30 de HSG Toan 6 Va Dap An

8/14/2019 30 de HSG Toan 6 Va Dap An

1/62

Tng hp thi hc sinh gii ton 6 s i

Thi gian lm bi 120 pht

Cu 1 : (2 im) Cho biu thc122

1223

23

++++

=aaa

aaA

a, Rt gn biu thcb, 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( = ncbaCu 3: (2 im)

a. Tm n n2 + 2006 l mt s chnh phngb. 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 snhnb

na

+

+

vb

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 tipnhau 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 sgiao im ca chng.

------------------------------------------------------------------- s ii


Cu1:

a. Tm cc s t nhin x, y. sao cho (2x+1)(y-5)=12b.Tm s t nhin sao cho 4n-5 chia ht cho 2n-1c. Tm tt c cc s B= 62xy427, bit rng s B chia ht

cho 99Cu 2.

a. chng t rng230

112

+

+

n

nl phn s ti gin.

V c Huy


2/62

Tng hp thi hc sinh gii ton 6b. Chng minh rng : 2

2

1+ 23

1+ 2

4

1+...+ 2100

1


3/62

Tng hp thi hc sinh gii ton 6b. Tia i ca mi tia Ox, Oy, Oz l phn gic ca gc hp

bi hai tia cn li.-------------------------------------------------------

s ivThi gian lm bi 120 pht

Cu 1. Tnh:a. A = 4 + 2 2 + 2 3 + 2 4 +. . . + 2 20b. tm x bit: ( x + 1) + ( x + 2) + . . . + ( x + 100) =

5750.Cu 2.

a. Chng minh rng nu: ( )egcdab ++ 11 th degabc 11.

b. Chng minh rng: 10 28 + 8 72.Cu 3.

Hai lp 6A;6B cng thu nht mt s giy vn bng nhau. Lp6A c 1 bn thu c 26 Kg cn li mi bn thu c 11 Kg ; Lp 6Bc 1 bn thu c 25 Kg cn li mi bn thu c 10 Kg . Tnh shc sinh mi lp bit rng s giy mi lp thu c trong khong200Kg n 300 Kg.Cu 4.

Tm 3 s c tng bng 210, bit rng7

6s th nht bng

11

9s th 2 v bng

3

2s th 3.

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 ctbn on thng AB, AC, AD, BC, BD, CD.

-------------------------------------------------------------- s v


Bi 1 (3):

a) So snh: 222333 v 333222b) Tm cc ch s x v y s 281 yx chia ht cho 36c) Tm s t nhin a bit 1960 v 2002 chia cho a c cng

s d l 28Bi 2 (2):

Cho : S = 30 + 32 + 34 + 36 + ... + 32002

a) Tnh S

V c Huy


4/62

Tng hp thi hc sinh gii ton 6b) 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 28Bi 4 (3):

Cho gc AOB = 1350

. C l mt im nm trong gc AOBbit gc BOC = 900a) Tnh gc AOCb) Gi OD l tia i ca tia OC. So snh hai gc AOD v

BOD------------------------------------------------

s vi.Thi gian lm bi 120 pht

Bi 1( 8 im1. Tm ch s tn cng ca cc s sau:

a) 571999 b) 931999

2. Cho A= 9999931999 - 5555571997. Chng minh rng A chiaht cho 5.

3 . Cho phn sb

a( a


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6/62

Tng hp thi hc sinh gii ton 6Bi 3 : (4)

Cho bng gm 2007 nh sau :

17 36 19

Phn u ca bng nh trn . Hy in s vo ch trng saocho 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 ?

-------------------------------------------------------- s ix

Thi gian lm bi: 120 pht

Bi 1:(1,5) Tm x, bit:a) 5x = 125; b) 32x = 81 ; c) 52x-3

2.52 = 52.3Bi 2 :(1,5) Cho a l s nguyn. Chng minh rng: a 5 5 5a< < b

a a>b a+m > b+n.

