MathDB

20

Part of 2017 AMC 12/AHSME

Problems(2)

Solving for the Log

Source: 2017 AMC 12A #20

2/8/2017
How many ordered pairs (a,b)(a, b) such that aa is a real positive number and bb is an integer between 22 and 200200, inclusive, satisfy the equation (log⁡ba)2017=log⁡b(a2017)(\log_b a)^{2017} = \log_b (a^{2017})?
<spanclass=′latex−bold′>(A) </span>198<spanclass=′latex−bold′>(B) </span>199<spanclass=′latex−bold′>(C) </span>398<spanclass=′latex−bold′>(D) </span>399<spanclass=′latex−bold′>(E) </span>597 <span class='latex-bold'>(A) \ </span>198\qquad <span class='latex-bold'>(B) \ </span> 199 \qquad <span class='latex-bold'>(C) \ </span> 398 \qquad <span class='latex-bold'>(D) \ </span>399\qquad <span class='latex-bold'>(E) \ </span> 597
AMCAMC 12AMC 12 A2017 AMC 12A
A Proof Without Words

Source: 2017 AMC 12B #20

2/16/2017
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}
AMCAMC 12AMC 12 Bfloor function2017 AMC 12B