = = + += = U _ cs 199911 ... 11 ; .......... ; 11 ; 1Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Cu 4 :C bao nhiu s c 4 ch s c tnh cht sau: Chia ht cho 11 v tng cc ch s ca n chia ht cho 11. P NBi 1: a)Cch 1 :Qui ng mu s ri so snh t. Cch 2: 2000200019992000200020001999199920001999< =Cch 3: 2000200019992000200019991200020001000020002000199920002000120001999< = + = + b) Bi 2: S hc sinh c lp l : 3 + 8 = 11 (phn) S hc sinh gii k I chim : 113hc sinh c lp S hc sinh gii k II chim : 229hc sinh c lp 6 hc sinh ng vi s phn c lp: 223113229= (c lp) S hc sinh c lp l: 6 :44223= hc sinh Vy s hc sinh 7A l 44 bn Bi 3:25% = 41 Khi lng ca 43nc trong thng l: 5,7 - 2,4= 3,3 (kg) Khi lng nc trong thng y nc l 3,3 : 43= 4,4 (kg) Khi lng thng khng l : 5,7 - 4,4 = 1,3 (kg) Bi 4:S phi tm l:A = abcd(0 + + + > + + + > + + + > + + + > e >= +n N nn n nnn nGenerated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh b)Tnh B = 14: (1 54 212 8 ) + 14. 1 24 3 Cu 2: (4)Tm x bit: a/3 + 2x -1 = 24 [42 (22 - 1)] b/ (x+1) + (x+2) + (x+3) + ...+ (x+100) = 205550 c/ 5 x = 18 + 2.(-8) d/(3x 24 ) .75 = 2.76.012009 Cu 3: (2) Tm cc s t nhin x, ysao cho :(2x+1)(y-5)=12 Cu 4: (4) a) Tnh tng: S= 2 2 2 2 2.......1.2 2.3 3.4 98.99 99.100+ + + + + b) Chng minh rng: ( )2 3 4 1003 3 3 3 ..... 3 40 + + + + + .Cu 5: (2)Cho biu thcA = 52 n a, Tm cc s nguyn n biu thc A l phn s. b, Tm cc s t nhin n biu thc A l s nguyn Cu 6: (4) Cho gc AMC = 600. Tia Mx l tia i ca tia MA, My l phn gic ca gcCMx, Mt l tia phn gic ca gc xMy. c.Tnh gc AMy. d.Chng minh rng MC vung gc vi Mt. P N Cu 1: (4) Mi cu 2 a/ Kt qu 18 b/Kt qu 141115 Cu 2: (4) a) 3 + 2x-1 = 24 [42 (22 - 1)] 3 + 2x-1 = 24 42 + 3 2x-1 = 24 42 2x-1 = 22 (0,5) x -1 = 2 x= 3 (0,5) b) ( x+1)+ (x+2)+ (x+3)+ ...+ (x+100)=205550 x+x+x+...+x+1+2+3+...+100=205550 100x+5050=205550 (0,5) 100x=200500 x=2005(0,5) c/ x=7 hoc x=3; (1 mi nghim 0,5 ) d/ x=30 (1) Cu 3: (2) Ta c 2x+1; y-5 L c ca 12 12= 1.12=2.6=3.4 (0,5) do 2x+1 l => 2x+1 =1 hoc 2x+1=3(0,5) 2x+1=1=> x=0; y-5=12 => y=17hoc 2x+1=3=> x=1; y-5=4=>y=9(0,5) vy (x,y) = (0,17); (1,9)(0,5) Cu 4: (4)S =2 2 2 2 2.......1.2 2.3 3.4 98.99 99.100+ + + + + Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh =2( 1 1 1 1 1.......1.2 2.3 3.4 98.99 99.100+ + + + +)(0,5) = 2 (1 1 1 1 1 1 1 1 1 1...1 2 2 3 3 4 98 99 99 100 + + + + + ) (0,5) = 2(1 11 100 ) = 2.99100 =99 49150 50= (1) Cu 5: (2)a/ neZ v n= 2(1) b/(n - 2 ) e( -5) = { } 1; 5 ( 0,5 ) 2 1 12 1 32 5 32 5 7n n Nn n Nn n Nn n N = = e = = e = = e = = e (0,5 ) Vy n = 1;3;7 Cu 6: (4) Hnh v:(0,5) a) Tia Mx l tia i ca tia MA gc AMx l gc bt:

