MathDB

10

Part of 2020 AMC 12/AHSME

Problems(2)

Nested Logarithms

Source: 2020 AMC 12A #10

1/31/2020
There is a unique positive integer nn such that log⁡2(log⁡16n)=log⁡4(log⁡4n).\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}. What is the sum of the digits of n?n?
<spanclass=′latex−bold′>(A)</span>4<spanclass=′latex−bold′>(B)</span>7<spanclass=′latex−bold′>(C)</span>8<spanclass=′latex−bold′>(D)</span>11<spanclass=′latex−bold′>(E)</span>13<span class='latex-bold'>(A) </span> 4 \qquad <span class='latex-bold'>(B) </span> 7 \qquad <span class='latex-bold'>(C) </span> 8 \qquad <span class='latex-bold'>(D) </span> 11 \qquad <span class='latex-bold'>(E) </span> 13
AMCAMC 12AMC 12 Alogarithms2020 AMC2020 AMC 12A
Squares, Circles, and Chords

Source: 2020 AMC 12B #10

2/6/2020
In unit square ABCD,ABCD, the inscribed circle ω\omega intersects CD‾\overline{CD} at M,M, and AM‾\overline{AM} intersects ω\omega at a point PP different from M.M. What is AP?AP?
<spanclass=′latex−bold′>(A)</span>512<spanclass=′latex−bold′>(B)</span>510<spanclass=′latex−bold′>(C)</span>59<spanclass=′latex−bold′>(D)</span>58<spanclass=′latex−bold′>(E)</span>2515<span class='latex-bold'>(A) </span> \frac{\sqrt5}{12} \qquad <span class='latex-bold'>(B) </span> \frac{\sqrt5}{10} \qquad <span class='latex-bold'>(C) </span> \frac{\sqrt5}{9} \qquad <span class='latex-bold'>(D) </span> \frac{\sqrt5}{8} \qquad <span class='latex-bold'>(E) </span> \frac{2\sqrt5}{15}
AMCAMC 12AMC 12 B2020 AMC 12Bgeometry2020 AMCsquare