MathDB

22

Part of 2016 AMC 10

Problems(2)

Counting Divisors

Source: 2016 AMC10A #22

2/3/2016
For some positive integer nn, the number 110n3110n^3 has 110110 positive integer divisors, including 11 and the number 110n3110n^3. How many positive integer divisors does the number 81n481n^4 have?
<spanclass=′latex−bold′>(A)</span>110<spanclass=′latex−bold′>(B)</span>191<spanclass=′latex−bold′>(C)</span>261<spanclass=′latex−bold′>(D)</span>325<spanclass=′latex−bold′>(E)</span>425<span class='latex-bold'>(A) </span>110 \qquad <span class='latex-bold'>(B) </span> 191 \qquad <span class='latex-bold'>(C) </span> 261 \qquad <span class='latex-bold'>(D) </span> 325 \qquad <span class='latex-bold'>(E) </span> 425
2016 AMC 10ADivisors2016 AMC 12AAMC 10AAMC 12AAMC 12number theory
Let's give this neat problem a &quot;round&quot; of applause

Source: AMC 10B 2016 #22

2/21/2016
A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won 1010 games and lost 1010 games; there were no ties. How many sets of three teams {A,B,C}\{A, B, C\} were there in which AA beat BB, BB beat CC, and CC beat A?A?
<spanclass=′latex−bold′>(A)</span> 385<spanclass=′latex−bold′>(B)</span> 665<spanclass=′latex−bold′>(C)</span> 945<spanclass=′latex−bold′>(D)</span> 1140<spanclass=′latex−bold′>(E)</span> 1330<span class='latex-bold'>(A)</span>\ 385 \qquad <span class='latex-bold'>(B)</span>\ 665 \qquad <span class='latex-bold'>(C)</span>\ 945 \qquad <span class='latex-bold'>(D)</span>\ 1140 \qquad <span class='latex-bold'>(E)</span>\ 1330
AMCAMC 10AMC 10 BAMC 12 B2016 AMC 10B