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"!# %$
&'&(
)+*-,/.1032-4657*-8:9@*=BA39C4D8E9C8GFH*4JI70LK-M NPO-QRK-S@8"A:2 ,}9>Ek jV032E=1.m*4b.J=@* .w.1032LA:9.m.J9Cn ,}2 d{.ID9C4D8:2a4*-8:o+>@4 T .D2a= o:9>@8p.1032@8C;:nEA:2a4m=];:= T 8:xw.1032d9 T 8:x|*-4D4m*-8:xa2-nE2-8r2=22|.10C*a.
D
6 (a, b) ≡ (2, 1) (mod 5) k Z032a8
n =
Q
n = c 2 + d2
a = 5k + 2 b = 5t + 1
Q->r032-4J2
Q>r032-4J2
c = 2k + t
Q
k t∈
*8:o
Q
d = k − 2t
k
kZ032-4J2 d9C4J2CQ
a2 + b2 = 5k2 + 4k + 5t2 + 2t + 1 = (2k + t + 1)2 + (k − 2t)2 5
d}4J9Cn >r0 T I70r>r2=22|.10C* .
n = c 2 + d2
Q>r032-4J2
c = 2k + t + 1
*8:o
Q
d = k − 2t
k
³3
D
6 (a, b) ≡ (3, 1) (mod 5) k Z032a8
n =
Q
a = 5k + 3 b = 5t + 1
D
I (a, b) ≡ (4, 2) (mod 5) k Z032a8
n = c 2 + d2
Q
a = 5k + 4 b = 5t + 2
kZ032-4J2 d9C4J2CQ
Q>r032-4J2
Q>r032-4J2
c = 2k − t + 1
Q
k t∈
*-8:o
Q
D
6 (a, b) ≡ (4, 3) (mod 5) k Z032a8
n = c 2 + d2
Q
a = 5k + 4 b = 5t + 3
Q>r032-4J2
Q>r032-4J2
kZ032-4J2 d9C4J2CQ
c = 2k + t + 2
Q
k t∈
k
d = 2t + k + 1
a2 + b2 = 5k2 + 8k + 5t2 + 4t + 4 = (2k + t + 2)2 + (2t − k)2 5
d}4J9Cn >r0 T I70r>r2=22|.10C* .
n =
Q
k t∈
a2 + b2 = 5k2 + 6k + 5t2 + 2t + 2 = (2k − t + 1)2 + (2t + k + 1)2 5
d}4J9Cn>r0 T I70|>r2=22z.10C* .
n =
Q>r032-4J2
*8:o
Q
d = 2t − k
k
kZ032-4J2 d9C4J2CQ
a2 + b2 = 5k2 + 8k + 5t2 + 6t + 5 = (k + 2t + 2)2 + (2k − t + 1)2 5
d}4J9Cn>r0 T I70|>r2=22z.10C* .
n = c 2 + d2
Q>r032-4J2
c = k + 2t + 2
*-8:o
d = 2k − t + 1
Q k
Z032a4J2ad9C4J2r032a4J2 Q Q Q:.1032-8 2 2 a b n ∈ n = c 2 + d2 d9C4?=9CnL2 Q k (a + b )/5 = n
% $#9c$#% d6), ∈> B ; : F / ( 1 #) , :): F C F.L L?:+HI$& / (% : 0',$B / ("> M#)H3'( $#)1 #( J " C L FEJ L+:(?&,$/'( 6,$#0 % &4#)(;>9:J!&I3 ?: F$2!&&IM?3 JK( & , (?2 2=# (&5N - % , A & , I&$& #%( '& #) & %4( : 2 %I#)( : F -"/ / ( M H 0' ,$B 1 J 1 C L F'J L+: ,$/;6),$0 &#)(':KJ!?: F$ F6) ,$ 0 @ / @ 'H , KH ;* :/ $&=;, > , (?/ & :/ F /0B/ / ,6 #/0"#, % % , E /, : F #: #)(; (?> ) E'# ( # LNM@ ! :# # 6,$0 :#: F$ / ("> L , ,(?6 & : , (?> 0 / B/ / > C T o T .10VID2-8 T .10'ID2-8r9C41YE> T .10G.1032 4J2-=1;3,P. T 8:x@21ug34J2-=7= T 9C8
X
4(x3 + y 3 ) ≥ x3 + 3x2 y + 3xy 2 + y 3
k
Z0 T =yID*-8|>r9C4DY:Q-A@= T .m=X2 ,{d6>r032a8*nL9C4J2 I19Cng3, T IJ*a.J2o27ug:4J2a== T 9C8ID*-8+A:2c4J2-*-4D4m*-8:xa2o T 8r2@o T oE2a*4m, T 2-4j6*-8:og3;3,P,:.10 T =?.D2a4Dns*g3*4b.k6)G2@xa2.
