1.Introduction

I found some parameter solutions for 5.6.6, 6.8.8, 8.10.10, 9.10.10 and 10.12.12.


2. Theorem
          
      (1).   There is a parameter solution for the equation.
         
         A15+ A25+ A35+ A45+ A55+ A65 = B15+ B25+ B35+ B45+ B55+ B65



 (1+a1x)5+(1-a1x)5+(1+a2x)5+(1-a2x)5+(1+a3x)5+(1-a3x)5-(1+b1x)5-(1-b1x)5-(1+b2x)5-(1-b2x)5-(1+b3x)5-(1-b3x)5
=(10a14-10b24-10b14+10a34-10b34+10a24)x4
+(-20b22-20b12-20b32+20a12+20a22+20a32)x2

It comes to solving the next simultaneous Diophantine equation.

     a12+a22+a32 = b12+b22+b32

     a14+a24+a34 = b14+b24+b34
 
There are many solutions of the above simultaneous Diophantine equation. 


      For example,  (a1,a2,a3,b1,b2,b3)=(10,1,9,11,5,6)

           102+  12+  92  = 112+  52+  62

           104+  14+  94  = 114+  54+  64

We obtain a parameter solution for (1).

(1+10x)5+(1-10x)5+(1+x)5+(1-x)5+(1+9x)5+(1-9x)5 = (1+11x)5+(1-11x)5+(1+5x)5+(1-5x)5+(1+6x)5+(1-6x)5

 
      (2).   There is a parameter solution for the equation.
         
         A16+ A26+ A36+ A46+ A56+ A66+ A76+ A86 = B16+ B26+ B36+ B46+ B56+ B66+ B76+ B86



 (1+a1x)6+(1-a1x)6+(1+a2x)6+(1-a2x)6+(1+a3x)6+(1-a3x)6+(1+a4x)6+(1-a4x)6-(1+b1x)^6-(1-b1x)6-(1+b2x)6-(1-b2x)6-(1+b3x)6-(1-b3x)6-(1+b4x)6-(1-b4x)6

=(2a36-2b26+2a16+2a26-2b16-2b36-2b46+2a46)x6
+(-30b24-30b14+30a44-30b34-30b44+30a34+30a14+30a24)x4
+(-30b32-30b42-30b22-30b12+30a12+30a42+30a22+30a32)x2


It comes to solving the next simultaneous Diophantine equation.

     a12+a22+a32+a42 = b12+b22+b32+b42

     a14+a24+a34+a44 = b14+b24+b34+b44

     a16+a26+a36+a46 = b16+b26+b36+b46
 
There are many solutions of the above simultaneous Diophantine equation. 


      For example,  (a1,a2,a3,a4,b1,b2,b3,b4)=(25,21,16,2,24,23,14,5)

           252+  212+  162+   22  = 242+  232+  142+   52

           254+  214+  164+   24  = 244+  234+  144+   54

           256+  216+  166+   26  = 246+  236+  146+   56

We obtain a parameter solution for (2).

  (1+25x)6+(1-25x)6+(1+21x)6+(1-21x)6+(1+16x)6+(1-16x)6+(1+2x)6+(1-2x)6

 =(1+24x)6+(1-24x)6+(1+23x)6+(1-23x)6+(1+14x)6+(1-14x)6+(1+5x)6+(1-5x)6

 
      (3).   There is a parameter solution for the equation.
         
      A18+ A28+ A38+ A48+ A58+ A68+ A78+ A88+A98+ A108 = B18+B28+ B38+ B48+B58+ B68+ B78+B88+ B98+B108



 (1+a1x)8-(1-a1x)8+(1+a2x)8-(1-a2x)8+(1+a3x)8-(1-a3x)8+(1+a4x)8-(1-a4x)8+(1+a5x)8-(1-a5x)8
-(1+b1x)8+(1-b1x)8-(1+b2x)8+(1-b2x)8-(1+b3x)8+(1-b3x)8-(1+b4x)8+(1-b4x)8-(1+b5x)8+(1-b5x)8

