answer61 - 6101 AB yB 1504 , 1 b < a 2008 B x = a 2...

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Unformatted text preview: 6101 AB yB 1504 , 1 b < a 2008 B x = a 2 - b 2 = (a + b)(a - b) 2008 x+y=a2, y=b2, a + b ( a - b) + 2 B B a+bB aB bN 3 a + b 2007 B (1), aB b =1, a + b a+bB 3, 5, 7, ..., 2007, 1003 P xB 1003 , 4 a + b 1004 B (2), aB b =2, a + b N a+bB 4, 6, 8, ..., 1004, 501 P xB 8, 12, 16, ..., 2008 , 501 , * (3) x P x = a2 - b2 a + b 3 B B aB b =, a+b =, (1), 2 2 x = a - b = (a + b)(a - b) 4 , aB b =, a+b =, a -b 2B a +b 4B x8B (2), N @ E @EK @ ' @ @ ' @ ExX =* * 8 8 q 0n @ 77' @ ExX\+p q E p 2, xB 1003+501=1504 , 13 <7 1, 2, 3, ..., 2008, B B x+y B * BR 12, 22, 32, ..., 20082, x " " P AB A P xB B BB B ? , N 6102 , a, b, c * N B , ? - 1+ 2 2 a = ( 2 - 1)c , b=c 2b - 2c 2a + 4c b + + a + b + 2c a + 2b + c a + b + c a a, b, c , * 1. a+b+2c = xB a+2b+c = yB a+b+c = zB x,* y, z B a = xy+3z, b = yz, c = xzB 2b2c = 2(yx) , 2a+4c = 2(xy+z) , b=yzB 2b - 2c 2a + 4c b + + a + b + 2c a + 2b + c a + b + c 2( y - x) 2( x - y + z ) y - z 2 y 2 x 2z y + + = =( + ) + ( + ) -5 x y z x y y z 2 y 2x 2z y ( + ) + ( + ) - 5 2 4 + 2 2 - 5 = 2 2 -1 2.h x y y z 2 y 2x 2z y = = , x = yB 2 z = y h x y y z B a + b + 2c = a + 2b + c , 2 (a + b + c) = a + 2b + c a = ( 2 - 1)c h b=c - 1+ 2 2 7d @Ex 8 = 0q*nc 6103 x - y + z - w = -2 x 2 - y 2 + z 2 - w 2 = -2 3 r* 8S 3 3 3 x - y + z - w = -8 x 4 - y 4 + z 4 - w 4 = -14 a, b, c A r 1 *йE A r (x, y, z, w) ? (1, 0, 1, 2) , (1, 2, 1, 0) , (1, 2, 1, 0) , (1, 0, 1, 2) [, 1] x - y + z - w = -2...(1) x 2 - y 2 + z 2 - w 2 = -2...(2) 1. 3 3 3 3 x - y + z - w = -8...(3) x 4 - y 4 + z 4 - w 4 = -14....(4) (2) - (1) ( x2x )( y2y )+( z2z )( w2w ) = 0 , ( x2x3 )( y2y3 )+( z2z3 )( w2w3 ) = 0...(5) (3) - (2) - 3 (1) ( x3 x23x)( y3y23y)+( z3z2 3z )( w3w2 3w ) = 0 , ( x2x3 ) x( y2y3 ) y +( z2z3 ) z( w2w3 ) w = 0 ...(6) (4) - (3) - 3 (2) ( x4 x33x2)( y4y33y2)+( z4z33z2 )( w4w33w2 ) = 0 , ( x2x3 ) x2( y2y3 ) y2 +( z2z3 ) z2( w2w3 ) w2 = 0 ...(7) (7) - (6) - 3 (5) ( x2x3 )2( y2y3 )2+( z2z3 )2( w2w3 )2= 0...(8) 2., m = x2x3, p = y2y3, n = z2z3, q = w2w3 m- p+n-q =0 m+n = p+q m + n = p + q 2 , (5), (8), 2 2 2 2 2 2 2 m - p + n - q = 0 m + n = p + q mn = pq m+n = p+q m+n = p+q B m = p, n = q B m = q, n = pB 2 2 (m - n) = ( p - q ) m - n = ( p - q ) X*i ,, (1) , m = p, n = qB m = p B x2x3= y2y3 x2y2 = xy (xy)(x+y1) = 0 (i), x y = i * ,, 0X (1), z w = 2, 2 2 ,,iX n = qB z z3= w w3 z2w2 = zw (zw)(z+w1) = 0 1 3 , z w 0, z + w = 1, z w = 2, z =- w = !r 2 2 , , * (3) S r (ii), x + y 1=0, , n = qB (zw)(z+w1) = 0, z w 0(, x + y =1 z + w =1 , (i), ), z + w = i * ,, 1X x - y + z - w = -2 B ! r , y = 1 x , z = x , w = 1 + x B B (3), x3(1x)3+(x)3(1+x)3 = 8 x2 = 1 x = 1, (x, y, z, w)= (1, 0, 1, 2), (1, 2, 1, 0) x + w =1 y + z =1 (1), B y = 1 + x , z = x , x - y + z - w = -2 B (3), x3(1+x)3+(x)3(1x)3 = 8 x2 = 1 x = (2) , m=q, n=pXi * ,, w = 1 x B ! r 1, , (x, y, z, w)= (1, 2, 1, 0) , (1, 0, 1, 2) 3. (x, y, z, w)=(1, 0, 1, 2), (1, 2, 1, 0), (1, 2, 1, 0), (1, 0, 1, 2) [, 2], a=x+zB b=xzB