Real or Nonreal

Calculus Level 5

Let S be the set of all (A,B) in the plane such that -20 < A < 20 and -20 < B < 20, and there exist distinct numbers a, b, c, d, either all real or all nonreal, such that

a + b + c + d = 6 , a + b + c + d = 6,
a 2 + b 2 + c 2 + d 2 = 12 , a^2 + b^2 + c^2 + d^2 = 12,
a 3 + b 3 + c 3 + d 3 = A , a^3 + b^3 + c^3 + d^3 = A,
a 4 + b 4 + c 4 + d 4 = B a^4 + b^4 + c^4 + d^4 = B

Find the area of S to 3 decimal places.

Inspired by:

https://brilliant.org/problems/cool-inequality-6-making-it-really-tough/

https://brilliant.org/problems/duplicate-roots-everywhere/?ref_id=1406216


The answer is 65.984.

Christopher Criscitiello by Christopher Criscitiello , 25, USA

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1 solution

a, b, c, d will be the roots of the polynomial

− A X 3 + 2 A − B 4 + X 4 − 6 X 3 + 12 X 2 − 36 -\frac{A X}{3}+2 A-\frac{B}{4}+X^4-6 X^3+12 X^2-36

which has discriminant

1 3 ( − A 4 + 2592 A 3 − 1278 A 2 B − 124128 A 2 + 216 A B 2 + 41472 A B + 1990656 A − 12 B 3 − 3537 B 2 − 334368 B − 10637568 ) \frac{1}{3} \left(-A^4+2592 A^3-1278 A^2 B-124128 A^2+216 A B^2+41472 A B+1990656 A-12 B^3-3537 B^2-334368 B-10637568\right)

The roots of a quartic polynomial are distinct, and all real or all nonreal iff the discriminant is positive. So numerically integrating over this region gives us the answer.

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