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Huỳnh Minh Khai: Gv: THCS thị Trấn cầu kè, Trà Vinh TUYỂN TẬP CÁC ĐỀ THI HS GIỎI MÔN TOÁN LỚP 6 (CÓ ĐÁP ÁN ĐẦY ĐỦ) Generated by Foxit PDF Creator © Foxit Software http://www.foxitsoftware.com For evaluation only.

Tuyển tập 60 đề thi học sinh giỏi toán 6

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Text of Tuyển tập 60 đề thi học sinh giỏi toán 6

  • 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 )

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  • 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 122

    1223

    23

    aaa

    aaA

    a, Rt gn biu thc b, Chng minh rng nu 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 s abc sao cho 12 nabc v 2)2( ncba

    Cu 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 N* Hy so snh nb

    na

    v

    b

    a

    b. Cho A = 110

    11012

    11

    ; B =

    110

    11011

    10

    . 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 s hoc 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 N

    Cu 1: Ta c: 122

    1223

    23

    aaa

    aaA =

    1

    1

    )1)(1(

    )1)(1(2

    2

    2

    2

    aa

    aa

    aaa

    aaa

    iu kin ng a -1 ( 0,25 im). Rt gn ng cho 0,75 im. b.Gi d l c chung ln nht ca a2 + 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 + 1 v 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 n31 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 ( a 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 nn khng 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 im

    Ta xt 3 trng hp 1ba

    1ba

    1ba

    (0,5 im).

    TH1: 1ba

    a=b th nbna

    th nbna

    = ba

    =1. (0 , v ,5 im).

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  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh

    TH1: 1ba

    a>b a+m > b+n.

    M nbna

    c phn tha so vi 1 l nbba

    ba

    c phn tha so vi 1 l bba

    , v nbba

    < bba

    nn nbna

    < ba

    (0,25 im).

    TH3: ba

    x=0; y-5=12 => y=17

    hoc 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)

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  • 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 c

    5(12n+1)-2(30n+2)=1 chia ht cho d (0,5) vy d=1 nn 12n+1 v 30n+2 nguyn t cng nhau

    do 230

    112

    n

    nl phn s ti gin (0,5)

    b. Ta c 22

    1 x = 2 c). 52x-3 2.52 = 52.3 52x: 53 = 52.3 + 2.52 52x: 53 = 52.5 52x = 52.5.53 52x = 56 => 2x = 6 => x=3

    Bi 2. V a l mt s t nhin vi mi a Z nn t a < 5 ta

    => a = {0,1,2,3,4}.

    Ngha l a ={0,1,-1,2,-2,3,-3,4,-4}. Biu din trn trc s ccc s ny u ln hn -5 v nh hn 5 do -5

  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Cu 5. Bn im A,B,C,Dkhng nm trn ng thng a . Chng t rng ng thng a hoc khng ct, hoc ct ba, hoc ct bn on thng AB, AC, AD, BC, BD, CD.

    P N

    Cu 1. a). 2A = 8 + 2 3 + 2 4 + . . . + 2 21.

    => 2A A = 2 21 +8 ( 4 + 2 2 ) + (2 3 2 3) +. . . + (2 20 2 20). = 2 21.

    b). (x + 1) + ( x + 2 ) + . . . . . . . . + (x + 100) = 5750

    => 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 5050 100 x = 700

    x = 7

    Cu 2. a) egcdababc 10010000deg = 9999 cdab 99 + egcdab 11. b). 10 28 + 8 9.8 ta 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) 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: 11

    9:

    7

    6 =

    22

    21 (s th hai)

    S th ba bng: 11

    9:

    3

    2 =

    22

    27 (s th hai)

    Tng ca 3 s bng 22

    272122 (s th hai) =

    22

    70(s th hai)

    S th hai l : 210 : 22

    70 = 66 ; s th nht l:

    22

    21. 66 = 63 ; s th 3 l:

    22

    27.66 = 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). Nu c 1 im ( Chng hn im A thuc na mt phng) ba im B, C, D thuc na mt phng i th 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 s 281 yx chia ht cho 36

    c) Tm s t nhin a bit 1960 v 2002 chia cho 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

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  • 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 bit gc BOC = 900 a) Tnh gc AOC b) Gi OD l tia i ca tia OC. So snh hai gc AOD v BOD

    HNG DN

    Bi 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) s 281 yx 36 ( 0 x, y 9 , x, y N )

    42

    9)281(

    y

    yx (0,5)

    9;7;5;3;142 yy (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 = 8

    132004 (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: a Ta 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) theo gi thit C nm trong gc AOB nn 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, D thng hng. Do gc DOA + gc AOC = 1800 (hai gc k b) => gc AOD = 1800 - gc AOC = 1800 - 450 => gc AOD = 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.

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  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh

    3 . Cho phn s b

    a (0 < a < b) cng thm m n v (m > 0) vo t v mu th phn s mi ln hn hay b

    hn b

    a?

    4. Cho s 16*4*710*155 c 12 ch s . chng minh rng nu thay cc du * bi cc chc s khc nhau trong ba ch s 1,2,3 mt cch tu th s lun chia ht cho 396. 5. chng minh rng:

    a) 3

    1

    64

    1

    32

    1

    16

    1

    8

    1

    4

    1

    2

    1 ; b)

    16

    3

    3

    100

    3

    99...

    3

    4

    3

    3

    3

    2

    3

    110099432

    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< a

    b) Xc nh im M trn tia Ox sao cho OM = 2

    1(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 xt 71999 Ta c: 71999 = (74)499.73 = 2041499. 343 Suy ra ch s tn cng bng 3 ( 0,25 im ) Vy s 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)

    mb

    ma

    b

    a

    4.(1 im ) Ta nhn thy , v tr ca cc ch s thay th ba du sao trong s trn u hng chn v v ba ch s i mt khc nhau, ly t tp hp 3;2;1 nn tng ca chng lun bng 1+2+3=6. Mt khc 396 = 4.9.11 trong 4;9;11 i mt nguyn t cng nhau nn ta cn chng minh

    A = 16*4*710*155 chia ht cho 4 ; 9 v 11. Tht vy : +A 4 v s to bi hai ch s tn cng ca A l 16 chia ht cho 4 ( 0,25 im ) + A 9 v tng cc ch s chia ht cho 9 : 1+5+5+7+1+4+1+6+(*+*+*)=30+6=36 chia ht cho 9 ( 0,25 im ) + A 11 v hiu s gia tng cc ch s hng chn v tng cc ch s hng l l 0, chia ht cho 11. {1+5+7+4+1)-(5+1+6+(*+*+*)}= 18-12-6=0 ( 0,25 im ) Vy A 396 5(4 im )

    a) (2 im ) t A= 65432 2

    1

    2

    1

    2

    1

    2

    1

    2

    1

    2

    1

    64

    1

    32

    1

    16

    1

    8

    1

    4

    1

    2

    1 (0,25 im )

    2A= 5432 2

    1

    2

    1

    2

    1

    2

    1

    2

    11 (0,5 im )

    2A+A =3A = 1- 12

    12

    2

    16

    6

    6

    (0,75 im )

    3A < 1 A < 3

    1 (0,5 im )

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  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh

    b) t A= 10099432 3

    100

    3

    99...

