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Difference between revisions of "AMC 12C 2020"

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==Problem 1==
 
==Problem 1==
  
−
What is the sum of the solutions to the equation <math>(x + 5)(x + 4) - (x + 5)(x - 6) = 0</math>?
+
What is the sum of the solutions to the equation <math>(x + 5)(x + 4) - (x + 5)(x - 6) = 0</math> without multiplicity?
  
  
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==Problem 2==
 
==Problem 2==
  
−
How many increasing subsets of <math>{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}}</math> contain no <math>2</math> consecutive prime numbers?
+
How many increasing subsets of <math>\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}</math> contain no <math>2</math> consecutive prime numbers?
  
  
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A field is on the real <math>xy</math> plane in the shape of a circle, centered at <math>(5, 6)</math> with a a radius of <math>8</math>. The area that is in the field but above the line <math>y = x</math> is planted. What fraction of the field is planted?
 
A field is on the real <math>xy</math> plane in the shape of a circle, centered at <math>(5, 6)</math> with a a radius of <math>8</math>. The area that is in the field but above the line <math>y = x</math> is planted. What fraction of the field is planted?
−
 
  
  
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<math>\mathrm{(A) \ } -1000\qquad \mathrm{(B) \ } 1290\qquad \mathrm{(C) \ } 4356\qquad \mathrm{(D) \ } 7840\qquad \mathrm{(E) \ } 8764</math>
 
<math>\mathrm{(A) \ } -1000\qquad \mathrm{(B) \ } 1290\qquad \mathrm{(C) \ } 4356\qquad \mathrm{(D) \ } 7840\qquad \mathrm{(E) \ } 8764</math>
−
 
  
  
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<math>\mathrm{(A) \ } 15\qquad \mathrm{(B) \ } 16\qquad \mathrm{(C) \ } 18\qquad \mathrm{(D) \ } 19\qquad \mathrm{(E) \ } 22</math>
 
<math>\mathrm{(A) \ } 15\qquad \mathrm{(B) \ } 16\qquad \mathrm{(C) \ } 18\qquad \mathrm{(D) \ } 19\qquad \mathrm{(E) \ } 22</math>
−
 
  
  
 
==Problem 6==
 
==Problem 6==
 
<math>10</math> candy canes and <math>9</math> lollipops are to be distributed among <math>8</math> children such that each child gets atleast <math>1</math> candy. What is the probability that once the candies are distributed, no child has both types of candies?
 
<math>10</math> candy canes and <math>9</math> lollipops are to be distributed among <math>8</math> children such that each child gets atleast <math>1</math> candy. What is the probability that once the candies are distributed, no child has both types of candies?
−
 
  
  
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Persons <math>A</math> and <math>B</math> can plough a field in <math>10</math> days, persons <math>B</math> and <math>C</math> can plough the same field in <math>7</math> days, and persons <math>A</math> and <math>C</math> can plough the same field in <math>15</math> days. In how many days can all of them plough the field together?
 
Persons <math>A</math> and <math>B</math> can plough a field in <math>10</math> days, persons <math>B</math> and <math>C</math> can plough the same field in <math>7</math> days, and persons <math>A</math> and <math>C</math> can plough the same field in <math>15</math> days. In how many days can all of them plough the field together?
−
 
  
  
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−
<math>\mathrm{(A) \ } -12\qquad \mathrm{(B) \ } 0\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \ } 16\qquad \mathrm{(E) \ } 24</math>
+
<math>\mathrm{(A) \ } 38\qquad \mathrm{(B) \ } 42\qquad \mathrm{(C) \ } 46\qquad \mathrm{(D) \ } 50\qquad \mathrm{(E) \ } 54</math>
−
 
 
−
 
 
  
 
==Problem 9==
 
==Problem 9==
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probability of seeing a shooting star in the second <math>15</math> minutes of an hour stargazing on a  
 
probability of seeing a shooting star in the second <math>15</math> minutes of an hour stargazing on a  
 
random night?
 
random night?
−
 
  
  
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<math>R(100) + R(99) + R(98) + R(97) + … + R(3) + R(2) + R(1) + R(0)</math>?
 
