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Real Number Picking

المؤلف:  American Mathematics Competitions

المصدر:  Problem B3 in "William Lowell Putnam Competition Archive, 1993" https://www.unl.edu/amc/a-activities/a7-problems/putnam/.

الجزء والصفحة:  ...

18-10-2020

1207

Real Number Picking

Pick two real numbers  and  at random in  with a uniform distribution. What is the probability  that , where  denotes the nearest integer function, is even?

The answer may be found as follows.

		{P(ay<x<by) for 0<=a<b<1; P(x/b<y<x/a) for 1<a<b" src="https://mathworld.wolfram.com/images/equations/RealNumberPicking/Inline9.gif" style="height:58px; width:203px" />	(1)
		{int_0^1int_(ay)^(by)dxdy=1/2(b-a) for 0<=a<b<1; int_0^1int_(x/b)^(x/a)dydx=1/(2a)-1/(2b) for 1<a<b" src="https://mathworld.wolfram.com/images/equations/RealNumberPicking/Inline12.gif" style="height:86px; width:274px" />	(2)

so

			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)

(OEIS A091651).

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REFERENCES:

American Mathematics Competitions. Problem B3 in "William Lowell Putnam Competition Archive, 1993" https://www.unl.edu/amc/a-activities/a7-problems/putnam/.

Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, p. 139, 2003.

Losinski, L. F.; Alexanderson, G. L.; and Larson, L. C. "The Fifty-Fourth William Lowell Putnam Mathematical Competition." Amer. Math. Monthly 101, 725-734, 1994.

Scholes, J. "54th Putnam 1993." https://www.kalva.demon.co.uk/putnam/putn93.html.

Sloane, N. J. A. Sequence A091651 in "The On-Line Encyclopedia of Integer Sequences."