    3

    4

    3

    3

    3

    2

    3

    1 3A= 1-

    9998332 3

    100

    3

    99...

    3

    4

    3

    3

    3

    3

    3

    2

    (0,5 im )

    4A = 1-100999832 3

    100

    3

    1

    3

    1...

    3

    1

    3

    1

    3

    1 4A< 1-

    999832 3

    1

    3

    1...

    3

    1

    3

    1

    3

    1 (1) (0,5 im )

    t B= 1-999832 3

    1

    3

    1...

    3

    1

    3

    1

    3

    1 3B= 2+

    98972 3

    1

    3

    1...

    3

    1

    3

    1 (0,5 im )

    4B = B+3B= 3- 993

    1 < 3 B <

    4

    3 (2)

    T (1)v (2) 4A < B < 4

    3 A <

    16

    3 (0,5 im )

    Bi 2 ( 2 im ) a) (1 im )V OB

  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Cho 20 im, trong c a im thng hng. C 2 im, ta v mt ng thng. Tm a , bit v c tt c 170 ng thng.

    P N

    A. PHN S HC

    Cu 1: a, Ta thy; 9999

    2323

    101.99

    101.23

    99

    23

    999999

    232323

    10101.99

    10101.23

    99

    23

    99999999

    23232323

    1010101.99

    1010101.23

    99

    23

    Vy; 99999999

    23232323

    999999

    232323

    9999

    2323

    99

    23

    b, Ta phi chng minh , 2. x + 3 . y chia ht cho 17, th 9 . x + 5 . y chia ht cho 17 Ta c 4 (2x + 3y ) + ( 9x + 5y ) = 17x + 17y chia ht cho 17 Do vy ; 2x + 3y chia ht cho 17 4 ( 2x +3y ) chia ht cho 17 9x + 5y chia ht cho 17 Ngc li ; Ta c 4 ( 2x + 3y ) chia ht cho 17 m ( 4 ; 17 ) = 1 2x + 3y chia ht cho 17 Cu 2 ; Ta vit li A nh sau :

    A=

    1009.7.23).1009

    1.

    7

    1.

    23

    1

    1009

    1

    7

    1

    23

    1(

    1009.7.23).1009

    1

    7

    1

    23

    1(

    +

    11611009).723(

    1

    = 17.231009.231009.7

    7.231009.231009.7

    +

    17.231009.71009.23

    1

    = 1

    Cu 3; a, 2

    1 (

    10.9

    1...

    4.3

    1

    3.2

    1

    3.2

    1

    2.1

    1 ) . x =

    45

    23

    )90

    1

    2

    1.(

    2

    1 . x =

    45

    23 x = 2

    b, 43

    30 =

    4

    13

    12

    11

    1

    13

    42

    11

    1

    30

    131

    1

    30

    43

    1

    => a =1 ; b = 2 ; c = 3 ; d = 4

    Cu 4; Ta c

    88.135

    58.120

    2

    1

    qa

    qa (q1, q2 N )

    704.10808

    52210809

    2

    1

    qa

    qa

    T ( 2 ) , ta c 9 . a = 1080 . q2 + 704 + a ( 3 ) Kt hp ( 1 ) vi ( 2 ) , ta c a = 1080 . q 180 V a nh nht, cho nn, q phi nh nht => q = 1 => a = 898 B- PHN HNH HC Cu 1; Gi Ot , Ot, l 2tia phn gic ca 2

    k b gc xOy v yOz Gi s , xOy = a ; => yOz = 180 a

    Khi ; tOy = 2

    1a t,Oy =

    2

    1( 180 a)

    => tOt, = )180(2

    1

    2

    1aa = 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, gi s khng c 3 im no thng hng.S ng thng v 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

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  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh

    S 8

    Thi gian lm bi : 120 Bi 1 : (3 ) Ngi ta vit cc s 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 199 Lp 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 v gm : 104, 114194 (c 10 s ) (05) cc s c 4 ch s lm hng chc l 140,141,142,..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) . 1 2 3 4 5 6 7 8 9 10 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)

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  • 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 = 18567 c) 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 = 40 Nn 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 * Vi Cch 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)

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  • 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 + ... + 71999 Chng 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 1998

    1...........

    3

    1

    2

    11

    n

    m 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 002000200020

    991999199919A v phn s

    2000

    1999B

    So 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 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 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:

    1998

    1...........

    3

    1

    2

    11

    n

    m. T 1 n 1998 c 1998 s Nn v phi c 1998 s hng ta ghp

    thnh 999 cp nh sau:

    1000

    1

    999

    1...........

    1996

    1

    3

    1

    1997

    1

    2

    1

    1998

    11

    n

    m

    1000.999

    1999.......

    1996.3

    1999

    1997.2

    1999.

    1998.1

    1999

    Quy ng tt c 999 phaan s ny ta c:

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  • 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.........1999.1999.1999 999998997321 aaaaaa

    n

    m

    Vi a1 , a2 , a3 , ........... , a998 , a999 N

    1998.1997.1996..............................3.2.1

    )..........(1999 999998997321 aaaaaa

    n

    m

    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:

    2000200000002000000000

    1999199900001999000000

    002000200020

    991999199919

    A

    B

    2000

    1999

    100010001.2000

    100010001.1999

    )110000100000000(2000

    )110000100000000(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 ng H Ni - Ph L th xe t t H Ni v i c 25 Km. Vy 5 ln 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. 729.7239.162.54.18234.9.3

    27.81.243729.218122

    b. 100.99

    1

    99.98

    1

    4.3

    1

    3.2

    1

    2.1

    1

    c. 1100

    1

    4

    1

    3

    1

    2

    12222

    d. 629199

    920915

    27.2.76.2.5

    8.3.494.5

    Cu ng Sai

    a. S -55

    1bng 5 +

    5

    1

    (0.25 im)

    . S 117

    3

    bng

    7

    80

    (0.25 im)

    c) S -114

    5bng 11-

    4

    5

    (0.25 im)

    d) Tng -35

    1+ 2

    3

    2bng -1

    15

    13

    (0.25 im)

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  • 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 3

    1 qung ng AB. Gi th 2 i km

    gi u l 12

    1 qung ng AB, gi th 3 i km gi th 2

    12

    1 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.