M nbna

+

+

c phn tha so vi 1 l nbba

+

b

a c phn tha so vi 1 lb

ba , vnb

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)=>* 2n-1=1 => n=1*2n-1=3=>n=2 (0,25)vy n=1;2 (0,25)c. (1) Ta c 99=11.9B 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 11x-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 ching 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 nhau

do 230

112

+

+

n

nl phn s ti gin (0,5)

b. Ta c 221

2x = 6 => x=3Bi 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 cccs ny u ln hn -5 v nh hn 5 do -5


25/62

Tng hp thi hc sinh gii ton 6Tch ring s dng cn 30 s chi lm 6 nhm. Theo bi tng cc s ca mi nhm u l s dng nn tng ca6 nhm u l s dng v do tng ca 31 s chou l s dng.Bi 5 (2): V c 11 tng m ch c th c 10 ch s tn

cng u l cc s t 0 , 1 ,2, ., 9 nn lun tm c haitng c ch s tn cng ging nhau nn hiu ca chng lmt s nguyn c tn cng l 0 v l s chia ht cho 10.Bi 6 (1,5).Ta c: ' 0 ' 060 , 60 x Oy x Oz = = v tia Ox nm gia hai

tia Oy, Oz nn ' ' 0120 yOz yOx x Oz = + = vy xOy yOz zOx= =

Do tia Ox nm gia hai tia Oy, Oz v ' ' x Oy x Oz = nn Ox l tiaphn gic ca gc hp bi hai tia Oy, Oz.Tng t tia Oy (tia i ca Oy) v tia Oz (tia i ca tia Oz)l phn gic ca gc xOz v xOy.

p n s iv

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

=> 2A A = 2 21 +8 ( 4 + 2 2 ) + (2 3 2 3) +. . . + (2 20 220). = 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 =5750100 x = 5750

5050100 x = 700

x = 7Cu 2. a) egcdababc ++= 10010000deg = 9999 cdab 99+ +

egcdab ++ 11.

V c Huy


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Tng hp thi hc sinh gii ton 6b). 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 72Cu 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. hsS 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 bng22

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 mtphng

Xt 3 trng hpa). 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 mtphng) ba im B, C, D thuc na mt phng i th ngthng a ct ba on thng AB, AC, AD

c). Nu c 2 im chng hn (A v B) thuc mt na mtphng hai im kia (C v D) thuc mi mt phng i th act bn on thng AC, AD, BC, BD

hng dn s v

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 )

V c Huy


27/62

Tng hp thi hc sinh gii ton 6

++++

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: aTa c a = 29q + 5 = 31p +28 (0,5) 29(q - p) = 2p

+ 23V 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 vOA=> gc AOC + gc BOC = gcAOB=> gc AOC = gc AOB - gc

BOC=> gc AOC = 1350 - 900 = 450

b) v OD l tia i ca tiaOC 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

V c Huy


28/62


29/62

Tng hp thi hc sinh gii ton 6Tht vy :+A 4 v s to bi hai ch s tn cng ca A l 16 chia htcho 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 tngcc 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 3965(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= 54322

1

2

1

2

1

2

1

2

11 ++

(0,5 im )

2A+A =3A = 1- 12

12

2

16

6

6 tOt, = )180(2

1

2

1aa + = 900

O

Cu 2; Gi s trong 20 im, khng c 3 im no thnghng. Khi , s ng thng v c l; 19 . 20:2 = 190Trong 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

p n s viii

V c Huy


32/62

Tng hp thi hc sinh gii ton 6Bi 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 cngch s hng trm .Lp th nht gm 100 s t 100 n 199Lp th hai gm 100 s t 200 n 299Lp 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 hngchc 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 slng 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 sc 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 8 910

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 bngnhau s 3 l 19 (0.5)100 - (17 + 19 + 36) = 28

V c Huy


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Tng hp thi hc sinh gii ton 6Bi 4 (2). Trong cc s cho t nht c 1 s dng vnu tri li tt c u l s m th tng ca 5 s btk trong chng s l s m tri vi gi thit.Tch ring s dng cn 30 s chi lm 6 nhm. Theo bi tng cc s ca mi nhm u l s dng nn

tng ca 6 nhm u l s dng v do tng ca 31s cho u l s dng.Bi 5 (2): V c 11 tng m ch c th c 10 ch stn cng u l cc s t 0 , 1 ,2, ., 9 nn lun tm -c hai tng c ch s tn cng ging nhau nn hiu cachng l mt s nguyn c tn cng l 0 v l s chia htcho 10.Bi 6 (1,5).Ta c: ' 0 ' 060 , 60 x Oy x Oz = = v tia Ox nm gia

hai tia Oy, Oz nn ' ' 0

120 yOz yOx x Oz = + = vy

xOy yOz zOx= =Do tia Ox nm gia hai tia Oy, Oz v ' ' x Oy x Oz = nn Ox ltia phn gic ca gc hp bi hai tia Oy, Oz.Tng t tia Oy (tia i ca Oy) v tia Oz (tia i ca tiaOz) l phn gic ca gc xOz v xOy.

