AMC 12C 2020 Problems

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==Problem 1

A plane flies at a speed of $590$ miles/hour. How many miles in two hours

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

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 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

A Regular Octagon has an area of $18 + 18\sqrt {2}$. What is the sum of the lengths of the diagonals of the octagon?

Problem 8

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 9

Let $E(x)$ denote the sum of the even digits of a positive integer and let $O(x)$ denote the sum of the odd digits of a positive integer. For some positive integer $N$, $3E(3N)$ = $4O(4N)$. What is the product of the digits of the least possible such $N$?

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 $\overline{E\omega}$? Source: JHMMC 2019

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

Problem 12

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 $k$?

Problem 13

An alien walks horizontally on the real number line starting at the origin. On each move, the alien can walk $1$ or $2$ numbers the right or left of it. What is the expected distance from the alien to the origin after $10$ moves?

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

Let $V$ and $F$ be the vertex and focus of the Parabola $P(x) = \frac{1}{8} x$ respectively. For a point $G$ lying on the directrix of $P(x)$, and a point $H$ lying on $P(x)$, $\overline {GH} = 10$ and Quadrilateral $VFGH$ is cyclic. If $VFGH$ has integral side lengths, what is the minimum possible area of $VFGH$?

Problem 17

Let $H(n)$ denote the $2nd$ nonzero digit from the right in the base - $10$ expansion of $(2n + 1)!$, for example, $H(2) = 1$. What is the sum of the digits of $\prod_{k = 1}^{2020}H(k)$?

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

$\frac{x^{4} + 2020}{108} = \lfloor \sqrt (x^{2} - x)\rfloor$?

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 $m + n$?

Problem 23

Problem 24

A sequence $(a_n)$ is defined as $a_1 = \frac{1}{\sqrt{3}}$, $a_2 = \sqrt{3}$, and for all $n > 1$,

$a_{n + 1} = \frac{2a_n-1}{1 - a_n^2}$

What is $\lfloor \ a_{2020}\rfloor$?

Problem 25

Let $P(x) = x^{2020} + 2x^{2019} + 3x^{2018} + … + 2019x^{2} + 2020x + 2021$ and let $Q(x) = x^{4} + 2x^{3} + 3x^{2} + 4x + 5$. Let $U$ be the sum of the $kth$ power of the roots of $P(Q(x))$. It is given that the least positive integer $y$, such that $3^{y} > U$ is $2021$. What is $k$?