    729.723162.6.2.9243.9.3

    9.813.243729.2181322 729.7231944.729243.729

    729729.2181 2

    12910.729

    2910.729

    )7231944243(729

    )7292181(729

    Cu b. Ta c:

    ;2

    1

    1

    1

    2.1

    1 ;

    3

    1

    2

    1

    3.2

    1 ;

    4

    1

    3

    1

    4.3

    1 ..; ;

    99

    1

    98

    1

    99.98

    1

    100

    1

    99

    1

    100.99

    1

    Vy

    100.99

    1

    99.98

    1

    4.3

    1

    3.2

    1

    2.1

    1

    100

    1

    99

    1

    99

    1

    98

    1

    4

    1

    3

    1

    3

    1

    2

    1

    2

    1

    1

    1

    100

    99

    100

    11 .

    Cu c. Ta c:

    ;2

    1

    1

    1

    2.1

    1

    2

    12

    ;3

    1

    2

    1

    3.2

    1

    3

    12

    ;100

    1

    99

    1

    100.99

    1

    100

    1;...;

    4

    1

    3

    1

    4.3

    1

    4

    122

    Vy 2222 010

    1

    4

    1

    3

    1

    2

    1

    100.99

    1

    4.3

    1

    3.2

    1

    2.1

    1

    1 1 1 1 1 1 11

    2 2 3 3 4 99 100

    1 991 1.

    2 100

    Cu d:

    3 0 1 8 2 2 0 2 7 2 9 1 8

    9 1 9 1 9 2 9 1 8 2 8 1 8

    5 .2 .3 2 .3 .2 2 .3 ( 5 .2 3 )2

    5 .2 .2 .3 7 .2 .3 2 .3 ( 5 .3 7 .2 )

    Cu 2: Qung ng i c trong 3 gi u l:

    1 1 1 1 1 1

    3 3 12 3 12 12

    1 1 1 1 1 1 11

    3 3 3 12 12 12 4

    Qung ng i trong gi th t l 4

    1 qung ng

    Cu 3: a. V on thng BC=5cm V cung trn (B;3cm) B C V cung trn (C;4cm) H Ly giao Im A ca hai cung trn. V cc on thng AB, AC ta c tam gic ABC.

    A A

    C

    I K

    B

    A

    O

    H

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  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh b. C 6 tam gic n l AOK; AOI; BOK; BOH; COH; v COI. C 3 tam gic Ghp i l AOB; BOC; COA. C 6 tam gic Ghp ba L ABH; BCI; CAK; ABI; BCK; CAH. C mt tam gic Ghp 6 l tam gic ABC. Vy trong hnh c tt c 6+3+1+6 = 16(Tam gic). Cu 4: a.Tm hai s tn cng ca 2100. 210 = 1024, bnh phng ca hai s c tn cng bng 24 th tn cng bng 76, c s tn cng bng 76 nng ln ly tha no( khc 0) cng tn cng bng 76. Do : 2100 = (210)10= 1024 = (10242)5 = (76)5 = 76. Vy hai ch s tn cng ca 2100 l 76. * Tm hai ch s tn cng ca 71991. Ta thy: 74=2401, s c tn cng bng 01 nng ln ly tha no cng tn cng bng 01. Do : 71991 = 71988. 73= (74)497. 343 = (01)497. 343 = (01) x 343 =43 Vy 71991 c hai s tn cng l 43. Tm 4 s tn cng ca 51992. 51992 = (54)498 =0625498=0625

    ------------------------------------------------------------------------------------------

    S 12 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.

    3 . Cho phn s b

    a ( a

  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh a(b+m) < b( a+m)

    mb

    ma

    b

    a

    4.(1 im ) Ta nhn thy , v tr ca cc ch s thay th ba du sao trong s trn u hng chn v v ba ch s i mt khc nhau, ly t tp hp 3;2;1 nn tng ca chng lun bng 1+2+3=6. Mt khc 396 = 4.9.11 trong 4;9;11 i mt nguyn t cng nhau nn ta cn chng minh

    A = 16*4*710*155 chia ht cho 4 ; 9 v 11. Tht vy : +A 4 v s to bi hai ch s tn cng ca A l 16 chia ht cho 4 ( 0,25 im ) + A 9 v tng cc ch s chia ht cho 9 : 1+5+5+7+1+4+1+6+(*+*+*)=30+6=36 chia ht cho 9 ( 0,25 im ) + A 11 v hiu s gia tng cc ch s hng chn v tng cc ch s hng l l 0, chia ht cho 11. {1+5+7+4+1)-(5+1+6+(*+*+*)}= 18-12-6=0 ( 0,25 im ) Vy A 396 5(4 im )

    a) (2 im ) t A= 65432 2

    1

    2

    1

    2

    1

    2

    1

    2

    1

    2

    1

    64

    1

    32

    1

    16

    1

    8

    1

    4

    1

    2

    1 (0,25 im )

    2A= 5432 2

    1

    2

    1

    2

    1

    2

    1

    2

    11 (0,5 im )

    2A+A =3A = 1- 12

    12

    2

    16

    6

    6

    (0,75 im )

    3A < 1 A < 3

    1 (0,5 im )

    b) t A= 10099432 3

    100

    3

    99...

    3

    4

    3

    3

    3

    2

    3

    1 3A= 1-

    9998332 3

    100

    3

    99...

    3

    4

    3

    3

    3

    3

    3

    2

    (0,5 im )

    4A = 1-100999832 3

    100

    3

    1

    3

    1...

    3

    1

    3

    1

    3

    1 4A< 1-

    999832 3

    1

    3

    1...

    3

    1

    3

    1

    3

    1 (1) (0,5 im )

    t B= 1-999832 3

    1

    3

    1...

    3

    1

    3

    1

    3

    1 3B= 2+

    98972 3

    1

    3

    1...

    3

    1

    3

    1 (0,5 im )

    4B = B+3B= 3- 993

    1 < 3 B <

    4

    3 (2)

    T (1)v (2) 4A < B < 4

    3 A <

    16

    3 (0,5 im )

    Bi 2 ( 2 im ) a) (1 im )V OB

  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Bi 1( 3 im) a, Cho A = 9999931999 - 5555571997. Chng minh rng A chia ht cho 5

    b, Chng t rng: 41

    1 +

    42

    1 +

    43

    1+ +

    79

    1 +

    80

    1 >

    12

    7

    Bi 2 ( 2,5 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 3

    2 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 3: (2 im). Tm s t nhin n v ch s a bit rng:

    1+ 2+ 3+ .+ n = aaa Bi4 ; (2,5 im) a, Cho 6 tia chung gc. C bao nhiu gc trong hnh v ? V sao. b, Vy vi n tia chung gc. C bao nhiu gc trong hnh v.

    P N Bi1: a, 1,5 im. chng minh A ta xt ch s tn cng ca A bng vic xt ch s tn cng ca tng s hng Ta c: 31999 = ( 34)499 . 33 = 81499 . 27

    Suy ra: 31999 c tn cng l 7 71997 = ( 74)499 .7 = 2041499 . 7 7 1997 C tn cng l 7

    Vy A c tn cng bng 0 A 5

    b, (1,5 im) Ta thy: 41

    1 n

    80

    1 c 40 phn s.

