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
-What is the the sum of the solutions of the equation <math>(x + 4)(x - 5)(x + 6)(x - 8) = 0</math>?
+==Problem 1
+A plane flies at a speed of <math>590</math> miles/hour. How many miles in two hours
-<math>\textbf{(A)}\ -3 \qquad\textbf{(B)}\ 0 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 7</math>
-==Problem 2==
-On a plane lie <math>2</math> points, <math>A(0, 7)</math>, and <math>B(12, 13)</math>. How many points <math>C</math> lie on the same plane such that <math>\bigtriangleup ABC</math> is an isosceles triangle with area <math>50</math>?
 ==Problem 3==
@@ Line 20: / Line 15: @@
 ==Problem 4==
-<math>10</math> cows can consume <math>40</math> kilograms of grass in <math>20</math> days. How many more cows are required such that all the cows together can consume <math>60</math> kilograms of grass in <math>10</math> days?
+A spaceship flies in space at a speed of <math>s</math> miles/hour and the spaceship is paid <math>d</math> dollars for each <math>100</math> miles traveled. It’s only expense is fuel in which it pays <math>\frac{d}{2}</math> dollars per gallon, while going at a rate of <math>h</math> hours per gallon. Traveling <math>3s</math> miles, how much money would the spaceship have gained?
 <math>\textbf{(A)}\ 20 \qquad\textbf{(B)}\ \ 21 \qquad\textbf{(C)}\ \ 22 \qquad\textbf{(D)}\ \ 23 \qquad\textbf{(E)}\ 24</math>
 ==Problem 5==
-Lambu the Lamb is tied to a post at the origin <math>(0, 0)</math> on the real <math>xy</math> plane with a rope that measures <math>6</math> units. <math>2</math> wolves are tied with ropes of length <math>6</math> as well, both of them being at points <math>(6, 6)</math>, and <math>(-6, -6)</math>. What is the area that the lamb can run around without being in the range of the wolves?
+Let <math>R(x)</math> be a function satisfying <math>R(m + n) = R(m)R(n)</math> for all real numbers <math>n</math> and <math>m</math>. Let <math>R(1) = \frac{1}{2}.</math> What is <math>R(1) + R(2) + R(3) + … + R(1000)</math>?
-<math>\textbf{(A)}\ 70 \qquad\textbf{(B)}\ 71 \qquad\textbf{(C)}\ 72 \qquad\textbf{(D)}\ 100 \qquad\textbf{(E)}\ 110</math>
 ==Problem 6==
@@ Line 42: / 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==
-The real value of <math>n</math> that satisfies the equation <math>ln(n) + ln(n^{2} - 34) = ln(72)</math> can be written in the form <cmath>a + \sqrt{b}</cmath> where <math>a</math> and <math>b</math> are integers. What is <math>a + b</math>?
+What is the value of <math>sin(1^\circ)sin(3^\circ)sin(5^\circ)…sin(179^\circ) - sin(181^\circ)sin(182^\circ)…sin(359^\circ)</math>?
-<math>\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 16 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ 24</math>
 ==Problem 9==
-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
+Let <math>E(x)</math> denote the sum of the even digits of a positive integer and let <math>O(x)</math> denote the sum of the odd digits of a positive integer. For some positive integer <math>N</math>, <math>3E(3N)</math> = <math>4O(4N)</math>. What is the product of the digits of the least possible such <math>N</math>?
-<math>R(20) + R(19) + R(18) + R(17) + … + R(3) + R(2) + R(1) + R(0)</math>?
-<math>\mathrm{(A) \ } 38 \qquad \mathrm{(B) \ } 42\qquad \mathrm{(C) \ } 46\qquad \mathrm{(D) \ } 50\qquad \mathrm{(E) \ } 54</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?
+In how many ways can <math>n</math> candy canes and <math>n + 1</math> lollipops be split between <math>n - 4</math> children if each child must receive atleast <math>1</math> candy but no child receives both types?
 ==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
-<math>\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  </math>
+<math>\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  </math>
 ==Problem 12==
-Rajbhog, Aditya, and Suman are racing a <math>1000</math> meter race. Aditya beats Rajbhog by <math>9</math> seconds and beats Suman by <math>250</math> meters. Given that Rajbhog beat Suman by <math>2</math> seconds, by how many meters would Aditya beat Rajbhog if they both were having a <math>3500</math> meter race?
+For some positive integer <math>k</math>, let <math>k</math> satisfy the equation
+<math>log(k - 2)! + log(k - 1)! + 2 = 2 log(k!)</math>.
+What is the sum of the digits of <math>k</math>?
 ==Problem 13==
-In how many ways can the first <math>15</math> positive integers; <math>\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}</math> in red, blue, and green colors if no <math>3</math> numbers <math>a, b</math>, and <math>c</math> are the same color with <math>a + b - c</math> being even?
