AMC 12C 2020 Problems

Problem 1

What is the sum of the solutions of the equation $(x + 4)(x - 5)(x + 6) = 0$ ?

$\textbf{(A)}\ -5 \qquad\textbf{(B)}\ 0 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 14$

Problem 2

What is the numerical value of the sum $\sum_{k = 1}^{11}(i^{3} + i^{2})$

$\textbf{(A)}\ 4000 \qquad\textbf{(B)}\ 4608 \qquad\textbf{(C)}\ 4862 \qquad\textbf{(D)}\ 5792 \qquad\textbf{(E)}\ 6100$

Problem 3

In a bag are $7$ marbles consisting of $3$ blue marbles and $4$ red marbles. If each marble is pulled out $1$ at a time, what is the probability that the $6th$ marble pulled out red?

$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ \frac{1}{8} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ \frac{4}{7} \qquad\textbf{(E)}\ 1$

Problem 4

$10$ cows can consume $40$ kilograms of grass in $20$ days. How many more cows are required such that all the cows together can consume $60$ kilograms of grass in $10$ days?

$\textbf{(A)}\ 20 \qquad\textbf{(B)}\ \ 21 \qquad\textbf{(C)}\ \ 22 \qquad\textbf{(D)}\ \ 23 \qquad\textbf{(E)}\ 24$

Problem 5

A lamb is tied to a post at the origin $(0, 0)$ on the real $xy$ plane with a rope that measures $6$ units. $2$ wolves are tied with ropes of length $6$ as well, both of them being at points $(6, 6)$ , and $(-6, -6)$ . What is the area that the lamb can run around without being in the range of the wolves?