    Vy 80

    1

    79

    1

    78

    1......

    43

    1

    42

    1

    41

    1

    = 60

    1

    59

    1......

    42

    1

    41

    1 +

    62

    1

    61

    1.+

    80

    1

    79

    1 (1)

    V .42

    1

    41

    1..>

    60

    1 v

    61

    1 >

    62

    1 >>

    80

    1 (2)

    Ta c 60

    1

    60

    1.+

    60

    1

    60

    1 +

    80

    1+

    80

    1 +.+

    80

    1

    80

    1

    = 12

    7

    12

    34

    4

    1

    3

    1

    80

    20

    60

    20

    (3)

    T (1) , (2), (3) Suy ra:

    80

    1

    79

    1

    78

    1......

    43

    1

    42

    1

    41

    1 >

    12

    7

    Bi 2: V s trang ca mi quyn v loi 2 bng 3

    2 s trang ca 1 quyn loi 1. Nn s trang ca 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 3 Do s trang ca 8 quyn loi 1 bng : 4 .8 : 2 = 16 ( quyn loi 3) S trang ca 9 quyn loi 2 bng 9 .4 : 3 = 12 (qun loi 3) Vy 1980 chnh l s trang ca 16 + 12+ 5 = 33(quyn loi 3) Suy ra: S trang 1 quyn v loi 3 l 1980 : 33 = 60 ( trang)

    S trang 1 quyn v loi 2 l 803

    4.60 (trang)

    S trang 1 quyn v loi1 l; 1202

    3.80 ( trang)

    Bi 3:

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  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh T 1; 2; ; n c n s hng

    Suy ra 1 +2 ++ n = 2

    ).1( nn

    M theo bi ra ta c 1 +2 +3+..+n = aaa

    Suy ra 2

    ).1( nn = aaa = a . 111 = a . 3.37

    Suy 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( nn c 3 ch s Suy ra n+1 < 74 n = 37 hoc n+1 = 37

    +) Vi n= 37 th 7032

    38.37 ( loi)

    +) Vi n+1 = 37 th 6662

    37.36 ( 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 l 152

    6.5 gc

    B, 1 im . T cu a suy ra tng qut. Vi n tia chung gc c n( 2

    1n) (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 9.45.54

    1.3.5 2.6.10 4.12.20 9.27.45

    b.Chng minh : Vi kN* ta lun c :

    1 2 1 1 3. 1k k k k k k k k . p dng tnh tng :

    S = 1.2 2.3 3.4 ... . 1n n . Bi 2: (3 im).

    a.Chng minh rng : nu 11ab cd eg th : deg 11abc . b.Cho A = 2 3 602 2 2 ... 2 . Chng minh : A 3 ; 7 ; 15. Bi 3(2 im). Chng minh :

    2 3 4

    1 1 1 1...

    2 2 2 2n < 1.

    Bi 4(2 im). a.Cho on thng AB = 8cm. im C thuc ng thng AB sao cho BC = 4cm. Tnh di

    on thng AC. b.Cho 101 ng thng trong bt c hai ng thng no cng ct nhau v khng c ba

    ng thng no cng i qua mt im. Tnh s giao im ca chng.

    P N Bi 1.

    a. 1.5.6 2.10.12 4.20.24 9.45.54

    1.3.5 2.6.10 4.12.20 9.27.45

    =

    1.5.6 1 2.2.2 4.4.4 9.9.9 1.5.62

    1.3.5 1 2.2.2 4.4.4 9.9.9 1.3.5

    .

    b.Bin i :

    1 2 1 1 1 2 1 3 1k k k k k k k k k k k k p dng tnh :

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  • 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 1n n n n n n n n

    Cng li ta c :

    1 2

    3. 1 23

    n n nS n n n S

    .

    Bi 2. a.Tch nh sau :

    deg 10000 100 9999 99abc ab cd eg ab cd ab cd eg . Do 9999 11;99 11 9999 99 11ab cd M : 11ab 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 : 2

    1 1 1 1.

    1 1n n n n n

    p dng : 2 2 2

    1 1 1 1 1 1 1 11 ; ;...; .

    2 2 3 2 3 1n n n

    2 3 4

    1 1 1 1...

    2 2 2 2n <

    11 1.

    n

    Bi 4. a.Xt hai trng hp : *TH 1: C thuc tia i ca tia BA.

    Hai tia BA, BC l hai tia i nhau B nm gia A v C AC = AB + BC = 12 cm.

    *TH 2 : C thuc tia BA. C nm gia A v B (V BA > 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 pht

    CBACBA

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  • 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 S126 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 2

    1

    n

    n

    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 N

    Cu 1. (2). a, Ta c 5S = 52 + 53 +54 ++52007 5S S = (52 + 53 +54 ++52007) (5 + 52 + 53 + + 52006) 4S = 52007-5

    Vy S = 20075 5

    4

    b, S = (5 + 54) + (52 + 55) +(53 + 56) +.. + (52003 +52006) Bin i c S = 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 5

    31 1 1 1

    n n n

    n n n n

    A c gi tr nguyn 5

    1n nguyn.

    M 5

    1n 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 B(18) 72 B(24) BCNN (18;24;72) = 72. Cu 5. (2) O D B A C x V A nm gia B v C nn BA +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

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  • 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 hp A = n N / n (n + 1) 12. B = x Z / x < 3. a. Tm giao ca 2 tp hp.

    b. c bao nhiu tch ab (vi a A; b B) c to thnh, cho bit nhng tch l c ca 6. Cu 2: ( 3 im). a. Cho C = 3 + 32 + 33 + 34 + 3100 chng 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 4 ch s chia ht cho 5 t 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 hn 3/8 tui em l 7 nm. Cu 4: (2 im). a. Cho gc xoy c s o 1000. V tia oz sao cho gc 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 -1 0 1 2 0 0 0 0 0 0 1 -2 -1 0 1 2 2 -4 -2 0 2 4 3 -6 -3 0 3 6 Nhng tch l c ca 6: +1; + 2 + 3 + 6 0,5 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 ) : 40 0,5 im b. Mi s c dng abc0, abc5. Vi abc0 - 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. Vi abc5 Cch chn tng t v cng c 180 s. Vy ta thit lp c 360 s c 4 ch s chia ht cho 5 t 6 ch s cho 0,5 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 nm 0,5 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

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  • 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 tui 0,5 im Cu 4: a, C 2 cch v tia OZ (c hnh v) Gc XOZ = 650 hoc 1350 1 im b, C th din t trung im M ca on thng AB bng 3 cch khc nhau M l trung im MA+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)

    * 5a 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)

    * 5ab . 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) V 3 dng trn bao gm tt c cc dng s phI m v 3 dng l phn bit.Nn s lng cc s t nhin c 3 ch 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)

