Find all positive integers $n$ for which the equation $x + y + u + v = n \sqrt{ xyuv }$ has a solution in positive integers.

Question

Find all positive integers $n$ for which the equation

$$ x + y + u + v = n \sqrt{ xyuv } $$

has a solution in positive integers.

This problem is taken from Vietnamese Mathematical Olympiad, 2002, Question B5

---

Context: Will Jaggy requested this problem and solution to be posted. See his comment

Answer 1

Solution from Selected Problems of the Vietnamese Mathematical Olympiad (1962–2009). (Volume 5, Mathematical Olympiad Series)

Answer 2

here is my list of (x+y+z+t)^2 = kxyzt in positive integers. Note k need not be a

square.


ordered and fundamental integers x >= y >= z >= t >= 1,

( x+y+z+t)^2 = kxyzt   and 2(x+y+z+t)  <= kyzt:

      k               x      y      z      t
      1              10     10      9      1  2sum: 60 kyzt:  90
      1              12      6      4      2  2sum: 48 kyzt:  48
      1              15     10      3      2  2sum: 60 kyzt:  60
      1              18      9      8      1  2sum: 72 kyzt:  72
      1              21     14      6      1  2sum: 84 kyzt:  84
      1              30     24      5      1  2sum: 120 kyzt:  120
      1               4      4      4      4  2sum: 32 kyzt:  64
      1               6      6      3      3  2sum: 36 kyzt:  54
      1               8      5      5      2  2sum: 40 kyzt:  50

      2              10      5      4      1  2sum: 40 kyzt:  40
      2              12      8      3      1  2sum: 48 kyzt:  48
      2               4      3      3      2  2sum: 24 kyzt:  36
      2               8      4      2      2  2sum: 32 kyzt:  32

      3               4      4      3      1  2sum: 24 kyzt:  36
      3               6      2      2      2  2sum: 24 kyzt:  24
      3               9      6      2      1  2sum: 36 kyzt:  36

      4               2      2      2      2  2sum: 16 kyzt:  32
      4               6      3      2      1  2sum: 24 kyzt:  24

      5              10      8      1      1  2sum: 40 kyzt:  40
      5               5      2      2      1  2sum: 20 kyzt:  20

      6               6      4      1      1  2sum: 24 kyzt:  24

      8               4      2      1      1  2sum: 16 kyzt:  16

      9               2      2      1      1  2sum: 12 kyzt:  18

     12               3      1      1      1  2sum: 12 kyzt:  12

     16               1      1      1      1  2sum: 8 kyzt:  16