Difference between revisions of "AMC 12C 2020 Problems"

Latest revision as of 14:02, 27 November 2022

1 plus 1

==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

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 $k$ ?

@@ Line 1: / Line 1: @@
-==Problem 1==
+plus 1
-A tank contains <math>20</math>% acid and <math>80</math>% water, which contains <math>l</math> gallons of liquid initially.  How much more <math>60</math>% acid and <math>40</math>% water, <math>2l</math> gallon solution should be added to the original solution to make a mixture consisting of <math>30</math>% acid and <math>70</math>% water?
+==Problem 1
-==Problem 2==
+A plane flies at a speed of <math>590</math> miles/hour. How many miles in two hours
-A plane flies at a speed of <math>590</math> miles/hour <math>60^\circ</math> north of west, while another plane flies directly in the east direction at a speed of <math>300</math> miles/hour. How far are apart are the the <math>2</math> planes after <math>3</math> hours?
 ==Problem 3==
@@ Line 33: / Line 31: @@
 ==Problem 7==
-The line <math>k</math> has an equation <math>y = 2x + 5</math> is rotated clockwise by <math>45^{\circ}</math> to obtain the line <math>l</math>. What is the distance between the <math>x</math> - intercepts of Lines <math>k</math> and <math>l</math>?
+A Regular Octagon has an area of <math>18 + 18\sqrt {2}</math>. What is the sum of the lengths of the diagonals of the octagon?
 ==Problem 8==
@@ Line 49: / Line 47: @@
 ==Problem 11==
-Let <math>ABCD</math> be an isosceles trapezoid with <math>\overline{AB}</math> being parallel to <math>\overline{CD}</math> and <math>\overline{AB} = 5</math>, <math>\overline{CD} = 15</math>, and <math>\angle ADC = 60^\circ</math>. If <math>E</math> is the intersection of <math>\overline{AC}</math> and <math>\overline{BD}</math>, and <math>\omega</math> is the circumcenter of <math>\bigtriangleup ABC</math>, what is the length of <math>\overline{E\omega}</math>?
+Let <math>ABCD</math> be an isosceles trapezoid with <math>\overline{AB}</math> being parallel to <math>\overline{CD}</math> and <math>\overline{AB} = 5</math>, <math>\overline{CD} = 15</math>, and <math>\angle ADC = 60^\circ</math>. If <math>E</math> is the intersection of <math>\overline{AC}</math> and <math>\overline{BD}</math>, and <math>\omega</math> is the circumcenter of <math>\bigtriangleup ABC</math>, what is the length of <math>\overline{E\omega}</math>? Source: JHMMC 2019
@@ Line 76: / Line 74: @@
 ==Problem 16==
+Let <math>V</math> and <math>F</math> be the vertex and focus of the Parabola <math>P(x) = \frac{1}{8} x</math> respectively. For a point <math>G</math> lying on the directrix of <math>P(x)</math>, and a point <math>H</math> lying on <math>P(x)</math>, <math>\overline {GH} = 10</math> and Quadrilateral <math>VFGH</math> is cyclic. If <math>VFGH</math> has integral side lengths, what is the minimum possible area of <math>VFGH</math>?
 ==Problem 17==
@@ Line 114: / Line 114: @@
 ==Problem 23==
-Let <math>P(x) = x^{2020} + 2x^{2019} + 3x^{2018} + … + 2019x^{2} + 2020x + 2021</math> and let <math>Q(x) = x^{4} + 2x^{3} + 3x^{2} + 4x + 5</math>. Let <math>U</math> be the sum of the <math>kth</math> power of the roots of <math>P(Q(x))</math>. It is given that the least positive integer <math>y</math>, such that <math>3^{y} > U</math> is <math>2021</math>. What is <math>k</math>?
 ==Problem 24==
@@ Line 125: / Line 122: @@
 What is <math>\lfloor \ a_{2020}\rfloor</math>?
+==Problem 25==
+Let <math>P(x) = x^{2020} + 2x^{2019} + 3x^{2018} + … + 2019x^{2} + 2020x + 2021</math> and let <math>Q(x) = x^{4} + 2x^{3} + 3x^{2} + 4x + 5</math>. Let <math>U</math> be the sum of the <math>kth</math> power of the roots of <math>P(Q(x))</math>. It is given that the least positive integer <math>y</math>, such that <math>3^{y} > U</math> is <math>2021</math>. What is <math>k</math>?

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