0180 AMx = => MC nm gia MA v Mx(0,5) nn:

AMC CMx AMx + =thay s:

0 060 180 CMx + ==>

0 0 0180 60 120 CMx = = (0,5) My l tia phn gic ca gc CMx nn: My nm gia MC v Mx v 0 01 1120 602 2xMy yMC xMC = = = =(0,5) Tia Mx l tia i ca tia MA gc AMx l gc bt:

0180 AMx = => My nm gia MA v Mx(0,5) nn:

AMy yMx AMx + =thay s: 0 060 180 yMx + ==> 0 0 0180 60 120 yMx = =(0,5) b) Do My l tia phn gic ca gc CMx nn Mx v MC nm trn hai na mt phng i nhau c b l tia My. Mt l phn gic ca gc yMx nn Mt nm trn cng mt na mt phng b cha tia My. Vy Mt v MC nm trn hai na mt phng i nhau c b cha tia My hay My nm gia MC v Mt nn:

CMy yMt CMt + = (*) (0,5) 600 A M C x y t Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Li c tia Mt l phn gic ca gc xMy nn:

0 01 160 302 2xMt tMy xMy = = = = thay s vo (*) ta c:

0 0 060 30 90 CMt = + =hay MCvung gc vi Mt. (ccm) (0,5) S 50 Bi 1(2 im). Mt dy s cng c 45 s hng. Bit s hng chnh gia l 50. Hy xc nh dy s cng. Bi 2:(2 im). Cho S = 5 + 52 + 53 + + 52006 a. Tnh S b. Chng minhS .126 Bi 3:(2 im). a.Chng minh rng : nu ( )11 ab cd eg + + .th :deg 11 abc . . b.Cho A = 2 3 602 2 2 ... 2 . + + + +Chng minh : A.3 ; 7 ; 15. Bi 4(2 im). Chng minh : 2 3 41 1 1 1...2 2 2 2n+ + + +< 1. Bi 5 (2 im). a.Cho on thng AB = 8cm. im C thuc ng thng AB sao cho BC = 4cm. Tnh di on thng AC. b.HaionthngABvCDkhngcngnmtrnmtngthng.Chngcthcmyimchung?V sao? P N Bi 1(2 im). Trc s hng chnh gia c 22 s hng, sau s hng chnh gia c 22 s hng. *Nu cng sai d=1 th u1=50-22=28(0,5) u45=50+22=72 Dy s l 28, 29, 30,...50,...71, 72. (0,5) *Nu cng sai d=2 th u1=50-22.2=6 u45=50+22.2=94 Dy s l 6, 8, 10,...50,...92, 94. (0,5) D thy cng sai d khng th ln hn 2.(0,5) Bi 2:(2 im). (0,5) a. (1.5) Ta c 5S =5(5 + 52 + 53 + + 52006) 5S = 52 + 53 +54 ++52007(0,5) 5S S = (52 + 53 +54 ++52007) (5 + 52 + 53 + + 52006) (0,5) 4S = 52007-5 Vy S = 20075 54 (0,5) b. (0,5) S = (5 + 54) + (52 + 55)+(53 + 56) +.. + (52003 +52006) S = 5(1+53)+52(1+53) +53(1+53)++ 52003(1+53)(0,25) S = 126.(5 + 52 + 53 ++ 52003) V 126 . 126 S . 