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¥Ñ yll ©ó ¬P´Á¤é ¥|¤ë 27, 2003 5:53 pm


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                 ¼ÊºÙ¡G___________ ½s¸¹¡G____________

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  ¡i°ÝÃD3¡jX+Y+2Z=-2
           X+2Y+3Z=a
           X+3Y+4Z=b
           X+4Y+5Z=b2

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  ¡i²Ä¤@ÃD¡j³]¨ç¼ÆFº¡¨¬F(1)=249¡A¥BF(n+1)=21+4F(n),n=1,2,3,¡K,¸Õ¨DF(2003)
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¥Ñ ---- ©ó ¬P´Á¤é ¥|¤ë 27, 2003 7:41 pm


1. xy+x+y=99
(x+1)(y+1)=100
100=2^2 * 5^2
¬G¦³(2+1)(2+1)=9­Ó¦]¼Æ
±Æ°£1 ©M100 ¨â­Ó¦@¦³7­Ó
§Yx+1 ¦³7­Ó¥i¯à©Ê¡A¬G¨½(x,y)¦³7²Õ¸Ñµª

2.³]¤@®Ú¬°a, ¥t¤@®Ú¬°2a¡A«h¦³
p=-3a
q=2a^2
§Y9q=2p^2

3.X+Y+2Z=-2 ...(1)
X+2Y+3Z=a ...(2)
X+3Y+4Z=b ...(3)
X+4Y+5Z=b^2...(4)

y+z=(2)-(1)=(3)-(2)=(4)-(3)
§Ya+2 = b-a = b^2-b
¥Ña+2=b-a ±o2a+2=b
§â2a+2=b¥N¤Jb-a=b^2-b
2a+2-a = 4a^2+8a+4-2a-2
4a^2+5a=0
a(4a+5)=0
¦]a«D¾ã¼Æ¡A¬Ga=-5/4

¥N¤Jb=2a+2,
b=-1/2
(a,b)=(-5/4, -1/2)

4.
³]¤T¨¤§Î¤TÃ䬰a,b,4
­Ya<4a+4>b
«h(a,b)=(2,5),(3,5),(3,6)¤T²Õ¸Ñ
­Ya<b<=4
a+b>4
«h(a,b)=(1,4),(2,3),(2,4),(3,2),(3,3),(3,4)¤»²Õ¸Ñ
¦@¦³9²Õ¸Ñ

5.
x^5+x^4+1
=x^5+x^4+x^3-x^3+1
=x^3(x^2+x+1)-(x-1)(x^2+x+1)
=(x^3-x+1)(x^2+x+1)

6.¥Ñ°t¤èªk¥iª¾
x±o³Ì¤p­È®Éµª®×¬°-b/2a =-2001/-1998 = 667/666
³Ì±µªñ¾ã¼Æ­È¬°1¡A¬Gx=1

7.3x^4+42x^3+204x^2+390x+225
=3(x^4+14x^3+68x^2+130x+75)
¥N¤Jx=-1 ¤W¦¡¬°0¡A§Yx+1 ¬°¦¹¦¡ªº¦]¤l¡A
¥Ñªø°£ªk±o¤W¦¡µ¥©ó
=3(x+1)(x^3+13x^2+55x+75)
¥N¤Jx=-3 ¤W¦¡¬°0¡A§Yx+3¬°¦¹¦¡¦]¤l¡A
¥Ñªø°£ªk±o¤W¦¡µ¥©ó
=3(x+1)(x+3)(x^2+10x+25)
=3(x+1)(x+3)(x+5)^2
¥N¤Jx=24
=3(25)(27)(29)^2
=3^4 * 5^2 * 29^2
¶}¤è«á±oA=3^2*5*29=1305

8.³]M¬OBÂI¦bMªº««¨¬¡C
EF*BH/2 = 40
BH=8/7
©úÅã¦a¡AABDE¬O­Ó¥­¦æ¥|Ãä§Î¡A¬G¨ä­±¿n¬°¡G
1*80/7 = 80/7

