MathDB

12

Part of 2021 AMC 12/AHSME Fall

Problems(2)

One Thousand and One Terms

Source: AMC 12A #12

11/11/2021
What is the number of terms with rational coefficients among the 10011001 terms of the expression (x23+y3)1000( x \sqrt[3]{2} + y \sqrt{3})^{1000}?
<spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 166<spanclass=′latex−bold′>(C)</span> 167<spanclass=′latex−bold′>(D)</span> 500<spanclass=′latex−bold′>(E)</span> 501<span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 166 \qquad<span class='latex-bold'>(C)</span>\ 167 \qquad<span class='latex-bold'>(D)</span>\ 500 \qquad<span class='latex-bold'>(E)</span>\ 501
Sums of Divisors

Source: 2021 Fall AMC 12B #12

11/17/2021
For nn a positive integer, let f(n)f(n) be the quotient obtained when the sum of all positive divisors of nn is divided by nn. For example, f(14)=(1+2+7+14)÷14=127.f(14) = (1 + 2 + 7 + 14) \div 14 = \frac{12}{7}. What is f(768)−f(384)?f(768) - f(384)?
<spanclass=′latex−bold′>(A)</span>1768<spanclass=′latex−bold′>(B)</span>1192<spanclass=′latex−bold′>(C)</span>1<spanclass=′latex−bold′>(D)</span>43<spanclass=′latex−bold′>(E)</span>83<span class='latex-bold'>(A) </span>\frac{1}{768}\qquad<span class='latex-bold'>(B) </span>\frac{1}{192}\qquad<span class='latex-bold'>(C) </span>1\qquad<span class='latex-bold'>(D) </span>\frac{4}{3}\qquad<span class='latex-bold'>(E) </span>\frac{8}{3}
AMCAMC 12AMC 12 B