½ = 1/x² + 1/y² +...+ 1/z²

All variables must be different, positive integers, and there must be a finite number of terms.

Is this the only solution? :)

I think the answer here is merely to multiply the answer by a common denominator and then you can square each term and add them. 90000 is the GCD for the given solution. Then the numerators become 22500, 10000, 5625, 3600,  2500, 625, 100, 36, 9, 4, 1, which add up to 45000, half of 90000.

One thing to note about these is the prime factorization for 300 is 2x2x3x5x5, 5 times the LCD of the first 5 numbers.

2 3 5

1-0-0 (2)

0-1-0 (3)

2-0-0 (4)

0-0-1 (5)

1-1-0 (6)

2-1-0 (12)

1-1-1 (30)

1-0-2 (50)

2-0-2 (100)

1-1-2 (150)

2-1-2 (300)