+An alien walks horizontally on the real number line starting at the origin. On each move, the alien can walk <math>1</math> or <math>2</math> numbers the right or left of it. What is the expected distance from the alien to the origin after <math>10</math> moves?
 ==Problem 14==
@@ Line 99: / Line 75: @@
 ==Problem 16==
-For some positive integer <math>k</math>, let <math>k</math> satisfy the equation
+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>?
-<math>log(k - 2)! + log(k - 1)! + 2 = 2 log(k!)</math>.
-What is the sum of the digits of <math>k</math>?
-<math>\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 19</math>
 ==Problem 17==
-In rectangle <math>ABCD</math>, <math>\overline{AB} = 10</math> and <math>\overline{BC} = 6</math>. Let the midpoint of <math>\overline{AB}</math> be <math>M</math> and let the midpoint of <math>\overline{BC}</math> be <math>N</math>. The centroids of Triangles <math>\bigtriangleup ADM</math>, <math>\bigtriangleup CDN</math>, and <math>\bigtriangleup DMN</math> are connected to from the minor triangle <math>\bigtriangleup JKL</math>. What is the length of largest altitude of <math>\bigtriangleup JKL</math>?
+Let <math>H(n)</math> denote the <math>2nd</math> nonzero digit from the right in the base - <math>10</math> expansion of <math>(2n + 1)!</math>, for example, <math>H(2) = 1</math>. What is the sum of the digits of <math>\prod_{k = 1}^{2020}H(k)</math>?
 ==Problem 18==
-<math>\bigtriangleup ABC</math> lays flat on the ground and has side lengths <math>\overline{AB} = 8, \overline{BC} = 15</math>, and <math>\overline{AC} = 17</math>. Vertex <math>A</math> is then lifted up creating an elevation angle with the triangle and the ground of <math>60^{\circ}</math>. A wooden pole is dropped from <math>A</math> perpendicular to the ground, making an altitude of a <math>3</math> Dimensional figure. Ropes are connected from the foot of the pole, <math>D</math>, to form <math>2</math> other segments, <math>\overline{BD}</math> and <math>\overline{CD}</math>. What is the volume of <math>ABCD</math>?
+<math>\bigtriangleup ABC</math> lays flat on the ground and has side lengths <math>\overline{AB} = 3, \overline{BC} = 4</math>, and <math>\overline{AC} = 5</math>. Vertex <math>A</math> is then lifted up creating an elevation angle with the triangle and the ground of <math>60^{\circ}</math>. A wooden pole is dropped from <math>A</math> perpendicular to the ground, making an altitude of a <math>3</math> Dimensional figure. Ropes are connected from the foot of the pole, <math>D</math>, to form <math>2</math> other segments, <math>\overline{BD}</math> and <math>\overline{CD}</math>. What is the volume of <math>ABCD</math>?
+<math>\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}</math>
 ==Problem 19==
-An urn left on a deserted island, consists of <math>12</math> golden blocks, <math>16</math> silver blocks, <math>20</math> zinc blocks, and <math>24</math> wooden blocks. <math>9</math> 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 <math>2020</math> times, that no pirate who has more than <math>4</math> golden blocks has more than <math>3</math> silver blocks?
+Let <math>P(x)</math> be a cubic polynomial with integral coefficients and roots <math>\cos \frac{\pi}{13}</math>, <math>\cos \frac{5\pi}{13}</math>, and <math>\cos \frac{7\pi}{13}</math>. What is the least possible sum of the coefficients of <math>P(x)</math>?
 ==Problem 20==
@@ Line 130: / Line 101: @@
 <math>\textbf{(A)}\ -3\qquad\textbf{(B)}\ -2\qquad\textbf{(C)}\ -1\qquad\textbf{(D)}\ 0\qquad\textbf{(E)}\ 1</math>
-==Problem 24==
+==Problem 21==
 Let <math>\lfloor x \rfloor</math> denote the greatest integer less than or equal to <math>x</math>. How many positive integers <math>x < 2020</math>, satisfy the equation
   <math>\frac{x^{4} + 2020}{108} = \lfloor \sqrt (x^{2} - x)\rfloor</math>?
+==Problem 22==
+A convex hexagon <math>ABCDEF</math> is inscribed in a circle. <math>\overline {AB}</math> <math>=</math> <math>\overline {BC}</math> <math>=</math> <math>\overline {AD}</math> <math>=</math> <math>2</math>. <math>\overline {DE}</math> <math>=</math> <math>\overline {CF}</math> <math>=</math> <math>\overline {EF}</math> <math>=</math> <math>4</math>. The measure of <math>\overline {DC}</math> can be written as <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m + n</math>?
+==Problem 23==
+==Problem 24==
+A sequence <math>(a_n)</math> is defined as <math>a_1 = \frac{1}{\sqrt{3}}</math>, <math>a_2 = \sqrt{3}</math>, and for all <math>n > 1</math>,
+<math>a_{n + 1} = \frac{2a_n-1}{1 - a_n^2}</math>
+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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