Problem of the Week #87: Monday April 21st, 2025
As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems.
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This problem is inspired by the 1975 USAMO Q1.
The floor function [a] denotes the greatest integer less than or equal to a. For x and y strictly between 0 and 1, the probability of [2x]+[2y] being 1 is 1/2:
a). If we choose x and y randomly between 0 and 1, what’s the probability that 2[x+y] is 0? 1? 2?
b). If we choose x and y randomly between 0 and 1, what’s the probability that 2[x+y] less than [2x]+[2y]? Equal to? Greater than?
c). If we choose x and y randomly between 0 and 1, what’s the probability that [2x+y] + [2y+x] is 0? 2? 3?
d). If we choose x and y randomly between 0 and 1, what’s the probability that [2x+y]+[2y+x] is less than [3x]+[3y]? Equal to? Greater than?
e). If we choose x and y randomly between 0 and 1, what’s the probability that 2[1.5x+1.5y] is less than [3x]+[3y]? Equal to? Greater than?
f). If we choose x and y randomly between 0 and 1, for a positive integer k, what’s the probability that 2[(k/2)x+(k/2)y] is less than [kx]+[ky]? Equal to? Greater than?
g). Share your own problem inspired by this one.
h). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.