Efficient way to solve equal sums $x_1^k+x_2^k+\dots+x_5^k=y_1^k+y_2^k+\dots+y_5^k$ with Mathematica?

This isn't really an answer, but it's too big for a comment.

Here's some code to brute force solve the problem (since there wasn't any code provided in the question). It lists all the possible numbers; finds the ones whose product matches; then of those, finds the ones whose sum of squares match, etc..

Module[{h = 50, r, product, squared, fourth, sixth},
  r = Flatten[
    Table[{a, b, c, d, e}, {a, h}, {b, a + 1, h}, {c, b + 1,h},
        {d, c + 1, h}, {e, d + 1, h}], 4];
  Print[Length@r];
  product =
    Select[GatherBy[r, Times @@ # &], Length@# > 1 &];
  Print[Length@product];
  squared =
    Flatten[Select[GatherBy[#, Total@(#^2) &], Length@# > 1 &] & /@ product, 1];
  Print[Length@squared];
  fourth =
    Flatten[Select[GatherBy[#, Total@(#^4) &], Length@# > 1 &] & /@ squared, 1];
  Print[Length@fourth];
  sixth =
    Flatten[Select[GatherBy[#, Total@(#^6) &], Length@# > 1 &] & /@ fourth, 1];
  sixth]

It's quite fast, but runs into memory issues fairly quickly after 50. I've managed to run it up to $1 \leq x_i \leq 60$ and found no more solutions than in the question.

I've got things to be and places to do, but hopefully somebody might find this useful as a starting point. On the other hand, if you want to check up to 300, there will be 20 billion combinations to sift through using this method.

EDIT

Using a C++ program to check up to 150, I've found the following solutions:

$$\begin{align} 2\cdot 16\cdot 25\cdot 45\cdot 48 &= 3\cdot 9\cdot 32\cdot 40\cdot 50\\ 2\cdot 43\cdot 52\cdot 99\cdot 114 &= 3\cdot 22\cdot 76\cdot 86\cdot 117\\ 4\cdot 32\cdot 50\cdot 90\cdot 96 &= 6\cdot 18\cdot 64\cdot 80\cdot 100\\ 8\cdot 33\cdot 38\cdot 68\cdot 87 &= 12\cdot 17\cdot 57\cdot 58\cdot 88\\ 6\cdot 37\cdot 41\cdot 87\cdot 110 &= 11\cdot 15\cdot 58\cdot 82\cdot 111\\ 8\cdot 50\cdot 51\cdot 100\cdot 123 &= 12\cdot 24\cdot 82\cdot 85\cdot 125\\ 7\cdot 56\cdot 57\cdot 102\cdot 116 &= 8\cdot 38\cdot 84\cdot 87\cdot 119\\ 11\cdot 42\cdot 58\cdot 100\cdot 105 &= 14\cdot 28\cdot 75\cdot 87\cdot 110\\ 14\cdot 44\cdot 56\cdot 93\cdot 123 &= 21\cdot 24\cdot 77\cdot 82\cdot 124\\ 4\cdot 50\cdot 57\cdot 110\cdot 129 &= 6\cdot 25\cdot 86\cdot 95\cdot 132\\ 15\cdot 43\cdot 75\cdot 116\cdot 124 &= 20\cdot 29\cdot 93\cdot 100\cdot 129\\ 6\cdot 48\cdot 75\cdot 135\cdot 144 &= 9\cdot 27\cdot 96\cdot 120\cdot 150 \end{align}$$

EDIT 2

$$\begin{align} 2\cdot 52\cdot 72\cdot 133\cdot 147 &= 3\cdot 28\cdot 98\cdot 117\cdot 152\\ 19\cdot 48\cdot 86\cdot 143\cdot 144 &= 26\cdot 32\cdot 99\cdot 129\cdot 152\\ 4\cdot 41\cdot 86\cdot 144\cdot 159 &= 9\cdot 16\cdot 106\cdot 129\cdot 164\\ 16\cdot 66\cdot 76\cdot 136\cdot 174 &= 24\cdot 34\cdot 114\cdot 116\cdot 176\\ 12\cdot 59\cdot 82\cdot 143\cdot 174 &= 22\cdot 26\cdot 116\cdot 123\cdot 177\\ 11\cdot 75\cdot 100\cdot 148\cdot 183 &= 12\cdot 61\cdot 125\cdot 132\cdot 185\\ 16\cdot 53\cdot 102\cdot 176\cdot 177 &= 24\cdot 32\cdot 118\cdot 159\cdot 187\\ 24\cdot 74\cdot 105\cdot 172\cdot 175 &= 28\cdot 56\cdot 129\cdot 150\cdot 185\\ 3\cdot 32\cdot 112\cdot 182\cdot 183 &= 7\cdot 13\cdot 122\cdot 168\cdot 192\\ 8\cdot 34\cdot 132\cdot 171\cdot 185 &= 11\cdot 24\cdot 148\cdot 153\cdot 190\\ 15\cdot 71\cdot 75\cdot 164\cdot 212 &= 20\cdot 41\cdot 100\cdot 159\cdot 213\\ 8\cdot 64\cdot 100\cdot 180\cdot 192 &= 12\cdot 36\cdot 128\cdot 160\cdot 200\\ 12\cdot 74\cdot 82\cdot 174\cdot 220 &= 22\cdot 30\cdot 116\cdot 164\cdot 222\\ 16\cdot 96\cdot 106\cdot 189\cdot 209 &= 19\cdot 64\cdot 154\cdot 159\cdot 216\\ 26\cdot 74\cdot 123\cdot 177\cdot 220 &= 30\cdot 59\cdot 143\cdot 164\cdot 222\\ 22\cdot 84\cdot 116\cdot 200\cdot 210 &= 28\cdot 56\cdot 150\cdot 174\cdot 220\\ 16\cdot 66\cdot 119\cdot 208\cdot 213 &= 24\cdot 39\cdot 142\cdot 187\cdot 224\\ 4\cdot 86\cdot 104\cdot 198\cdot 228 &= 6\cdot 44\cdot 152\cdot 172\cdot 234\\ 9\cdot 65\cdot 118\cdot 200\cdot 228 &= 20\cdot 25\cdot 152\cdot 177\cdot 234\\ 24\cdot 84\cdot 122\cdot 209\cdot 217 &= 31\cdot 56\cdot 154\cdot 183\cdot 228\\ 28\cdot 88\cdot 112\cdot 186\cdot 246 &= 42\cdot 48\cdot 154\cdot 164\cdot 248\\ 28\cdot 58\cdot 133\cdot 213\cdot 222 &= 37\cdot 42\cdot 142\cdot 203\cdot 228\\ 8\cdot 29\cdot 142\cdot 219\cdot 228 &= 12\cdot 19\cdot 146\cdot 213\cdot 232\\ 16\cdot 100\cdot 102\cdot 200\cdot 246 &= 24\cdot 48\cdot 164\cdot 170\cdot 250\\ 14\cdot 112\cdot 114\cdot 204\cdot 232 &= 16\cdot 76\cdot 168\cdot 174\cdot 238\\ 30\cdot 77\cdot 126\cdot 205\cdot 244 &= 35\cdot 61\cdot 140\cdot 198\cdot 246\\ 10\cdot 80\cdot 125\cdot 225\cdot 240 &= 15\cdot 45\cdot 160\cdot 200\cdot 250 \end{align}$$