1.
F(n+1)=4F(n)+21
F(n+1)+7=4F(n)+28
F(n+1)+7=4(F(n)+7)
³]g(n)=F(n)+7
g(n+1)=4g(n)
g(n)=4^(n-1) * g(1)
=4^(n-1) * (249+7)
=4^(n-1)*256
=4^(n-1)*4^4
=4^(n+3)
g(2003)=4^2006 = (4^3)^668 * 4^2 = (1024)^668 * 16
= 24^668 * 16 (mod 100)
= 576^334 * 16 (mod 100)
= 76*16 (mod 100)
= 16 (mod 100)
F(2003)+7=16(mod 100)
F(2003)=9(mod 100)
¬Gf(2003)ªº¥½¨â¦ì¼Æ¬O09

2.­Yb=1®É¡A1*aªºªø¤è§Î¥i§@¥H¤U¤ÀªR¡G
1*aªºªø¤è§Îªº­Ó¼Æ¬O1­Ó
1*(a-1)ªºªø¤è§Îªº­Ó¼Æ¬O2­Ó
1*(a-2)ªºªø¤è§Îªº­Ó¼Æ¬O3­Ó
...
1*1ªºªø¤è§Îªº­Ó¼Æ¬Oa­Ó¡A
¬G¦@¦³ªø¤è§Î(1+...+a)­Ó
¨Ì¦P¼Ë­ì²z¡A·ía=1®É¡A1*bªºªø¤è§Î¦@¦³(1+...+b)­Ó
§Ya*bªºªø¤è§Î¦@¦³¯x§Î(1+...+a)(1+...+b)=ab(a+1)(b+1)/4­Ó

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¥Ñ kevin ©ó ¬P´Á¤é ¥|¤ë 27, 2003 9:30 pm


¶ñ¥R
1.6
2.
3.
4.
5.
6.
7.
8.15
­pºâ
1.
F(n)=21+4*21+4*4*21+..............+4^(n-2)*21+4^(n-1)*249
F(2003)=21+4*21+4*4+21+.........+4^2001*21+4^2002*249
¥Ñ©ó4+4*4+4*4*4+........+4^10ªº§À¼Æ2¦ì=00
©Ò¥H±q4*21+4*4*21+......+4^2000*21ªº§À¼Æ¤]¬O00
©Ò¥H¤£¥Î¦Ò¼{³o³¡¤À
21+4^2001*21+4^2002*249
=201+4*21+16*249=4089
¦]¦¹§À¼Æ¬°89
2.
ªø¤À¦¨a¥÷.¼e¤À¦¨b¥÷
ªø¥ô¨ú­Y¤z¥÷¦@¦³a(a+1)/2ºØ
¼e¥ô¨ú­Y¤z¥÷¦@¦³b(b+1)/2ºØ
ªø©M¼e³£¥ô¨ú­Y¤z¥÷«K¥i¥H²Õ¦¨¯x§Î
¦]¦¹¦@¦³a(a+1)b(b+1)/4­Ó¯x§Î
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kevin
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µù¥U®É¶¡: 2002-12-22

¥Ñ ming ©ó ¬P´Á¤é ¥|¤ë 27, 2003 10:48 pm


­×§ï
1.9
2.q=2/9p^2
3.a=-5/4,b=-1/2
4.8
5.(x^3-x+1)(x^2+x+1)
6.x=1
7.1305
8.15

¤@89.f(1)§À¼Æ¬O49,f(2)§À¼Æ¬O17,¸òµÛ¬O89,77...57,f(11)¤S¬O49,­«½ÆµÛ.¦]¦¹f(2003)=f(3)=89(mod100)
¤G:¤@­Ó¤@­Ó¯x§Î¥h¼Æ,¦³ab­Ó,
¨â­Ó¨â­Ó¥h¼Æ,¦³a(b-1)+b(a-1)­Ó,
¤T­Ó¤T­Ó¥h¼Æ,¦³a(b-2)+b(a-2)­Ó,
¥|­Ó¥|­Ó¥h¼Æ,¦³a(b-3)+b(a-3)+(a-1)(b-1)
¤»­Ó¤»­Ó¼Æ¦³a(b-5)+b(a-5)+(a-1)(b-2)+(a-2)(b-1)
³]n¦³n1,n2...nk,k­Ó¦]¼Æ,n1*nk=n2*n(k-1)....
¥Hn­Ón­Ó¼Æ,¦³,(a-n1+1)(b-nk+1)+(a-nk+1)(b-n1+1)...
¥Hab­Óab¥h¼Æ,¦³(a-a+1)(b-b+1)=1­Ó
µª®×¬O³o©Ò¦³¼Æ¦rªº©M

ming
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¥Ñ Herbie ©ó ¬P´Á¤@ ¥|¤ë 28, 2003 1:03 pm


go go go
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¥Ñ E.T ©ó ¬P´Á¤@ ¥|¤ë 28, 2003 2:28 pm


