AMC 12C 2020 Problems

Problem 1

A tank is filled with $20$ % acid and $80$ % water, which contains $l$ gallons of liquid initially. How much more $60$ % acid and $40$ % water, $2l$ gallon solution should be added to the original solution to make a mixture consisting of $30$ % acid and $70$ % water?

Problem 2

$\bigtriangleup ABC$ has side lengths of $7$ , $8$ , and $9$ , and $\angle ABC = 60^{\circ}$ . What is the smallest possible measure of $\angle BAC$ ?

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

A spaceship flies in space at a speed of $s$ miles/hour and the spaceship is paid $d$ dollars for each $100$ miles traveled. It’s only expense is fuel in which it pays $\frac{d}{2}$ dollars per gallon, while going at a rate of $h$ hours per gallon. Traveling $3s$ miles, how much money would the spaceship have gained?

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

Problem 5

A plane flies at a speed of $590$ miles/hour $60^\circ$ north of west, while another plane flies directly in the east direction at a speed of $300$ miles/hour. How far are apart are the the $2$ planes after $3$ hours?

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 $R(x)$ be a function satisfying $R(m + n) = R(m)R(n)$ for all real numbers $n$ and $m$ . Let $R(1) = \frac{1}{2}.$ What is $R(1) + R(2) + R(3) + … + R(1000)$ ?

Problem 10

In how many ways can $n$ candy canes and $n + 1$ lollipops be split between $n - 4$ 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 $\overline{E\omega}$ ?

$\textbf{(A)} \frac {31}{12}\sqrt{2} \qquad \textbf{(B)} \frac {35}{12}\sqrt{3} \qquad \textbf{(C)} \frac {37}{12}\sqrt{5} \qquad \textbf{(D)} \frac {39}{12}\sqrt{7} \qquad \textbf{(E)} \frac {41}{12}\sqrt{11} \qquad$

Problem 12

What is the value of $sin(1^\circ)sin(3^\circ)sin(5^\circ)…sin(179^\circ) - sin(181^\circ)sin(182^\circ)…sin(359^\circ)$ ?

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

A $6$ by $6$ glass case of $36$ glass boxes are to be filled with $3$ purple balls and $3$ red balls such that each row and column contains exactly $1$ of each a red and purple ball. In how many ways can this arrangement be done?

Problem 18

$\bigtriangleup ABC$ lays flat on the ground and has side lengths $\overline{AB} = 3, \overline{BC} = 4$ , and $\overline{AC} = 5$ . 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

Let $P(x)$ be a cubic polynomial with integral coefficients and roots $\cos \frac{\pi}{13}$ , $\cos \frac{5\pi}{13}$ , and $\cos \frac{7\pi}{13}$ . What is the least possible sum of the coefficients of $P(x)$ ?

Problem 20

What is the maximum value of $\sum_{k = 1}^{6}(2^{x} + 3^{x})$ as $x$ varies through all real numbers to the nearest integer?

$\textbf{(A)}\ -3\qquad\textbf{(B)}\ -2\qquad\textbf{(C)}\ -1\qquad\textbf{(D)}\ 0\qquad\textbf{(E)}\ 1$

Problem 21

Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$ . How many positive integers $x < 2020$ , satisfy the equation

Problem 22

A convex hexagon $ABCDEF$ is inscribed in a circle. $\overline {AB}$ $=$ $\overline {BC}$ $=$ $\overline {AD}$ $=$ $2$ . $\overline {DE}$ $=$ $\overline {CF}$ $=$ $\overline {EF}$ $=$ $4$ . The measure of $\overline {DC}$ can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m + n$ ?

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