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.

			(1)
			(2)

so

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

(OEIS A091651).

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."