MathDB
A Proof Without Words

Source: 2017 AMC 12B #20

February 16, 2017
AMCAMC 12AMC 12 Bfloor function2017 AMC 12B

Problem Statement

Real numbers xx and yy are chosen independently and uniformly at random from the interval (0,1)(0,1). What is the probability that ⌊log⁡2x⌋=⌊log⁡2y⌋\lfloor \log_2{x} \rfloor=\lfloor \log_2{y} \rfloor, where ⌊r⌋\lfloor r \rfloor denotes the greatest integer less than or equal to the real number rr?
<spanclass=′latex−bold′>(A)</span> 18<spanclass=′latex−bold′>(B)</span> 16<spanclass=′latex−bold′>(C)</span> 14<spanclass=′latex−bold′>(D)</span> 13<spanclass=′latex−bold′>(E)</span> 12<span class='latex-bold'>(A)</span>\ \frac{1}{8}\qquad<span class='latex-bold'>(B)</span>\ \frac{1}{6}\qquad<span class='latex-bold'>(C)</span>\ \frac{1}{4}\qquad<span class='latex-bold'>(D)</span>\ \frac{1}{3}\qquad<span class='latex-bold'>(E)</span>\ \frac{1}{2}
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