A358614 Decimal expansion of 9*sqrt(2)/32.

3, 9, 7, 7, 4, 7, 5, 6, 4, 4, 1, 7, 4, 3, 2, 9, 8, 2, 4, 7, 5, 4, 7, 4, 9, 5, 3, 6, 8, 3, 9, 7, 7, 5, 8, 4, 5, 9, 7, 7, 2, 0, 2, 1, 4, 9, 4, 9, 7, 6, 6, 6, 4, 5, 5, 8, 0, 9, 4, 1, 1, 7, 6, 3, 0, 9, 8, 9, 3, 5, 0, 9, 5, 6, 7, 4, 6, 7, 6, 0, 4, 6, 7, 6, 6, 7, 1, 4, 9, 4, 0, 2, 9, 6, 4, 9, 1, 9, 2

Offset: 0

Author

Bernard Schott, Dec 05 2022

Keywords

• nonn
• cons
• easy

Comments

Smallest constant M such that the inequality

|a*b*(a^2 - b^2) + b*c*(b^2 - c^2) + c*a*(c^2 - a^2)| <= M * (a^2 + b^2 + c^2)^2

holds for all real numbers a, b, c.

Equality stands for any triple (a, b, c) proportional to (1 - 3*sqrt(2)/2, 1, 1 + 3*sqrt(2)/2), up to permutation.

This constant is the answer to the 3rd problem, proposed by Ireland during the 47th International Mathematical Olympiad in 2006 at Ljubljana, Slovenia (see links).

Equivalently |(a - b)(b - c)(c - a)(a + b + c)| / (a^2 + b^2 + c^2)^2 <= M with (a,b,c) != (0,0,0).

Examples

			0.3977475644174329824...
		

Links

• Evan Chen, IMO 2006/3, IMO 2006 Solution Notes.
• The IMO compendium, Problem 3, 47th IMO 2006.
• Index to sequences related to Olympiads.

Crossrefs

Cf. A002193, A010474, A010503, A230981.

Programs

Maple

evalf(9*sqrt(2)/32), 100);

Mathematica

RealDigits[9*Sqrt[2]/32, 10, 120][[1]] (* Amiram Eldar, Dec 05 2022 *)

Formula

Equals (3/16) * A230981 = (3/32) * A010474 = (9/32) * A002193 = (9/16) * A010503.