    * Cc tha s 5 trong 100! ( khi phn tch cc tha s chia ht cho 5 ) l: 100 100

    245 25

    ( tha s)

    (1 im) * Cc tha s 2 c trong 100! l:

    100 100 100 100 100 100

    2 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! l 20 ch s 0. Cu 3: (1,5 im)

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  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh a/. V 6 ngy bo ph kn 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 = 1

    2 (ao)

    Vi x = 4, ta c: 1

    2 : 2 =

    1

    4 (ao)

    Vi x = 3, ta c: 1

    4 : 2 =

    1

    8 (ao)

    Vi x = 2, ta c: 1

    8 : 2 =

    1

    16 (ao)

    Vi x = 1, ta c: 1

    16 : 2 =

    1

    32 (ao) (0,5 im)

    Vy sau ngy th nht th bo ph c: 1

    32 (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:

    X 1 2 3 5 9 y 90 45 30 18 10 T suy ra a v b c cc trng hp sau:

    a 10 20 30 50 90 y 900 450 300 180 100 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 b 34

    1938

    n

    nA

    a. 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 BM v CK = 1 cm. Tnh di BK. Cu 5: (1)

    Tnh tng: B = 100.97

    2....

    10.7

    2

    7.4

    2

    4.1

    2

    P N Cu 1: (2) Ta c: p4 - q4 = (p4 1 ) (q4- 1); 240 = 8 .2.3.5

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  • 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 p2 l 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. 34

    1872

    34

    187)34(2

    34

    1938

    nn

    n

    n

    nA

    A N th 187 4n + 3 => 4n +3 187;11;17 (0,5) + 4n + 3 = 11 -> n = 2 + 4n +3 = 187 --> n = 46 + 4n + 3 = 17 -> 4n = 14 -> khng c n 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 N) -> n 17m + 12 (m N) (0,5)

    c) n = 156 -> ;19

    77A

    n = 165 -> 39

    89A

    n = 167 -> 61

    139A (0,5)

    Cu 3: (2) Do 4 = 12 . (- 4) = 22.(-1) n c cc trng hp sau:

    a.

    1

    3

    1

    12

    43

    1)2( 2

    y

    x

    y

    x

    y

    x (0,5)

    hoc

    1

    1

    1

    12

    y

    x

    y

    x (0,5)

    b.

    2

    4

    2

    22

    13

    2)2( 22

    y

    x

    y

    x

    y

    x (0,5)

    hoc

    2

    0

    2

    22

    y

    x

    y

    x (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 -> CAM = BAM - BAC = 200 (0,75)

    B

    A

    M

    x K C

    y

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  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh

    c. C xAy = x AC + CAy = 2

    1 BAC +

    2

    1 CAM

    = 2

    1 ( BAC + CAM) =

    2

    1 BAM =

    2

    1.80 = 400 (0,75)

    d. + Nu K tia CM -> C nm gia B v K1 -> BK1 = BC + CK1 = 6 (cm) (0,5) + Nu K tia CB -> K2 nm gia B v C -> BK2 = BC = CK2 =4 (cm) (0,5 ) Cu 5: (1)

    Ta c )4

    1

    1

    1(

    3

    2

    4.1

    2)

    4

    1

    1

    1(

    3

    1

    4.1

    1 );....

    10

    1

    7

    1(

    3

    2

    10.7

    2);

    7

    1

    4

    1(

    3

    2

    7.4

    2

    ......; )100

    1

    99

    1(

    3

    2

    100.97

    2 (0,5)

    B= )100

    1

    99

    1.....

    10

    1

    7

    1

    7

    1

    4

    1

    4

    1

    1

    1(

    3

    2

    B= 50

    33

    100

    99.

    3

    2)

    100

    1

    1

    1(

    3

    2 (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

    104.2

    65.213.210

    1212 +

    49

    1010

    2.3

    5.311.3

    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 s Bi 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 im C sao cho BC = 3 cm .So snh AB vi AC

    HNG DN CHM

    Bi Hng dn chm im

    1

    a, 2A A = 221 27 A 128

    b, = 104.2

    78.210

    12

    + 16.3

    16.39

    10

    = 3 + 3 = 6

    0.5 0.5 0.5 0.5

    2

    a, Tm c n = 2010

    b, Gi s phi tm l abc theo bi ra ta c a + b + c 9 v 2b = a + c nn 3b 9 b 3 vy b 9;6;3;0 abc 5 c 5;0

    1 0.5

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  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh

    Xt s abo ta c s 630 Xt s 5ab ta c s 135 ; 765

    0.5

    3

    P c dng 3k + 1; 3k + 2 kN Dng p = 3k + 2 th p + 4 l hp s tri vi bi

    p = 3k + 1 p + 8 = 3k + 9 3 p + 8 l hp s

    0.5 0.5 0.5 0.5

    4

    Gi 2 s phi tm l a v b ( a b) ta c (a,b) = 1 nn a = 6a b= 6b trong (a,b) = 1 ( a,b,a,bN) a + b = 14

    a 1 3 5 a 13 11 9 A 6 18 30 B 78 66 54

    0.5 0.5 1

    5

    xO BC A

    Hai im A v B trn tia Ox m 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 tng S = 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. Hng xut pht t A, Dng xut pht t B. H cng khi hnh lc 8 gi cng n C vo 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 15

    8. Thm 4 n v vo phn s th nht th tch mi l

    15

    56. 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:

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  • 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 gi khong 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 : )/(5024

    60.20

    60

    24hkm

    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 d MA) c th kt hp vi 2005 on thng cn li v cc on thng tng ng trn AB to thnh 2005 tam gic.

    Do 2006 on thng s to thnh 2005 . 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 15

    8. Thm 4 n v vo phn s th nht th tch mi l

    15

    56 suy ra tch mi hn

    tch c l 15

    56 -

    15

    8 =

    15

    48 y chnh l 4 ln phn s th hai. Suy ra phn s th hai l

    15

    48 : 4 =

    15

    12 =

    5

    4 .

    T suy ra phn s th nht l: 15

    8 :

    5

    4 =

    3

    2

    -------------------------------------------------------------------

    S 21

    Thi gian lm bi: 120 pht Cu 1: (1.5) Chng minh cc phn s sau y bng nhau:

    53

    25;

    5353

    2525 ;

    535353

    252525

    Cu 2: (1,5) Khng quy ng mu hyo snh hai phn s sau:

    67

    37 v

    677

    377

    Cu 3: (2) Tm s t nhin x, bit:

    5100

    20

    100

    30)5(

    xx

    Cu 4: (3)

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  • 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:

    53

    25

    101.53

    101.25

    5353

    2525 (0.5)

    53

    25

    10101.53

    10101.25

    535353

    252525 (0.5)

    Vy 535353

    252525

    5353

    2525

    53

    25 (0.5)

    Cu 2:

    677

    300

    670

    300 m

    677

    300

    67

    30

    67

    30

    670

    300 (1) (0.5)

    Ta c : 67

    30

    67

    371 v

    677

    300

    677

    3771 (2) (0.5)

    T (1) v (2) 67

    37

    677

    377 (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

    n

    m (1) v 10

    1

    n

    m (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 yOn = 150 ; mOy = 750 (1) Ch ra cch v v v ng. (0.5) b.Tnh c mOn = 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....321

    63.37373737.636363

    2) B= 237373735

    124242423.