c=y+wB d=yw a - c = -2 (a 2 - c 2 ) - 2(b - d ) = -2 2 2 a (a - 3b) - c (c - 3d ) = -8 (a 2 - 2b) 2 - 2b 2 - (c 2 - 2d ) 2 + 2d 2 = -14 B a c = 2 a - c = -2 (a - c )(a + c ) - 2(b - d ) = -2 3 (a - c) + 3ac (a - c) - 3ab + 3cd = -8 (a 2 + c 2 - 2b - 2d )(a 2 - c 2 - 2b + 2d ) - 2(b 2 - d 2 ) = -14 a - c = -2 ( a + c ) + (b - d ) = 1 2ac + ab - cd = 0 [(a - c) 2 + 2ac - 2b - 2d ][(a + c )(a - c) - 2b + 2d ] - 2(b + d )(b - d ) = -14 a - c = -2 a + c + b - d =1 2ac + ab - cd = 0 [4 + 2ac - 2b - 2d ][a + c + b - d ] + (b + d )(b - d ) = 7 a b = -1 - 2 c = a + 2 ' d = 3a 2 a=0 b = -1 c=2 d =0 ,, aB b, c, d B B , @EA x+z =0 xz = -1 x =1 x = -1 y = 2 x = 0 B B B B B z = -1 z = 1 w = 0 z = 2 y + w = 2 yw = 0 (x, y, z, w)= (1, 0, 1, 2) , (1, 2, 1, 0) , (1, 2, 1, 0) , (1, 0, 1, 2), @7 7 E E @ @'@ @ 7 E 3 * ` *4 c ,, 6104 A' A P R D B' B ,, B AB B Q 1, C ABCD B BB ,, A ' B ' B AD CD B D RB B RBB BB PQ B AB ,, 1 A P S D' B A' R N D B' Q C C' (a), B B AB B A ' DS h ,, ' ABCD B PQ BB CD B ,,S A' B 'C ' D ' B B C B,, S h D SB A ' B ' B AD h ,,S R B B ' CQ BC ' Q B ' C = BC ' (b), * ,,B h i A' B ' B N , BNB ' C ' B S ,, NB ' = BC ' = B ' C BCB ' = 90 = BNB ' (c), BB ' = BB ' CB ' = NB ' V BCB ' BNB ' BC = BN RAB = 90 = RNB (d), BA = BC = BN BR = BR V BRA BRN RA = RN D (e) RBB = NR + RD + DB ' + B ' N = AR + RD + DB ' + B ' C = AD + DC = 1+1 =2 2 ,, A P A' R N D B' B Q C BR, BBB (a) , BN RBB NB B B'B B B PQ ,, PQ B BB ' B (b) , *S ' Q = ' BQ BB B B ' B = ' Q - ' Q = 90 ' Q = 90 ' BQ = ' B V NB NB BB - BB - B CB (c) , BCB ' = 90 = BNB ' V BB ' = BB ' BNB ' BCB ' AAS, BN = BC = 1 = BA V NB ' = B ' C RAB = 90 = RNB V AB = NB (d) , ABR VNBR V RHS, AR = RN D (e) RBB = NR + RD + DB ' + B ' N = AR + RD + DB ' + B ' C = AD + DC = 1+1 =2 , 3 CQ = x, DB ' = y QB ' = BQ = 1 - x QCB ' CB ' = 1 - y (1 - x ) = x + (1 - y ) 1 - 2 x + x2 = x2 + 1 - 2 y + y 2 y 2 = x 2- y 2 2 2 BR = BR (c), (d), A' A P R D y B' 1-y B 1-x Q x C QCB ' ~ B ' DR RBB D = B ' QC B y x y x = ((1 - x) + x + (1 -* y )) = (2 - y ) =2 2 2- y @E'xX\+p @ 0q*n c = 11 77 E( @N @EPxX\+p @ =U\+] d 7, =U\+ ] 8 *йEA 2A *йE 11 5 E@ n 0q*= 8 6, @'E'xX\+p H* @'E'xX\+p , 2, @'E'xX\+p @EMxX\+p' 1, @E77 @ENxX\+pN 0, n 0q*= 8 * йE 6105 2008, , 1 , 2008 @Ex 7d 504 , (1) , 1 , 2008 , 2008 ~ 501 , 2 ~ *i i 2 + 2006, 4 + 2004, 6 + 2002, 8 + 2000, ..., 1000 + 1008, 1002 + 1006 (2)~ 2, 4, 6, 8, ..., 1002, 1006, 1008, ..., 2002, 2004, 2006 , 1002 , * i 502 8 + \ U = ] 2008, i~ (3) , (2), 2, 4, 6, 8, ..., 1002, 1004, 1006, 1008, ..., 2002, 2004, 2006, 2008 , 1004 , 504 8 + \ U = ] 2008, 7 ' , 1. E * 8 a E @ 2. , 7, 7 7 @ '@ @ 7 (11 , ), @'@ ~ E + p \ X E H 77 @ * 17 ~ i AP = x, RB ' = 1 - x * йE N B Q H* , E @ 77 @ 1004 , 2008 7 ' 1004 , 2008 , 2008~ * i 1 , 2, @ N + 5, ' @ a , 3, = (1 , ) (5 , ), $ N # E E p + \ X ...
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This note was uploaded on 03/25/2011 for the course MATH 1232 taught by Professor Dr.jiangzhengchien during the Spring '11 term at Nanjing University.

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