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Difference between revisions of "AMC 12C 2020 Problems"

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==Problem 1==
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<math>2</math> glass bowls hang on <math>2</math> sides of a balance each having a weight of <math>6</math> pounds, <math>1</math> bowl having <math>l</math> lemons and the other bowl having <math>2l</math>. If lemons weigh <math>m</math> pounds each, how many lemons should be added to the lighter bowl to balance the scale?
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==Problem 1
  
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<math>\textbf{(A)}\ -3 \qquad\textbf{(B)}\ 0 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 7</math>
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A plane flies at a speed of <math>590</math> miles/hour. How many miles in two hours
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==Problem 2==
 
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<math>\bigtriangleup ABC</math> has side lengths of <math>7</math>, <math>8</math>, and <math>9</math>, and <math>\angle ABC = 60^{\circ}</math>. What is the smallest possible measure of <math>\angle BAC</math>?
 
  
 
==Problem 3==
 
==Problem 3==
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==Problem 5==
 
==Problem 5==
  
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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?
+
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>?
  
 
==Problem 6==
 
==Problem 6==
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==Problem 7==
 
==Problem 7==
  
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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==
 
==Problem 8==
  
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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>?
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<math>\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 16 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ 24</math>
 
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==Problem 9==
 
==Problem 9==
  
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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>?
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<math>R(20) + R(19) + R(18) + R(17) + … + R(3) + R(2) + R(1) + R(0)</math>?
 
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<math>\mathrm{(A) \ } 38 \qquad \mathrm{(B) \ } 42\qquad \mathrm{(C) \ } 46\qquad \mathrm{(D) \ } 50\qquad \mathrm{(E) \ } 54</math>
 
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==Problem 10==
 
==Problem 10==
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==Problem 11==
 
==Problem 11==
  
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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
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<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==
 
==Problem 12==
  
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An ant is lost inside a square <math>ABCD</math> with an unknown side length. The ant is <math>7</math> units away from <math>A</math>, <math>35</math> units away from <math>B</math>, and <math>49</math> units away from <math>C</math>. By how many units is the ant away from <math>D</math>?
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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==
 
==Problem 13==
  
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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?
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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?
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==Problem 14==
 
==Problem 14==
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==Problem 16==
 
==Problem 16==
  
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For some positive integer <math>k</math>, let <math>k</math> satisfy the equation
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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>?
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<math>log(k - 2)! + log(k - 1)! + 2 = 2 log(k!)</math>.
 
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What is the sum of the digits of <math>k</math>?
 
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<math>\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 19</math>
 
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==Problem 17==
 
==Problem 17==
  
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A <math>6</math> by <math>6</math> glass case of <math>36</math> glass boxes are to be filled with <math>3</math> purple balls and <math>3</math> red balls such that each row and column contains exactly <math>1</math> of each a red and purple ball. In how many ways can this arrangement be done?
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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==
 
==Problem 18==
  
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<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>\theta^{\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>?
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<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>?
  
  
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Let <math>P(x)</math> be a cubic polynomial with integral coefficients and roots <math>\cos \frac{\pi}{7}</math>, <math>\cos \frac{\pi}{11}</math>, and <math>\cos \frac{\pi}{13}</math>. What is the least possible sum of the coefficients of <math>P(x)</math>?
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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==
 
==Problem 20==
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==Problem 22==
 
==Problem 22==
  
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A convex hexagon <math>ABCDEF</math> is inscribed in a circle. <math>\overline {AB}</math> <math>=</math> <math>\overline {BC}</math>
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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>?
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==Problem 23==
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==Problem 24==
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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>
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 +
What is <math>\lfloor \ a_{2020}\rfloor</math>?
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==Problem 25==
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 +
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>?

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 

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


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 $k$?

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