    2006

    5

    19

    5

    17

    55

    2006

    4

    19

    4

    17

    44

    :

    53

    3

    37

    3

    3

    13

    53

    12

    37

    12

    19

    1212

    .41

    61

    Cu II : 2

    Tm cc cp s (a,b) sao cho : 4554 ba

    Cu III : 2

    O

    m

    y

    n

    x z

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  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Cho A = 31 +32+33 + .....+ 32006

    a, Thu gn A b, Tm x 2A+3 = 3x Cu IV : 1

    So snh: A = 12005

    120052006

    2005

    v B =

    12005

    120052005

    2004

    Cu V: 2

    Mt hc sinh c quyn sch trong 3 ngy. Ngy th nht c c 5

    2 s trang sch; ngy th 2 c c

    5

    3 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....321

    63.37373737.636363

    =

    2006....321

    )63.10101.(37)37.10101.(63

    =

    2006....321

    )1010110101.(63.37 0

    2) B = 237373735

    124242423.

    2006

    5

    19

    5

    17

    55

    2006

    4

    19

    4

    17

    44

    :

    53

    3

    37

    3

    3

    13

    53

    12

    37

    12

    19

    1212

    .41

    61

    =1010101.5.47

    1010101.3.41.

    2006

    1

    19

    1

    17

    115

    2006

    1

    19

    1

    17

    114

    :

    53

    1

    37

    1

    19

    113

    53

    1

    37

    1

    19

    11.12

    .41

    47

    = 5.47

    3.41).

    4

    5.4.(

    41

    47 = 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 + .....+ 32006 3A =32+33 +34+ .....+ 32007 3A A = 32007 -3 A = 2

    332007

    (1)

    b) Ta c : 2. 2

    332007 +3 = 3x =>

    32007 -3 +3 = 3x => 32007 = 3x => x = 2007 (1) CU IV: 1

    A = 12005

    120052006

    2005

    <

    200412005

    2004120052006

    2005

    =

    )12005(2005

    )12005(20052005

    2004

    =

    12005

    120052005

    2004

    = B Vy A < B

    CU V : 2 Gi x l s trang sch, x N

    Ngy 1 c c l x5

    2 trang

    S trang cn li l x- x5

    2 = x

    5

    3 trang

    Ngy 2 c c l 5

    3.

    5

    3x = x

    25

    9 trang

    S trang cn li l x5

    3- x

    25

    9 = x

    25

    6 trang

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  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh

    Ngy th 3 c c l : x25

    6.80% +30 =

    125

    24x + 30

    Hay : x5

    2+ x

    25

    9+

    125

    24x + 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 s 50 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 1

    101 100 99 98 ... 3 2 1

    b. D = 3737.43 4343.37

    2 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. 380 830 (0,5) c. 803

    Bi 2 (2): a) (1)

    A = 50 chu so

    333...3 x 50 chu so

    1 00..0 - 1

    = 50 chu so 50 chu so 50 chu so

    33...300...0 - 33...3 (0,5)

    =

    49 chu so 49 chu so

    33 ... 33 00 ... 00

    33 ... 33

    33 ...32 66 ... 67

    (0,25). Vy A = 49 chu so 49 chu so

    33 ...3266 ... 67 (0,25)

    A B C D

    a1

    a2

    a3

    b1

    b2

    c1

    c2

    c3

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  • 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 1

    101 100 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 = 5151

    10151

    (0,25)

    b) (0,75)

    B = 3737.43 4343.37

    2 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 = 10

    102 = 102410 = 5

    21024 (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 = 181614....642

    2.550135450027

    b. So snh: A = 12007

    120062007

    2006

    v B =

    12006

    120062006

    2005

    Bi 2 (2) a. Chng minh rng: C = 2 + 22 + 2 + 3 + + 299 + 2100 chia ht cho 31 b. Tnh tng C. Tm x 22x -1 - 2 = C Bi 3 (2)

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  • 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 c 42 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 N

    Bi 1

    a. S = 270.450 270.550 270(450 550) 270000

    3000(2 18).9 90 90

    2

    b. Ta c nu 1a

    b th *( )

    a a nn N

    b b n

    2006 2006

    2007 2007

    2006 1 2006 1 2005

    2006 1 2006 2005 1A

    2006 2005 2005

    2007 2006 2006

    2006 2006 2006(2006 1) 2006 1

    2006 2006 2006(2006 1) 2006 1B

    Vy A < B Bi 2 a. C = 2 + 22 + 23 + .. + 299 + 2

    100 = 2(1 +2 + 22+ 23+ 24) + 26(1 + 2 + 22+ 23+ 24)++ (1 + 2 + 22+ 23+ 24).296 = 2 . 31 + 26 . 31 + + 296 . 31 = 31(2 + 26 ++296). Vy C chia ht cho 31

    b. C = 2 + 22 + 23 + .. + 299 + 2100 2C = 22 + 23 + 24 + + 2

    100 + 2101 Ta c 2C C = 2101 2 2101 = 22x-1 2x 1 = 101 2x = 102 x = 51

    Bi 3: Gi s cn tm l A: A = 4q1 + 3 = 17q2 + 9 = 19q3 + 13 (q1, q2, q3 thuc N) A + 25 = 4(q1 +7) = 17(q2 +2) = 19(q3 + 2) A + 25 chia ht cho 4; 17; 19 A + 25 =1292k A = 1292k 25 = 1292(k + 1) + 1267 khi chia A cho 1292 d 1267 Bi 4 Tng s im ca 10 lp 6A l (42 - 39) . 1 + (39 - 14) . 2 + (14 - 5) . 3 + 5 . 4 = 100(im 10) Bi 5:

    C 24 25

    3002

    ng thng. Vi n im c

    ( 1)

    2

    n n ng thng

    S 25 Thi gian lm bi: 120 pht

    1. Tnh cc gi tr ca biu thc. a. A = 1+2+3+4+.........+100

    b. B = -1 .

    2003

    5

    19

    5

    17

    55

    2003

    4

    19

    4

    17

    44

    :

    53

    3

    37

    3

    3

    13

    )53

    3

    7

    3

    3

    13(4

    .5

    1

    c. C = 100.99

    1...

    5.4

    1

    4.3

    1

    3.2

    1

    2.1

    1

    2. So snh cc biu thc : a. 3200 v 2300

    b. A = 1717

    404

    17

    2

    171717

    121212 vi B =

    17

    10.

