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There is no possible answer.

For example,

-84 + 100 = 16 and -842 + 1002 = 7056 + 10000 = 17056

-184 + 200 = 16 and -1842 + 2002 = 33856 + 40000 = 73856

-284 + 300 = 16 and -2842 + 3002 = 80656 + 90000 = 170656

There is no limit to how large the sum of squares can be.

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Q: What are two numbers whose sum is 16 and whose squares have the greatest sum?
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