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

p n s xi:

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

729729.21812

++

+

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:

V c Huy


35/62

Tng hp thi hc sinh gii ton 6 ;

2

1

1

1

2.1

1

2

12

=< ;3

1

2

1

3.2

1

3

12

=<

;1 0 0

1

9 9

1

1 0 0.9 9

1

1 0 0

1; . . . ;

4

1

3

1

4.3

1

4

122

=


36/62

Tng hp thi hc sinh gii ton 6* 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

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

p n s xii

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 cngca tng s :a) 571999 ta xt 71999

Ta c: 71999 = (74)499.73 = 2041499. 343 Suy ra ch s tn cngbng 3 ( 0,25 im )Vy s 571999 c ch s tn cng l : 3b) 931999 ta xt 31999

Ta c: 31999 = (34)499. 33 = 81499.27Suy ra ch s tn cng bng 7(0,25 im )

2. Cho A = 9999931999 - 5555571997 . chng minh rng A chia htcho 5 chng minh A chia ht cho 5 , ta xt ch s tn cng ca Abng vic xt ch s tn cng ca tng s hng.Theo cu 1b ta c: 9999931999 c ch s tn cng l 7Tng t cu 1a ta c: (74)499.7 =2041499.7 c ch s tn cng l7 ( 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


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Tng hp thi hc sinh gii ton 6Ta nhn thy , v tr ca cc ch s thay th ba du sao trongs trn u hng chn v v ba ch s i mt khcnhau, ly t tp hp { }3;2;1 nn tng ca chng lun bng1+2+3=6.Mt khc 396 = 4.9.11 trong 4;9;11 i mt nguyn t cng

nhau nn ta cn chng minhA = 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 htcho 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 tngcc 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 3965(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= 54322

1

2

1

2

1

2

1

2

11 ++

(0,5 im ) 2A+A =3A = 1- 1

2

12

2

16

6

6>

80

1(2)

V c Huy

B A xO


39/62

Tng hp thi hc sinh gii ton 6Ta 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++++++

> 127

Bi 2: V s trang ca mi quyn v loi 2 bng3

2s trang ca

1 quyn loi 1. Nn s trang ca 3 quyn loi 2 bng s trangca 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 bng 9 .4 : 3 = 12 (qun loi3)

Vy 1980 chnh l s trang ca 16 + 12+ 5 = 33(quynloi 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: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 ra2

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

Suy ra: n (n+1) = 2.3.37.aV tch n(n+1) Chia ht cho s nguyn t 37 nn n hoc n+1Chia ht cho 37

V s2

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

n+1 = 37+) Vi n= 37 th 703

2

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 = 666Bi 4 :A, 1,5 im

V c Huy


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Tng hp thi hc sinh gii ton 6V 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 c5.6 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 cn(2

1n) (gc).

----------------------------------------------------------p n s xiv

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 :( )

( )

( )

( ) ( ) ( ) ( ) ( )

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 11M M ( )9999 99 11ab cd + MM : ( ) 11ab cd eg + + M (theo bi ra) nn : deg 11.abc Mb.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.+ + + M*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+ + + M .*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+ + + + + + + + + + + + =

V c Huy


41/62

Tng hp thi hc sinh gii ton 6= ( )5 5715. 2 2 ... 2 15.+ + + M

Bi 3. Ta c :( )2

1 1 1 1.

1 1n n n n n< =

p dng : 2 2 21 1 1 1 1 1 1 1

1 ; ;...; .2 2 3 2 3 1n 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 tora 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.L u : Hc sinh gii cch khc ng vn cho im ti a. Bihnh khng v hnh khng chm im.

---------------------------------------------------------------p n s xv

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 126Cu 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, 6BCNN(3;4;5;6) = 60 . nen x + 2 = 60.nDo x = 60.n 2 (n = 1;2;3..)

V c Huy

cba

cba


42/62

Tng hp thi hc sinh gii ton 6Mt khc x 11 ln lt cho n = 1;2;3.Ta thy n = 7 th x = 418 11Vy s nh nht phi tm l 418.

Cu 3. (1). Ta c3 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.