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  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh 3. Cho 1s c 4 ch s: *26* in cc ch s thch hp vo du (*) c s c 4 ch s khc nhau chia ht cho tt c 4s : 2; 3 ; 5 ; 9. 4. Tm s t nhin n sao cho : 1! +2! +3! +...+n!. l s chnh phng? 5. Hai xe t khi hnh t hai a im A,B i ngc chiu nhau. Xe th nht khi hnh t A lc 7 gi. Xe th hai khi hnh t B lc 7 gi 10 pht. Bit rng i c qung ng AB . Xe th nht cn 2 gi , xe th hai cn 3 gi. Hi sau khi i 2 xe gp nhau lc my gi?

    6. Cho gc xOy c s o bng 1200 . im A nm trong gc xOy sao cho: 0 AOy =75 . im B nm ngoi

    gc xOy m : 0BOx =135 . Hi 3 im A,O,B c thng hng khng? V sao?

    P N Cu 1 : Tnh gi tr biu thc : a) Tng : S =1 +2 +3 +...+100 c 100 s hng . S = ( 1+ 100) + (2 +99) + (3 + 98) + ... + 950 + 51) c 50 cp . = 50 . 10 = 5050

    b) A =

    2003

    5

    19

    5

    17

    55

    2003

    4

    19

    4

    17

    44

    :

    )53

    3

    37

    3

    3

    13(

    )53

    3

    37

    3

    3

    13(4

    .5

    11

    Ta c : A = -

    )2003

    1

    19

    1

    17

    11(5

    )2003

    1

    19

    1

    17

    11(4

    :1

    4.

    5

    6

    = -

    6 4 4 6 4.5. : . 6

    5 1 5 5 4

    c). B = 3.2

    1 +

    4.3

    1 +

    5.4

    1 +

    6.5

    1+............+

    100.99

    1

    Ta c : B = 1 - 2

    1 +

    2

    1 -

    3

    1+

    3

    1 -

    4

    1+........+

    99

    1 -

    100

    1 = 1 -

    100

    1=

    100

    99

    2) Cu2. So snh . a) Ta c : 3200 =(32)100 = 9100 2300 =(23)100 =8100 V 9100 > 8100 Nn 3200 > 2300

    b) A = 101:1717

    101:404

    17

    2

    10101:171717

    10101:121212

    1717

    404

    17

    2

    171717

    121212

    17

    4212

    17

    4

    17

    2

    17

    12 A

    Vy A = 17

    10 hay A =B =

    17

    10

    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 th n! = 1.2.3.........n l mi s chn .Nn 1!+2!+......+n! =33 cng vi mt s chn bng sc ch s tn cng ca tng l ch s 3 .Nn n khng phi l s chnh phng. Vy ch c 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 c 2

    1 qung ng AB.

    1 gi xe th 2 i c 3

    1 qung ng AB .

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  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh

    1 gi c 2 xe i c 2

    1+

    3

    1=

    6

    5 qung ng AB.

    Sau 10 pht = 6

    1gi : Xe th nht i c

    6

    1.

    2

    1=

    12

    1 qung ng AB.

    Qung ng cn li l:

    1 - 12

    1

    12

    11 (ca AB)

    Thi gian hai xe cng i qung ng cn li l:

    12

    11:

    6

    5=

    10

    11 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 = 45 im 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 0 BOA = BOx + xOA =180 . Nn 3 im A,O,B thng hng. (0,75)

    +, Cn ti B th : xOB'= 1350 < 1800, 0 0 0 AOB' = 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 3 100

    1 1 1 1...

    3 3 3 3A

    Cu 2: Tm s t nhin a, b, c, d nh nht sao cho: 5

    3

    a

    b ;

    12

    21

    b

    c ;

    6

    11

    c

    d

    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 c 3A = 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 ra A= (3100-1) )/ 2.3100

    Cu 2: Ta c 12/21= 4/7, cc phn s 3/5, 4/5, 6/11 ti gin nn tn ti cc s t nhin k, l, m sao cho a=3k, b=5k, b=4n, c=7n, c= 6m, d=11m. T cc ng thc 5k=4n, v 7k = 6m ta c 4n5 v 7n 6 m (4,5)=1; (7,6)=1 nn n5, 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 nn 0< a-b < 25, khng th xy ra a-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

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  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh cu 4: (Hc sinh t v hnh)

    Ta thy : 0AOB + BOC + AOD >180

    v 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 bng v bi, 13 hc sinh thch bi v bng chuyn, 15 hc sinh thch bng v bng chuyn, 10 hc sinh thch c ba mn, 12 hc sinh khng thch mn no. Tnh xem 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 cng ca 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 N

    Cu 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 s khng 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).

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  • 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 - 5

    4

    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 + 3 chia 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 - 1 1 3 9 n - 5 - 3 - 7/3 - 5/3 - 1 1

    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 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: 1

    1052

    n 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 pht

    Bi 1:(2,25 im) Tm x bit

    a) x+1 7

    5 25 b) x-

    4 5

    9 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 1

    2 6 12 20 30 42

    c) C = 1 1 1 1

    ... ...1.2 2.3 1989.1990 2006.2007

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  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Bi 4:(1 im)

    Cho: A= 2001 2002

    2002 2003

    10 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 N

    Bi 1:(2,25 im)

    a) x=7 1 2

    25 5 25 ; b) x=

    5 4 45 44 89

    11 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 6

    1 12 2 3 3 4 4 5 5 6 6 7 7 7

    c) C = 1 1 1 1 1 1 1 1 2006

    1 ... ... 12 2 3 1989 1990 2006 2007 2007 2007

    Bi 4:(1 im)

    Ta c: 10A = 2002

    2002 2002

    10 10 9 = 1 +

    10 1 10 1

    (1)

    Tng t: 10B = 2003

    2003 2003

    10 10 9 = 1 +

    10 1 10 1

    (2)

    T (1) v (2) ta thy : 2002 2003

    9 9

    10 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 = AB hay AK + 2 = 7 AK = 5 cm. Trn tia AB c im I v K m AI < AK (v 4

  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Bi 2 (2 im )

    a. Tnh tng: M = 1400

    10....

    260

    10

    140

    10

    56

    10

    b. Cho S = 14

    3

    13

    3

    12

    3

    11

    3

    10

    3 . Chng minh rng : 1< S < 2

    Bi 3 ( 2 im) Hai ngi i mua go. Ngi th nht mua go np , ngi th hai mua go t. Gi go t r hn gi go np l 20%. Bit khi lng go t ngi th hai mua nhiu hn khi lng go np l 20%. Hi ngi no tr tin t hn? t hn mya % so vi ngi kia? Bi 4 ( 3 im) Cho 2 im M v N nm cng pha i vi A, nm cng pha i vi B. im M nm gia A v B. Bit AB = 5cm; AM = 3cm; BN = 1cm. Chng t rng: a. Bn im A,B,M,N thng hng b. im N l trung im ca on thng MB c. V ng trn tm N i qua B v ng trng tm A i qua N, chng ct nhau ti C, tnh chu vi ca CAN .

