Difference between revisions of "AMC 12C 2020 Problems"

Revision as of 17:33, 21 April 2020

Problem 1

What is the sum of the solutions of the equation $(x + 4)(x - 5)(x + 6)(x - 8) = 0$ ?

$\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 15$

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?

$\textbf{(A)}\ 70 \qquad\textbf{(B)}\ 71 \qquad\textbf{(C)}\ 72 \qquad\textbf{(D)}\ 100 \qquad\textbf{(E)}\ 110$

Problem 6

How many increasing(lower to higher numbered) subsets of $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ contain no $2$ consecutive prime numbers?

Problem 7

Let $T(n)$ denote the sum of the factors of a positive integer $n$ . What is the sum of the $3$ least possible values of $x$ such that $T(x) + T(2x) = 8$ ?

Problem 8

The real value of $n$ that satisfies the equation $ln(n) + ln(n^{2} - 34) = ln(72)$ can be written in the form $a + \sqrt{b}$ where $a$ and $b$ are integers. What is $a + b$ ?

$\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 16 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ 24$

Problem 9

Let $R(x)$ denote the number of trailing $0$ s in the numerical value of the expression $x!$ , for example, $R(5) = 1$ since $5! = 120$ which has $1$ trailing zero. What is the sum

$R(20) + R(19) + R(18) + R(17) + … + R(3) + R(2) + R(1) + R(0)$ ?

$\mathrm{(A) \ } -12\qquad \mathrm{(B) \ } 0\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \ } 16\qquad \mathrm{(E) \ } 24$

Problem 10

In how many ways can $10$ candy canes and $9$ lollipops be split between $8$ children if each child must receive atleast $1$ candy but no child receives both types?

@@ Line 63: / Line 63: @@
 Let <math>R(x)</math> denote the number of trailing <math>0</math>s in the numerical value of the expression <math>x!</math>, for example, <math>R(5) = 1</math>   since <math>5! = 120</math> which has <math>1</math> trailing zero. What is the sum
-<math>R(100) + R(99) + R(98) + R(97) + … + R(3) + R(2) + R(1) + R(0)</math>?
+<math>R(20) + R(19) + R(18) + R(17) + … + R(3) + R(2) + R(1) + R(0)</math>?
+<math>\mathrm{(A) \ } -12\qquad \mathrm{(B) \ } 0\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \ } 16\qquad \mathrm{(E) \ } 24</math>
 ==Problem 10==
 In how many ways can <math>10</math> candy canes and <math>9</math> lollipops be split between <math>8</math> children if each child must receive atleast <math>1</math> candy but no child receives both types?

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