AMC 12C 2020 Problems

Problem 1

What is the sum of the solutions to the equation $(x - 3)(x + 8)(x - 5)(x + 3) = 0$ ?

$\textbf{(A)}\ -3 \qquad\textbf{(B)}\ 0 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 7$

Problem 2

On a plane lie $2$ points, $A(0, 7)$ , and $B(12, 13)$ . How many points $C$ lie on the same plane such that $\bigtriangleup ABC$ is an isosceles triangle with area $50$ ?

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

Lambu the 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

The line $k$ has an equation $y = 2x + 5$ is rotated clockwise by $45^{\circ}$ to obtain the line $l$ . What is the distance between the $x$ - intercepts of Lines $k$ and $l$ ?

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

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

$\mathrm{(A) \ } 38 \qquad \mathrm{(B) \ } 42\qquad \mathrm{(C) \ } 46\qquad \mathrm{(D) \ } 50\qquad \mathrm{(E) \ } 54$

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?

Problem 11

Let $ABCD$ be an isosceles trapezoid with $\overline{AB}$ being parallel to $\overline{CD}$ and $\overline{AB} = 5$ , $\overline{CD} = 15$ , and $\angle ADC = 60^\circ$ . If $E$ is the intersection of $\overline{AC}$ and $\overline{BD}$ , and $\omega$ is the circumcenter of $\bigtriangleup ABC$ , what is the length of $\overline{E\omega}$ ?

$\textbf{(A)} \frac {31}{12}\sqrt{3} \qquad \textbf{(B)} \frac {35}{12}\sqrt{3} \qquad \textbf{(C)} \frac {37}{12}\sqrt{3} \qquad \textbf{(D)} \frac {39}{12}\sqrt{3} \qquad \textbf{(E)} \frac {41}{12}\sqrt{3} \qquad$

Problem 12

Rajbhog, Aditya, and Suman are racing a $1000$ meter race. Aditya beats Rajbhog by $9$ seconds and beats Suman by $250$ meters. Given that Rajbhog beat Suman by $2$ seconds, by how many meters would Aditya beat Rajbhog if they both were having a $3500$ meter race?

Problem 13

In how many ways can the first $15$ positive integers; $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}$ in red, blue, and green colors if no $3$ numbers $a, b$ , and $c$ are the same color with $a + b - c$ being even?

Problem 14

Let $K$ be the set of solutions to the equation $(x + i)^{10} = 1$ on the complex plane, where $i = \sqrt -1$ . $2$ points from $K$ are chosen, such that a circle $\Omega$ passes through both points. What is the least possible area of $\Omega$ ?

Problem 15

Let $N = 10^{10^{100…^{10000…(100 zeroes)}}}$ . What is the remainder when $N$ is divided by $629$ ?

Problem 16

For some positive integer $k$ , let $k$ satisfy the equation

$log(k - 2)! + log(k - 1)! + 2 = 2 log(k!)$ . What is the sum of the digits of $k$ ?

$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 19$

Problem 17

In rectangle $ABCD$ , $\overline{AB} = 10$ and $\overline{BC} = 6$ . Let the midpoint of $\overline{AB}$ be $M$ and let the midpoint of $\overline{BC}$ be $N$ . The centroids of Triangles $\bigtriangleup ADM$ , $\bigtriangleup CDN$ , and $\bigtriangleup DMN$ are connected to from the minor triangle $\bigtriangleup JKL$ . What is the length of largest altitude of $\bigtriangleup JKL$ ?

Problem 18

$\bigtriangleup ABC$ lays flat on the ground and has side lengths $\overline{AB} = 8, \overline{BC} = 15$ , and $\overline{AC} = 17$ . Vertex $A$ is then lifted up creating an elevation angle with the triangle and the ground of $60^{\circ}$ . A wooden pole is dropped from $A$ perpendicular to the ground, making an altitude of a $3$ Dimensional figure. Ropes are connected from the foot of the pole, $D$ , to form $2$ other segments, $\overline{BD}$ and $\overline{CD}$ . What is the volume of $ABCD$ ?

$\textbf{(A) } 180\sqrt{3} \qquad \textbf{(B) } 15 + 180\sqrt{3} \qquad \textbf{(C) } 20 + 180\sqrt{5} \qquad \textbf{(D) } 28 + 180\sqrt{5} \qquad \textbf{(E) } 440\sqrt{2}$

Problem 19

An urn left on a deserted island, consists of $12$ golden blocks, $16$ silver blocks, $20$ zinc blocks, and $24$ wooden blocks. $9$ pirates come to the island seeing the urn. Without noticing blocks are made of different materials, each of the pirates randomly grab an equal number of blocks from the urn, each at a time. The pirates then place the blocks back into the urn and then repeat the same process again. What is the probability that after the pirates repeat the same process $2020$ times, that no pirate who has more than $4$ golden blocks has more than $3$ silver blocks?