126(0,25) Bi 3:(2 im).a.1 Tch nh sau : ( ) ( )deg 10000 100 9999 99 abc ab cd eg ab cd ab cd eg = + + = + + + + . (0,5) = ) 101 ( 99 cd ab + +( ) eg cd ab + + ( )9999 99 11 ab cd + .Theo bi ra ( )11 ab cd eg + + . nn :deg 11. abc .(pcm)(0,5) b. (1) *A= ) 2 2 ...( ) 2 2 ( ) 2 2 (60 59 4 3 2+ + + + += ) 2 1 ( 2 ... ) 2 1 ( 2 ) 2 1 ( 259 3+ + + + +Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh =( )3 593 2 2 ... 2 3. + + + .(0,5) *A = ( ) ( ) ( )2 3 4 5 6 58 59 602 2 2 2 2 2 ... 2 2 2 + + + + + + + + += =( ) ( ) ( )2 4 2 58 22. 1 2 2 2 . 1 2 2 ... 2 . 1 2 2 + + + + + + + + + = ( )4 587 2 2 ... 2 7 + + + . . (0,25) *A = ( ) ( ) ( )2 3 4 5 6 7 8 57 58 59 602 2 2 2 2 2 2 2 ... 2 2 2 2 + + + + + + + + + + + + = =( ) ( ) ( )2 3 5 2 3 57 2 32 1 2 2 2 2 1 2 2 2 ... 2 1 2 2 2 + + + + + + + + + + + + = =( )5 5715. 2 2 ... 2 15. + + + . (0,25) Bi 4:(2 im).Ta bit :( )21 1 1 1.1 1 n n n n n< = (0,5) Nn : 221 3(a(. 3, 33. 3=>5(b(.3 0,25 m (3, 5) =1 =>(b(.3 0,25 3(a(+ 5(b(= 33 =>5(b( 33 =>(b( 6,6 (2) 0,25 T (1), (2) v b nguyn => (b(e{0; 3; 6} 0,25 -Nu |b| =0 th 3(a(= 33=>(a(= 11 => a = 11; b = 0 0,25 Ta c cc cp (0; 11), (0; -11) -Nu |b| =3 th 3(a(= 33 15 =18 =>(a(= 6 => a = 6; b = 30,25 Ta c cc cp (6; 3), (6; -3), (-6; 3), (-6; -3) -Nu |b| = 6 th 3(a(= 33 30 =3 =>(a(= 1 => a = 1; b = 6 0,25 Ta c cc cp (1; 6), (1; -6), (-1; 6), (-1; -6) KL: Ta c cc cp (0; 11), (0; -11), (6; 3), (6; -3), (-6; 3), (-6; -3) (1; 6), (1; -6), (-1; 6), (-1; -6) tho mn bi. 0,25 b) Cho n l s t nhin, tm s nguyn t p c 2 ch s sao chop = CLN(2n - 3; 3n +15)(1im) v p = CLN(2n - 3; 3n +15)=> +..2n 3 p3n 15 p 0, 25 +..6n 9 p6n 30 p 0, 25 ( ) ( ) + . 6n 30 6n 9 p 0, 25 . 39 pdo p l s nguyn t c 2 ch s => p = 13 0, 25 c) Cho S = 1 + 5+ 52 + 53 +54 + + 52010 Tm cc s d khi chia S cho 2, cho10, cho 13. (2 im) -S gm 2011 s hng u l s l nn S l => S chia cho 2 d 1. 0, 5 -S gm 2010 s hng chia ht cho 5 v mt s hng chia cho 5 d 1 => S chia cho 5 d 1. 0, 25 => S c tn cng l 6 hoc 1 m S l nn S c tn cng l 1. Vy S chia cho 10 d 1. 