1 (k + 1) − k k+1 k 1 1 = = − = − k(k + 1) k(k + 1) k(k + 1) k(k + 1) k k+1
k
³3S ¡|9>QCg34J93I12-2o T 8:x*= T 8E9r2@xa2 . 2007 X k=1
2007 X 1 1 1 1 2007 = − = 1− = k(k + 1) k k+1 2008 2008
k
k=1
eV= T n T ,{*4 T o32-*r*-g:gC, T 2-=?.D9g:4J93or0 T I70"4J2ag:4J2a=X2a8@4 T .m.D2a8E9 T .10"9C4b.10393I12a8 '# t @M
kc T 8:oE.1032x 4J2-* D. 2a=1.B4J2a*-,8C;:nLA32-4 =1;RI70E.10C*a.XQRd9C4B2 2a4W[+g:9C= T . T/ 2 Q Q*-8:o K u v > T .10 2 Q u > 4vw
u2 − 4vw
51;:=1. T dP[c[C9 T .104J2-*-=X9C83=QC*a,{, T 8 rt @M
^M
k~Z032w8C;:nEA:2a4m= Q Q Q*-8:o *4J2w*a,{, T 8:x|4J2a,{*a. T 9C83=039r2a·,P·,}q7·Y= 8:9r>@p8=4J2a−1 ,{*a. T 9C83=6A:r2p+1 .b>r2= 2a8r4Jp+2 939.m== *8:·o n P
i=1
ri = −a1
*8:o
P
r i r j = a2
1≤i<j≤n
n *-,P;R2-=9r2@93Ar 2@15 xa2. cos 15 = 2pq sin 30 = pq 4BCD
4 = p2 + q 2 − 2pq cos(θ + 90◦ ) = p2 + q 2 + 2pq sin θ
k
³:K 9C83=2r2.J;:4D8.J9=X9CnE2r=X9 T ID2 T 8.10312 2 5 g364J93oC7;RIJ.z9r9cd}*IJ.J9C4m=-k d9C4D3n&49r0 T ,}2*8[ k.10C4J≥ < 22G39r2 T 2 2 2 y (b2 − y0 )Q2>r= 0 (b −.10Cy*a0.y)(b − yT .J0;R− 1)d}4J9C =n 0 T =2-C;:b*a2,:6= y0k 930Ar293A T .10
*-8:o
( ABCD ABQ:: 2AD kw´ /T 2a)8E.10C* . QR*-8:oo T *xa9C83*a, =8:oE .1032c*-4J2-* BC = 2CD ∠BAD = α AC = d 9r0 T I70 T .z4J2-*o T ,/[d9@=z.10C*a. k]) T .10L\WK`jf*-8:o\W`j1Q 8(AD 2 + CD 2 ) = 5d2 .10 T = T nnL2-o T a* .J2a, [r[ T 2a,ho3= [ABC] =
4 5
· 58 d2 · sin α =
»+$+ $
Z0C* .VID9CnEgC,}2.D2a=v.1032 =X9r9@*-4Jor.J9c4J2-*o T 8:xB.10328:2`>FH*4b. T 8 ´ *-4Jo38:2a4pA39393Y+.10C*a.cIJ*nE2.J9".1032+g3;RAr0 T I70=* . T =1dP[ 2 k 2 =.10C z* 2 . 9C83=D.D4J;RIJ.|x*y , T 8:2=X2-x znL2a8 T .10,}2a8:x`.10 kG±-2 . A32*8[Gg:x9 T 8 & 3 / & & 3' -"0 #)% ,$5 / -, 2 / %.2 / , # $ & 3 ,D5 # 0, E , ( ,$0/ % 0, %I&& 3 / & / - %3 / ( /0>)I& 4E'# 0A , # $ 24)0 % 2 &5 # $4%I,$#'02 , (?& 2 , C!% 3 E/ 3? & % / >) /#'59>) , & ,$> 0 % , !#& )3 (G/ ( # & C3 :2 '&3 (", & >)> #(?)%#'&?E 6,/