=(16a27-16b47+16a37-16b57+16a17+16a57-16b37-16b17+16a47-16b27)x7
+(112a35+112a15-112b55+112a25-112b35-112b45-112b25-112b15+112a55+112a45)x5
+(-112b23+112a53+112a43-112b13+112a23+112a33+112a13-112b43-112b53-112b33)x3
+(-16b1+16a5+16a4-16b2-16b4+16a2-16b5+16a3-16b3+16a1)x



It comes to solving the next simultaneous Diophantine equation.

     a1+ a2+ a3+ a4+ a5 = b1+ b2+ b3+ b4+ b5

     a13+a23+a33+a43+a53 = b13+b23+b33+b43+b53

     a15+a25+a35+a45+a55 = b15+b25+b35+b45+b55

     a17+a27+a37+a47+a57 = b17+b27+b37+b47+b57
 
There is a solution of the above simultaneous Diophantine equation. 


For example,  (a1,a2,a3,a4,a5,b1,b2,b3,b4,b5)=(-59,-5,-1,33,57,-55,-23,13,39,51)(by G.Palama)

           -59+   (-5)+   (-1)+    33+    57  =  -55+   (-23)+   13+   39+    51

           -593+  (-5)3+  (-1)3+   333+   573  = -553+  (-23)3+  133+   393+   513

           -595+  (-5)5+  (-1)5+   335+   575  = -555+  (-23)5+  135+   395+   515

           -597+  (-5)7+  (-1)7+   337+   577  = -557+  (-23)7+  137+   397+   517

We obtain a parameter solution for (3).

 (1-59x)8-(1+59x)8+(1-5x)8-(1+5x)8+ (1-x)8-(1+x)8+(1+33x)8-(1-33x)8+(1+57x)8-(1-57x)8
=(1-55x)8-(1+55x)8+(1-23x)8-(1+23x)8+(1+13x)8-(1-13x)8+(1+39x)8-(1-39x)8+(1+51x)8-(1-51x)8
    

      (4).   There is a parameter solution for the equation.
         
      A19+ A29+ A39+ A49+ A59+ A69+ A79+ A89+A99+ A109 = B19+B29+ B39+ B49+B59+ B69+ B79+B89+ B99+B109



 (1+a1x)9+(1-a1x)9+(1+a2x)9+(1-a2x)9+(1+a3x)9+(1-a3x)9+(1+a4x)9+(1-a4x)9+(1+a5x)9+(1-a5x)9
-(1+b1x)9-(1-b1x)9-(1+b2x)9-(1-b2x)9-(1+b3x)9-(1-b3x)9-(1+b4x)9-(1-b4x)9-(1+b5x)9-(1-b5x)9

=(18a18+18a38+18a58-18b28+18a28+18a48-18b18-18b38-18b58-18b48)x8
+(168a36+168a56+168a46+168a16-168b46+168a26-168b56-168b16-168b26-168b36)x6
+(252a34+252a24+252a14-252b24-252b44+252a54+252a44-252b14-252b34-252b54)x4
+(72a32+72a12+72a52+72a22+72a42-72b12-72b22-72b52-72b42-72b32)x2


It comes to solving the next simultaneous Diophantine equation.

     a12+a22+a32+a42+a52 = b12+b22+b32+b42+b52

     a14+a24+a34+a44+a54 = b14+b24+b34+b44+b54

     a16+a26+a36+a46+a56 = b16+b26+b36+b46+b56

     a18+a28+a38+a48+a58 = b18+b28+b38+b48+b58

There is a solution of the above simultaneous Diophantine equation. 


For example,  (a1,a2,a3,a4,a5,b1,b2,b3,b4,b5)=(12,20231,11881,20885,23738,20449,436,11857,20667, 23750)(by A.Letac)

           122+ 202312+ 118812+ 208852+ 237382= 204492+ 4362+ 118572+ 206672+ 237502

           124+ 202314+ 118814+ 208854+ 237384= 204494+ 4364+ 118574+ 206674+ 237504
           
           126+ 202316+ 118816+ 208856+ 237386= 204496+ 4366+ 118576+ 206676+ 237506

           128+ 202318+ 118818+ 208858+ 237388= 204498+ 4368+ 118578+ 206678+ 237508

We obtain a parameter solution for (4).