M5

1n nguyn 5 (n-1) hay n-1 l c ca 5

Do 5 = {1; 5}Ta tm c n =2

n =0n =6n = -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 cmTa c DB + BA = DADB +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)--------------------------------------------------------

p n s xvi

Cu 1: Lit k cc phn t ca 2 tp hpa. A = { 0, 1, 2, 3} B = { - 2, -1, 0, 1, 2, } 0,5im

V c Huy


43/62

Tng hp thi hc sinh gii ton 6A B = { 0, 1, 2,} 0,5

im.b. C 20 tch c to thnh

-2 -1 0 1 20 0 0 0 0 0

1 -2 -1 0 1 22 -4 -2 0 2 43 -6 -3 0 3 6

Nhng tch l c ca 6: +1; + 2 + 3 + 6 0,5imCu 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.Vi abc0

- C 5 cch chn ch s hng nghn (v ch s hngnghn 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 lpc 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 tuianh 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. 0,5im3/4 tui em = 32 14 = 18 tui0,5 imTui em l: 18:3/4 = 24 tui0,5 imCu 4:a, C 2 cch v tia OZ (c hnh v)

V c Huy


44/62

Tng hp thi hc sinh gii ton 6Gc XOZ = 650 hoc 1350 1imb, C th din t trung im M ca on thng AB bng 3cch khc nhauM l trung im MA+MB=AB

MA=MB=AB/2Ca on thng AB MA=MB

-------------------------------------------------------p n s xvii

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 cchchn.Nn cc s thuc loi ny c : 8.9 = 72 ( s )

(0,5 im) V 3 dng trn bao gm ttc cc dng s phI m v 3 dng l phn bit.Nn s lng ccs 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)

V c Huy


45/62

Tng hp thi hc sinh gii ton 6Tch 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)

a/. V 6 ngy bo ph kn ao v c sau 1 ngy din tchbo tng ln gp i nn ph kn na ao th phI sau ngyth 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: 18

: 2 = 116

(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 ) (0,5im) 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 = 90Ta c cc trng hp sau:

X 1 2 3 5 9

y 90 45 30 18 10T suy ra a v b c cc trng hp sau:

a 10 20 30 50 90

y 900450

300

180

100

V c Huy


46/62

Tng hp thi hc sinh gii ton 6Cu 5: (1 im)

Ta c s :

-----------------------------------------------------------------p n s xviii

Cu 1: (2) Ta c: p4 - q4 = (p4 1 ) (q4- 1); 240 = 8 .2.3.5Chng 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 -13 (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 240Tng t ta cng c q4 - 1 240 (0,25)Vy: (p4 - 1) (q4 1) = p4 q4 240Cu 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; 46b.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

77=A

n = 165 ->39

89=A

V c Huy


47/62

Tng hp thi hc sinh gii ton 6n = 167 ->

61

139=A (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)

V c Huy


48/62


49/62

Tng hp thi hc sinh gii ton 65 + 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:

* chn v 2 + 6 + 1 + * chia 3 d 1; t tm c * = 4(1)

Cu 2 S = 1.2 + 2.3 + 3.4 + ... + 99.1003.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

+ 99.100.101S = 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 cchDng l 20 km, lc Ninh gp Dng nn Ninh cng cch Hng20 km.

n 9 gi 24 pht, Ninh gp Hng do tng vn tc ca

Ninh v Hng l:20 : )/(50

24

60.20

60

24hkm==

Do vn tc ca Ninh bng 1/4 vn tc ca Hng nn vn tcca 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 2006on thng ni t M n cc im .

V c Huy

0,5

1

1

0,5

0,5


50/62

Tng hp thi hc sinh gii ton 6Mi on thng (v d MA) c th kt hp vi 2005 on

thng cn li v cc on thng tng ng trn AB tothnh 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 tamgic ny ch l 1)Do s tam gic thc c l: 4022030 : 2 = 2011015

Cu 5: (1)

Tch ca hai phn s l15

8. Thm 4 n v vo phn s

th nht th tch mi l15

56suy ra tch mi hn tch c l

15

56-

15

8=

15

48y 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:

15

8:

5

4=

3

2

-----------------------------------------------------------p n s xxi

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 nghtr 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)

V c Huy


51/62

Tng hp thi hc sinh gii ton 6p s: S ngi trong i vn ngh l: 7Cu 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)

