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A352484
Decimal expansion of the probability that when three real numbers are chosen at random, uniformly and independently in the interval [0,1], they can be the lengths of the sides of a triangle whose altitudes are also the sides of some triangle.
2
3, 0, 5, 8, 3, 6, 7, 2, 2, 2, 5, 0, 7, 8, 8, 8, 7, 5, 6, 3, 4, 3, 5, 9, 5, 8, 1, 7, 0, 1, 9, 7, 8, 1, 7, 2, 1, 6, 0, 3, 2, 2, 4, 2, 0, 1, 4, 3, 4, 2, 6, 6, 0, 6, 7, 8, 3, 8, 7, 5, 0, 5, 8, 6, 0, 1, 1, 9, 9, 0, 4, 5, 9, 0, 4, 0, 4, 3, 4, 3, 2, 6, 8, 0, 5, 0, 0, 5, 9, 1, 5, 5, 7, 9, 9, 9, 2, 8, 7, 6, 0, 4, 7, 8, 5
OFFSET
0,1
COMMENTS
Without the condition on the altitudes the probability is 1/2.
LINKS
Murray S. Klamkin, Problem 1494, Crux Mathematicorum, Vol. 15, No. 10 (1989), p. 298; Solution to Problem 1494, by P. Penning, ibid., Vol. 17, No. 2 (1991), pp. 53-54.
Eric Weisstein's World of Mathematics, Altitude.
FORMULA
Equals 2*log(sqrt(5)-1) + 1 - sqrt(5)/2.
EXAMPLE
0.30583672225078887563435958170197817216032242014342...
MATHEMATICA
RealDigits[2*Log[Sqrt[5] - 1] + 1 - Sqrt[5]/2, 10, 100][[1]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Mar 18 2022
STATUS
approved