 (1+12x)9+(1-12x)9+(1+20231x)9+(1-20231x)9+(1+11881x)9+(1-11881x)9+(1+20885x)9+(1-20885x)9+(1+23738x)9+(1-23738x)9
=(1+20449x)9+(1-20449x)9+(1+436x)9+(1-436x)9+(1+11857x)9+(1-11857x)9+(1+20667x)9+(1-20667x)9+ (1+23750x)9+ (1-23750x)9
 
         
 
      (5).   There is a parameter solution for the equation.
         
      A110+ A210+ A310+ A410+ A510+ A610+ A710+ A810+ A910+ A1010+ A1110+ A1210
    = B110+ B210+ B310+ B410+ B510+ B610+ B710+ B810+ B910+ B1010+ B1110+ B1210



 (1+a1x)10+(1-a1x)10+(1+a2x)10+(1-a2x)10+(1+a3x)10+(1-a3x)10+(1+a4x)10+(1-a4x)10+(1+a5x)10+(1-a5x)10+(1+a6x)10+(1-a6x)10
-(1+b1x)10-(1-b1x)10-(1+b2x)10-(1-b2x)10-(1+b3x)10-(1-b3x)10-(1+b4x)10-(1-b4x)10-(1+b5x)10-(1-b5x)10-(1+b6x)10-(1-b6x)10

=(-2b310+2a310-2b510+2a510+2a210-2b110-2b410+2a410+2a110+2a610-2b610-2b210)x10
+(-90b68-90b58-90b38+90a68-90b28+90a28-90b48-90b18+90a38+90a18+90a48+90a58)x8
+(-420b16+420a66-420b26+420a36-420b66-420b56+420a26-420b36+420a56-420b46+420a46+420a16)x6
+(-420b64-420b34+420a34-420b24-420b14-420b54+420a24+420a14+420a44-420b44+420a64+420a54)x4
+(-90b32-90b22-90b12-90b42-90b62+90a62+90a32-90b52+90a52+90a22+90a42+90a12)x2



It comes to solving the next simultaneous Diophantine equation.

     a12+a22+a32+a42+a52+a62 = b12+b22+b32+b42+b52+b62

     a14+a24+a34+a44+a54+a64 = b14+b24+b34+b44+b54+b64

     a16+a26+a36+a46+a56+a66 = b16+b26+b36+b46+b56+b66

     a18+a28+a38+a48+a58+a68 = b18+b28+b38+b48+b58+b68

     a110+a210+a310+a410+a510+a610 = b110+b210+b310+b410+b510+b610

There is a solution of the above simultaneous Diophantine equation. 


For example,(a1,a2,a3,a4,a5,a6,b1,b2,b3,b4,b5,b6)=(22,61,86,127,140,151,35,47,94,121,146,148)
(by Nuutti Kuosa, Jean-Charles Meyrignac and Chen Shuwen)


    222+612+862+1272+1402+1512 = 352+472+942+1212+1462+1482

    224+614+864+1274+1404+1514 = 354+474+944+1214+1464+1484

    226+616+866+1276+1406+1516 = 356+476+946+1216+1466+1486

    228+618+868+1278+1408+1518 = 358+478+948+1218+1468+1488 

    2210+6110+8610+12710+14010+15110 = 3510+4710+9410+12110+14610+14810




We obtain a parameter solution for (5).

 (1+22x)10+(1-22x)10+(1+61x)10+(1-61x)10+(1+86x)10+(1-86x)10+(1+127x)10+(1-127x)10+(1+140x)10+(1-140x)10+(1+151x)10+(1-151x)10
=(1+35x)10+(1-35x)10+(1+47x)10+(1-47x)10+(1+94x)10+(1-94x)10+(1+121x)10+(1-121x)10+(1+146x)10+ (1-146x)10+(1+148x)10+(1-148x)10


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