­«µª ><
¡i°ÝÃD1¡j9
¡i°ÝÃD2¡j2p^2 /9 = q
¡i°ÝÃD3¡j( 1, 2)
¡i°ÝÃD4¡j11
¡i°ÝÃD5¡j( x^3 - x + 1 )( x^2 + x + 1)
¡i°ÝÃD6¡j1
¡i°ÝÃD7¡j1305
¡i°ÝÃD8¡j15



¤G¡B¡i²Ä¤@ÃD¡jF(1) = 249 , F(2) = 1017 , F(3) = 4089 , F(4) = 16377
F(5) = 65529 , F(6) = 262137 , F(7) = 1048569 , F(8) =4194297
F(9) = 16777209 , F(10) = 67108857 , F(11) = ...49 , F(12) = ...17 , ....

So F(2003) = F(3) = ¥½¨â¦ì¼Æ 89 ( mod 100 )

¡i²Ä¤GÃD¡j( a+b-1)(a-b+1)
¡¸Click this : [>> ¦³½ìªº¬ì¾Ç(Science's World) <<]

Let's go to discuss ~*

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E.T

 
¤å³¹: 1486
µù¥U®É¶¡: 2003-03-29
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¥Ñ Raceleader ©ó ¬P´Á¤T ¥|¤ë 30, 2003 10:00 pm


¡i°ÝÃD1¡j6²Õ

¡i°ÝÃD2¡j2p2=9q

¡i°ÝÃD3¡ja=-5/4,b=-1/2

¡i°ÝÃD4¡j8

¡i°ÝÃD5¡j(x3-x+1)(x2+x+1)

¡i°ÝÃD6¡jx=1

¡i°ÝÃD7¡j1305

¡i°ÝÃD8¡j15

¡i²Ä¤@ÃD¡j
F(1)=249¡AF(n+1)=21+4F(n)¡An=1,2,3,...
¦pªG¥½¨â¦ì¼Æ©T©w¡A¥¼¥¦¼Æ¥i¥ô·N¡A¨º»ò¥½¨â¦ì¼Æªº4­¿¤]¬O©T©w
³]G(n)=F(n)¤§­Èªº¥½¨â¦ì¼Æ
G(1)=49
G(2)=17
G(3)=89
G(4)=77
G(5)=29
G(6)=37
G(7)=69
G(8)=97
G(9)=09
G(10)=57
G(11)=49
...

©Ò¥H:
G(10k+1)=49
G(10k+2)=17
G(10k+3)=89
G(10k+4)=77
G(10k+5)=29
G(10k+6)=37
G(10k+7)=69
G(10k+8)=97
G(10k+9)=09
G(10k+10)=57
k=0,1,2,...

G(2003)=G(10*200+3)=G(30)=89
F(2003)¤§­Èªº¥½¨â¦ì¼Æ=89

¡i²Ä¤GÃD¡j
°²³]am=bn¡A(m,n)¨S¦³¾ã¼Æ¸Ñ

¦pªG¯x§Î¤À¦¨ªøaµ¥¥÷¡A¼ebµ¥¥÷
³]F(h,k)¬°ªøhµ¥¥÷¡A¼ekµ¥¥÷ªº¤p¯x§Îªº¼Æ¥Ø
¨º»òF(h,k)=(a-h+1)(b-k+1)

F(1,1)=ab
F(1,2)=a(b-1)
F(1,3)=a(b-2)
...
F(1,b)=a(b-b+1)=a
S(1)=F(1,1)+F(1,2)+F(1,3)+...+F(1,b)=a[1+2+3+...+b]=(1/2)ab(b+1)

