December 2002
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The solution is 386, 2114, 3970, 10430. These numbers satisfy the conditions below.
386+2114=2500=50^2
386+3970=4356=66^2
386+10430=10816=104^2
2114+3970=6084=78^2
2114+10430=12544=112^2
3970+10430=14400=120^2
386+2114+3970+10430=16900=130^2
Sketch of a non-brute-force solution:
The sum of all four (a+b+c+d) is the square of the right-hand side of a Pythagorean triple in three different ways, so it has to be divisible by a handful of prime factors, each of which is 1 mod 4 (or else the prime 2). Since it cannot exceed sqrt(4*12000) < 220, there are not that many candidates. The right one turns out to be 130=2*5*13.
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