0,25 Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh BI 2 : (Tip) S = 1 + 5+ 52 + 53 +54 + + 52010 S =1 + 5+ 52 +( 53 +54 + 55 +56) +( 57 +58 + 59 +510) + +( 52007 +52008 + 52009 +52010) 0,5 S =1 + 5+ 25 +53 (1 + 5+ 52 + 53) + 57 (1 + 5+ 52 + 53) + +52007 (1 + 5+ 52 + 53) 0,25 S =26 + 5 +53 .156 + 57 .156 + +52007 .156 Ta c 26 v 156 u chia ht cho 13 vy Schia cho 13 d 5. 0,25 BI 3 : (5 IM) a) Cho a, b, c, d l cc s t nhin khc 0 v biu thc:M = a b c da b c a b d a c d b c d+ + ++ + + + + + + +.Hi M c gi tr l s t nhin hay khng? V sao ? (2 im) V a, b, c, d e N* a+b+c >+ + + + +a aa b c a b c d 0,5 Tng t :>+ + + + +b ba b d a b c d >+ + + + +c ca c d a b c d >+ + + + +d db c d a b c d 0,25 M > + + +=+ + +a b c d1a b c d0,25 V a, b, c, d e N* a + b + c > a + b A c 104 im.0,25 Lp lun c 104.103/2 = 5356 on thng ni 2 trong 104 im ca A0,25 Ni 2 u ca mi on thng vi 1 im thuc 102 im cn li (khng phi l cc mt ca on thng ) c 102 tam gic. 0,25 Vy c 5356.102 tam gic. Nhng nh th th mi tam gic c tnh 3 ln, vy ta c5356.102 : 3 = 182104 tam gic. 0,25 Bi 5 :(1 im) Bi 5 :(1 im)Cho mt li vung kch thc 55. Ngi ta in vo mi ca li mt trong cc s -1; 0; 1. Xt tng ca cc s c tnh theo tng ct, theo tng hng v theo tng ng cho. Hy chng t rng trong tt c cc tng lun tn ti hai tng c gi tr bng nhau. V li vung c kch thc 55 thc 5 ct, 5 hng v 2 ng cho, do c tt c 12 tng. Do chn in vo cc s -1, 0 ,1 nn gi tr mi tng S l mt s nguyn tho mn : -5 S 5 .0,5 Vy c 11 s m 12 tng , theo nguyn tc i-rch-l tn ti hai tng c gi tr bng nhau. 0,5 Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh S 54 Olimpic huyn nm hc 2006 2007 (Thi gian lm bi 120 pht) Bi 1. Tm ch sx: a)137 +x 3chia ht cho 13. b)x 137 x 137chia ht cho 13.Bi 2. a) So snh phn s: 30115Vi49925 b) So snh tng S = 2 3 20071 2 3 2007... ...2 2 2 2 2nn+ + + + + +vi 2. ( ne N*) Bi 3. Vi gi tr no ca s t nhin a th: a)1 a 419 a 8++ c gi tr nguynb) 23 a 417 a 5 c gi tr ln nht. Bi 4. Tm ch s tn cng ca s 62006, 72007 Bi 5.Trong mt cuc thi c 50 cu hi. Mi cu tr li ng c 20 im, cn tr li sai b tr 15 im. Mt hc sinh c tt c 650 im. Hi bn tr li c my cu ng ? P N Bi 1.Tm ch sx: a)137 +x 3chia ht cho 13. A = 137 +x 3 = 137 + 30 + x = 12. 13 + (11 + x) => A.13 Khi 11 + x .13 V x l ch s t 0- > 9 => x = 2 b)x 137 x 137chia ht cho 13.