-------------------------------------------------------------------pnsxxii

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. 2332007

+3 = 3x =>32007 -3 +3 = 3x => 32007 = 3x => x = 2007 (1)

Cu IV: 1

A =12005

120052006

2005

++ x =625 trang

S 625 trang-----------------------------------------------------------------

p n s xxiii

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...31 2 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

1 2 3(0,25). Vy A = {

49 chu so 49 chu so

33 ...32 66 ... 671 2 3

(0,25)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

+ + + + + + + + + + +

V c Huy


53/62

Tng hp thi hc sinh gii ton 6Ta 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) + 11 4 4 4 4 4 4 2 4 4 4 4 4 4 3 = 50 + 1 = 51

(0,25)

Vy C =5151

10151

= (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 76Bi 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 b2c3; (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 b2c3; (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 b1c1; 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 99im cn li, ta v c 99 ng thng (0,5)

V c Huy


54/62

Tng hp thi hc sinh gii ton 6Lm 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)---------------------------------------------------------

p n s xxivBi 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

+ + += 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

10hay A =B =

17

10

3). Bi 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 0hoc 5.S va chia ht cho 3 v9 .Nn s phi c tngcc ch s chia ht cho 9.Vy : Ch s tn cng ca s l 0 *260 . Ch s u ls 1

V c Huy


56/62

Tng hp thi hc sinh gii ton 6Do s cho l 12604 ) Bi 4. Tm s t nhin n . M 1! +2!+3! +...+n! l bnh ph-ng 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! =33Vi 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 catng 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 c2

1qung ng AB.

1 gi xe th 2 i c31 qung ng AB .

1 gi c 2 xe i c2

1+

3

1=

6

5qung 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

11gi = 1 gi 6 pht.

Hai xe gp nhau lc 7 gi 10 pht + 1 gi 6 pht = 8 gi 16pht .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 tiaOA nm gia hai tia Ox v Oy.Ta c : 0 0 0xOA = 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 0BOx + xOA = 135 + 45 = 180 . Do 0BOA = BOx + xOA =180 . Nn 3im 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)

V c Huy


57/62

Tng hp thi hc sinh gii ton 6-------------------------------------------------------------------------------

hng dn Gii s xxvi

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 ra A= (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. T cc ng thc 5k=4n, v 7k = 6m tac 4n5 v 7n 6 m (4,5)=1; (7,6)=1 nnn5, 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 nnh nht bng 30 suy ra: k =24, m=35vy 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. Gis a>b.

a.Gi d thuc C(a,b) th a-b d ta s chng minh d 25tht vy gi s d>25 th b>25 ta c a 50 m b>25 nn 0< a-b < 25, khng th xy raa-b d ; d=25 xy ra khi a=50; b=25vy 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 tgi tr ln nht l 50 v 49cu 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

----------------------------------------------------------p n s xxvii

Cu 1: (3).a. V c s cho (1,5).- S hc sinh thch ng 2 mn bng v bi: 14 10 = 4(hs)

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Tng hp thi hc sinh gii ton 6- 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 sT 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. Trongs A c 6 ch s 0 nhng c 5 ch s 0 ng trc cc ch s 5152 53 . 58 59 60. Trong s nh nht c 5 ch s 0 ng trc s nh nht ls 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 nglin nhau th s l: 99999960S ny ch c 8 ch s khng tha mn. S ln nht ch c 5 ch s 9 lin nhau s c dng99999.Cc ch s cn li 78 59 60.

Vy s ln nht: 99999785860.Cu 2: (2,5).a.(1,5).A = 5 + 52 + + 5965A =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 l0.Vy: Ch s tn cng ca A l 0.b. (1).C: 6n + 3 = 2(3n + 6) 9 6n + 3 chia ht 3n + 62(3n + 6) 9 chia ht 3n + 6

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Tng hp thi hc sinh gii ton 6c) 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 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 20039 910 1 10 1>+ + 10A > 10BA > BBi 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


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Tng hp thi hc sinh gii ton 6 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= 16Vy 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)

=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 1 < S < 2

Bi 3:Gi gi go np l a (ng/kg) ; khi lng go np mua

l b (kg) (0,25 im)

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Tng hp thi hc sinh gii ton 6Suy 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 10096

..100

120

..100

80

=ba a.b(0.75im)Vy ngi th hai tr t tin hn ngi th nht . T l % t hnl:

%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 nmtrn 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)

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30 de HSG Toan 6 Va Dap An

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