F(2,1)=(a-1)b
F(2,2)=(a-1)(b-1)
F(2,3)=(a-1)(b-2)
...
F(2,b)=(a-1)(b-b+1)=a-1
S(2)=F(2,1)+F(2,2)+F(2,3)+...+F(2,b)=(a-1)[1+2+3+...+b]=(1/2)(a-1)b(b+1)

F(3,1)=(a-2)b
F(3,2)=(a-2)(b-1)
F(3,3)=(a-2)(b-2)
...
F(3,b)=(a-2)(b-b+1)=a-2
S(3)=F(3,1)+F(3,2)+F(3,3)+...+F(3,b)=(a-2)[1+2+3+...+b]=(1/2)(a-2)b(b+1)

...

F(a,1)=b
F(a,2)=(b-1)
F(a,3)=(b-2)
...
F(a,b)=(a-a+1)(b-b+1)=1
S(a)=F(a,1)+F(a,2)+F(a,3)+...+F(a,b)=[1+2+3+...+b]=(1/2)b(b+1)

Á`¼Æ=S(1)+S(2)+S(3)+...+S(a)=(1/2)b(b+1)[1+2+3+...+a]=(1/4)ab(a+1)(b+1)

Raceleader
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¥Ñ µY~~ ©ó ¬P´Á¤­ ¤­¤ë 02, 2003 4:52 pm


¼ÊºÙ¡G___µY~~________ ½s¸¹¡G____________

¡i°ÝÃD1¡j
1, 50    
3, 24
4,19
9,9
19,4
24,3
50,1

¦@7²Õ

¡i°ÝÃD2¡j
Let 2 root be a and 2a

3a=-p
a= -p/3


2a 2 =q

a2 = q/2

a= (q/2)^1/2

-p/3= (q/2)^1/2



¡i°ÝÃD3¡j

Y+Z=a+2
2Y+2Z=b +2
3Y+3Z=b2 +2

2(a+2)= b +2
2a+4= b +2
2a+2=b-------(1)

3(a+2)= b2 +2
3a+6= b2 +2
3a+8= b2 -------(2)

4a2+8a+4=3a+8
4a2  +5a-4+0

a= (-5  +/- ( 25-4*4*(-4)) !/2)  / 2*4
  = (-5 +/-(89) !/2  ) /8

b = 2((-5 +/-(89) !/2  ) /8)+2
   =(-5 +/-(89) !/2  ) /4+2


¡i°ÝÃD4¡j
WHEN¤TÃäªø, 3,4,5



¡i°ÝÃD5¡j
   x5+(x2+1)( x2-1)
= x5+(x2+1)( x+1)(x-1)

¡i°ÝÃD6¡j


¡i°ÝÃD7¡j1305

¡i°ÝÃD8¡j


¤G¡B
¡i²Ä¤@ÃD¡j¥BF(n+1)=21+4F(n),n=1,2,3,¡K,¸Õ¨DF(2003)
249 = 21+4F(n),
F(n)=57

F(2003)=57*2003=114171

The anawer =71

¡i²Ä¤GÃD¡j
¤j¯x§Î+ ab+ ab/2­Ó*a*2b(¤W¤U2­Ó¤p¯x§Î¦¨¤@­Ó¯x§Î)+ ab/2­Ó*2a*b(¥ª¥k2­Ó¤p¯x§Î¦¨¤@­Ó¯x§Î)+¡K

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¥Ñ ¯Á¬¥¦è ©ó ¬P´Á¤é ¤­¤ë 04, 2003 4:56 pm


1.  ¥u¦³1²Õ
2.  ¥O¸û¤pªº¤@®Ú¬°y,«hp=2[y+(3/4)]2+(9/8)-q
3.  
4.  8­Ó
5.
6.
7.  1512000®Ú¸¹5
8.  15
9.  F(2)¥½2¦ì=21+4*49=>17
     F(3)¥½2¦ì=21+4*17=>89
     F(3)¥½2¦ì=21+4*89=>77
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¥Ñ Herbie ©ó ¬P´Á¤» ¤»¤ë 07, 2003 7:56 pm


µª®×¤½§G¦b

http://netcity5.web.hinet.net/UserData/84536923/mathtest02.doc

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