10001 . x 7 ) 10 10 .( 13x 7 10 . 13 10 . x 7 10 . 13 x 137 x 137 B2 62 4 6+ + =+ + + = = 10001 khng chia ht cho 13 => B.13 Khix 7 .13 => x = 8 Bi 2. a) So snh phn s: 30115Vi49925 49925500252013001530115< = = < . Vy 30115 2 ta c: n 1 n n22 n21 n2n ++=+. T ta c: S =2220092 )2200922008( ..... )2524( )2423(212007 2007 2006 3 2 2< = + + + + . Vy S a = 0, a = 4 b) 23 a 417 a 5 c gi tr ln nht.

5 17 20 68 5(4 23) 47 5 474 23 4(4 23) 4(4 23) 4 4(4 23)a a aa a a a += = = + Nh vy bi ton a v tm s t nhin a 4a 23 l s t nhin nh nht. Vy a = 6 => 23 a 417 a 5= 13 Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Bi 4. Tm ch s tn cng ca s 62006, 72007 Ta c: 62 = 36 6 (mod10), vy 6n 6 (mod10)s nguyn dng n => 62006 6 (mod10) => ch s tn cng ca62006l 6 74 = 2401 1 (mod10),m 72007 = 74.501.73

(74)501 1 (mod10) => ch s tn cng ca 72004 l 1, M ch s tn cng ca 73 l 3 => ch s tn cng ca 72007 l 3 Bi 5. Nu bn tr li c 50 cu th tng s im l 50 x 20 = 1.000 (im) Nhng bn ch c 650 im cn thiu 1.000 650 = 350 (im). Thiu 350 im v trong s 50 cu bn tr li sai mt s cu. Gia cu tr li ng v tr li sai chnh lch nhau 20 + 15 = 35(im). Do cu tr li sai ca bn l 350:35 =10 (cu) Vy s cu bn tr li ng l 50 10 = 40 (cu) Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh S 55 Bi 1: Hy chn Kt qu ng. Tm x bit rng: 61) 3 (1...11 . 818 . 51=++ + +x x a. x = 27c. x = 25 b. x = 35d. x = 205 Bi 2:Hy chn Kt qu ng. Gc xOy c hai tia phn gic khi: a. Gc xOy l gc bt. b. Gc xOy l gc t. c. Gc xOy l gc vung.d. Gc xOy l gcnhn. Bi 3: Hy chn Kt qu ng. Cho 2 s: x =222222222221;y =444445444443; ta c: a. x = yc. x < y b. x > y Bi 4:So snh gi tr ca biu thc: A = 000 . 109999...9843+ + + vi s 99. Bi 5: Mt ngi i xe p t A n B, i t A vi vn tc 10km/ h, nhng t chnh gia ng n B vi vn tc 15km/h. Tnh xem trn c qung ng ngi i vi vn tc trung bnh l bao nhiu. Bi 6: Tm cp s nguyn dng (x;y) sao cho (x- 1) (5y + 2) = 16. Bi 7: Xt hnh v bn: a. C nhng tam gic no c cnh NC. b.Cttcbaonhiugccnhl N; hy k ra. c.Nu bitgcZMPB=600,ZNPC = 500

thPNclphngiccagcMPC hay khng ? v sao? A MKN B P C P N Bi 1: 2 im Chn cu a:x = 27 Bi 2: 2 imChn cu a: Bi 3: 2 imChn cu c:x < y Bi 4: 2 imBin i: A =)1000011 ( ... )911 ( )411 ( + + + = )10011 ( ... )311 ( )211 (2 2 2 + + + = 99 -)1001...3121(2 2 2+ + += 99 - B Trong B =)1001...413121(2 2 2 2+ + + +V B > 0nn A < 99 0.5 0.5 0.5 0.5 Bi 5: 3 imTrnqungngABc2kmthc1kmivivntc10km/h(ht 1/10h); 1km i vi vn tc 15km/h (ht 1/15h)Nn c 2km ngi i ht: 1.0 1.0 0.5 K H I Generated by Foxit PDF Creator Foxit Softwarehttp://www.foxitsoftware.com For evaluation only.Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh 61151101= + (h) Vy vn tc trung bnh ca ngi l: 2 : 1/6 = 12km/h 0.5 Bi 6: 3 imV x,y nguyn dng nn x - 1 l uc ca 16 M (16) = {1;2;4;8;16} 0.5 1.0 Ta c:x -1 = 1 x -1 = 2 x -1 = 4 x -1 = 8 x -1 = 16 x = 2 x = 3 x = 5 x = 9 x = 17 Thay ln lt cc gi tr ca x va tm c vo (x - 1) (5y + 2) = 16 x = 2 ta c: 5y + 2 = 16 y = 14/5 loi x = 3 ta c: 2 (5y + 2) = 16 y = 6/5 loi x = 5 ta c: 4(5y + 2) = 16 y = 2/5 loi x = 9 ta c: 8 (5y + 2) = 16 y = 0 x = 17 ta c: 16 (5y + 2) = 16 y = - 1/5 loi Kt lun: Cp s nguyn dng cn tm l(9;0) 1.0 0.5 Bi 7:6 im a. Nhng tam gic c cnh NC: A NCI; A NCP; A NCK; ANCB. b. Nhng gc c nh l N: ZANC, ZANB, ZANP ZBNP, ZBNC, ZPNC c. Ta c tia PM v PN nm gia hai tia PB v PCNn ZBPM + ZMPN + ZNPC = ZBPC = 1800 M ZBPM = 600 ;ZMPC = 500 Suy ra: ZMPN = 1800 - 600 - 500 = 700 Ta thy: ZMPN = ZNPC Nn PN khng phi l phn gic ca gc MPC. 2.0 2.0 0.5 0.5 1.0 S 56 Hy khoanh trn ch a, b, c hoc d nu l cu ng. Bi 1: Cho 2 s nguyn m v n:a. m + n = ,m, + ,n, vi mi m v n. b. m + n = ,m, + ,n, vi mi m v n cng du. c. m + n = ,m, + ,n, vi mi m v n tri du. d. m +