    P N

    Bi 1 ( 3 im) a.(1 im) Ta c 405n = .5 ( 0,25 im) 2405 = 2404. 2 = (.6 ).2 = .2 ( 0,25 im) m2 l s chnh phng nn c ch s tn cng khc 3. Vy A c ch s tn cng khc khng A 10 b. ( 1im)

    B = 2

    264

    2

    317592

    2

    317

    2

    5

    2

    92

    n

    n

    n

    nnn

    n

    nn

    nn

    n ( 0,25 im)

    B = 2

    184

    2

    18)2(4

    2

    264

    nn

    n

    n

    n (0,25 im )

    B l s t nhin th 2

    18

    n l s t nhin

    18 (n+2) => n+2 ( 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 16;7;4;1;0 th B N (0,25im ) c. (1 im) Ta c 55 =5.11 m (5 ;1) = 1 (0,25 im)

    Do C = yx1995 55

    11

    5

    C

    C

    1

    2

    (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 = 1400

    10...

    260

    10

    140

    10

    56

    10 =

    28.25

    5...

    13.10

    5

    10.7

    5

    7.4

    5 (0,25 im)

    =

    28

    1

    25

    1...

    13

    1

    10

    1

    10

    1

    7

    1

    7

    1

    4

    1.

    3

    5 ( 0, 25 im)

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  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh

    = 14

    5

    28

    6.

    3

    5

    28

    1

    4

    1.

    3

    5

    ( 0,5 im)

    b. (1 im)

    S = 15

    3

    15

    3

    15

    3

    15

    3

    15

    3

    14

    3

    13

    3

    12

    3

    11

    3

    10

    3 => S > 1

    15

    15 (1) ( 0,5im)

    S=10

    3

    10

    3

    10

    3

    10

    3

    10

    3

    14

    3

    13

    3

    12

    3

    11

    3

    10

    3 => S < 2

    10

    20

    10

    15 (2) ( 0,5 im)

    T (1) v (2) => 1 < S < 2 Bi 3:

    Gi gi go np l a (ng/kg) ; khi lng go np mua l b (kg) (0,25 im)

    Suy ra gi go t l a.10

    80; khi lng go t mua l b.

    100

    120 ( 0,25 im)

    S tin ngi th nht phi tr l a.b (ng) (0,25 im)

    S ting ngi th hai phi tr l 100

    96..

    100

    120..

    100

    80ba a.b (0.75im)

    Vy ngi th hai tr t tin hn ngi th nht . T l % t hn l:

    %4.:..100

    96.

    bababa (0,5 im)

    BI 4 V 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 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 nn AC = AN = AM + MN = 4 cm (0.25 im) Chu vi CAN = AC + CN = NA = 4 + 4+1= 9 (cm) (0,5 im)

    S 30 Bi 1 : Tm x bit a ) x + (x+1) +(x+2) +...... +(x +30) = 620 b) 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 Bi 3: Cho S = 1+3+32 +33+.........+348 +349

    a ) chng t S chia ht cho 4 b) Tm ch s tn cng ca S

    c) Chng t S =2

    1350

    Bi 4 : Tm 2 s a ,b N tho mn : 12a + 36b = 3211

    Bi 5 : Cho (2a + 7b) 3 ( a,b N ) Chng t : (4a + 2b ) 3 Bi 6 : Ly 1 t giy ct ra thnh 6 mnh .Ly 1 mnh bt k ct ra thnh 6 mnh khc . C nh th tip tc nhiu ln a) 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 CA CB Bi 8 : V on thng AB =5 cm .Ly 2 im C ,D nm gia A v B sao cho : AC +BD=6 cm a) chng t im C nm gia B v D b) Tnh di on thng CD

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  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh

    P N Bi 1 :

    a) 31x + 6202

    30)301(

    15515.3162031 x

    x= 155 :31 = 5

    b) 2102

    )22(

    xx 210)1( xx 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

    - Nu x = 3k ( tho mn ) .Nu x= 3k +1 th x+2 =3k+1+2 =(3k +3 ) 3

    - Nu x = 3k +2 th x +1 = 3k+1 +2 = (3k +3 ) 3 Vy trong 3 s t nhin lin tip c 1 s chia ht cho 3

    b )Nhn thy 17n , 17n +1 , 17n + 2 l 3 s t nhin lin tip m 17n khng chia ht cho 3 ,Nn trong 2 s cn li 1 s phi 3 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 = .............4 Vy S c tn cng bng 4

    c ) S = 1+3+32 +33+.........+348 +349 3S = 3 +3+32 +33+.........+348 +349+ 350

    ss 3 = 350 1

    2S = 350 1 Suy ra S =2

    1350

    Bi 4 : Nhn thy 12 a 4 v 36 b 4 m 3211 khng chia ht cho 4 , Vy khng c 2 s t nhin no tho mn Bi 5 : 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 giy Bi 7 :

    - Gi M l trung im ca AB suy ra MA = MB v M AB Xt 3 trng hp a ) C M ta c MA = MB suy ra CA = CB b ) C nm gia A v M CA < MA CA < MB (1) M nm gia C v B nn MB < CB (2) T (1) & (2) CA < CB c ) C nm gia M v B CB < MB CB < MA ( 3) M nm gia A v C nn MA < CA (4) T (3) v (4) CA < CB

    A B M C

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  • Hunh Minh Khai: Gv: THCS th Trn cu k, Tr Vinh Tm li C MA th ta lun c CA CB Bi 8 :

    C nm gia A v B nn : AC + CB = AB = 5 V AC + BD = 6 AC + CB < AC + BD CB < BD C nm gia D v B b ) BD = BC + CD v AC + BD = 6 nn AC + BC + CD = 6 (BC + AC) + CD = 6 CD = 6 AB = 6 -5 =1 Vy CD = 1

    S 31 Thi gian lm bi: 150 pht

    Nm hc 2009 - 2010 Cu 1 (2 im) Tnh

    a/ A = 123...9899100101123...9899100101

    b/ B = 423134846267.423133423133846267.423134

    Cu 2 (2 im) a/ Chng minh rng: 1028 + 8 chia ht cho 72 b/ Cho A = 1 + 2 + 22 + 23 + . . . + 22001 + 22002 B = 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: 12100

    1...

    24

    123

    122

    1

    P N

    Cu 1:

    a/ A = 10151

    51.101 (1 im)

    b/ B = 1423134846267.423133

    423133846267846267.423133

    (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 c 3 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 N; k > 0) + Nu p = 5k th do p nguyn t nn k = 1 => p = 5 + Nu p = 5k + 1 => p + 14 =