<math>R(100) + R(99) + R(98) + R(97) + … + R(3) + R(2) + R(1) + R(0)</math>?
−
 
−
 
  
  
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A line of hunters walk into a jungle where the distance between the first and last hunter is <math>125</math> meters which maintains constant throughout their walk as the hunters walk at a constant speed of <math>5</math> meters per second. A butterfly starts from the front of their line and flies to the back as they come forward and then turns and comes back as soon as it reaches the back of the line. When the butterfly is back at the front of the line, the hunter finds out that the butterfly has travelled a distance of <math>625</math> meters. What was the speed of the butterfly?
 
A line of hunters walk into a jungle where the distance between the first and last hunter is <math>125</math> meters which maintains constant throughout their walk as the hunters walk at a constant speed of <math>5</math> meters per second. A butterfly starts from the front of their line and flies to the back as they come forward and then turns and comes back as soon as it reaches the back of the line. When the butterfly is back at the front of the line, the hunter finds out that the butterfly has travelled a distance of <math>625</math> meters. What was the speed of the butterfly?
−
 
  
  
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The pentagon <math>ABCDE</math> rolls on a straight line as each side of the pentagon touches the ground at <math>1</math> stage in the entire cycle. What is the length of the path that vertex <math>C</math> travels throughout <math>1</math> whole cycle?
 
The pentagon <math>ABCDE</math> rolls on a straight line as each side of the pentagon touches the ground at <math>1</math> stage in the entire cycle. What is the length of the path that vertex <math>C</math> travels throughout <math>1</math> whole cycle?
−
 
  
  
 
==Problem 14==
 
==Problem 14==
  
−
Let <math>P(x)</math> be a polynomial with integral coefficients and <math>S(x) = \frac{P(x)}{x(1 - x)}</math> for all nonzero values of <math>x</math>. If <math>P(2) = P(3) = 8</math>, what is the numerical value of <math>P(100)</math>?
+
Let <math>P(x)</math> be a polynomial with integral coefficients and <math>S(x) = \frac{P^{2}(x)}{x(2 - x)}</math> for all nonzero values of <math>x</math>. If <math>P(2) = P(3) = 8</math>, what is the sum of the digits in the numerical value of <math>P(100)</math>?
−
 
 
  
  
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Let <math>N</math> be <math>10^{100^{1000^{10000…}}}</math>. (All the way till the number consisting of <math>100</math> zeroes starting with a <math>1</math>. What is the remainder when N is divided by <math>629</math>?
 
Let <math>N</math> be <math>10^{100^{1000^{10000…}}}</math>. (All the way till the number consisting of <math>100</math> zeroes starting with a <math>1</math>. What is the remainder when N is divided by <math>629</math>?
 +
 +
 +
==Problem 16==
 +
 +
An urn consists of <math>7</math> golden blocks and <math>8</math> silver blocks. <math>3</math> pirates find the urn and randomly split the blocks equally. In how many ways can the pirates split the blocks such that no pirate who has more than <math>2</math> golden blocks has more than <math>2</math> silver blocks?
 +
 +
 +
==Problem 17==
 +
 +
How many solutions does the trigonometric equation <math>tan(cos(x)) = cos((x\pi^{2} - sin(x))</math> have in the interval <math>[-\pi, \pi]</math>?
 +
 +
 +
 +
==Problem 18==
 +
 +
The triangle <math>ABC</math> with <math>\overline {AB} = 8</math>, <math>\overline {BC} = 15</math>, and <math>\overline {AC} = 17</math> is lifted up with an elevation angle of <math>60^\circ</math>. A pole is dropped from <math>A</math> perpendicular to the ground with an altitude of 6, at point <math>D</math>. Ropes are created to connect the points on the triangle to make segments <math>\overline {BD}</math>, and <math>\overline {CD}</math>. What is the volume of <math>ABCD</math>?
 +
 +
 +
 +
==Problem 19==
 +
 +
A regular isocehedron(the polyhedron consisting of <math>20</math> equilateral triangle faces) floats in empty space in which <math>8</math> ants are on <math>8</math> of the edges in the polyhedron, each edge chosen at random. (Note that there are a total of <math>30</math> edges with <math>5</math> edges meeting at each of the <math>12</math> vertices, as shown in the figure below). Each minute the following happens: When a bell rings(each minute), each of the <math>8</math> ants pick a random adjacent edge to crawl onto from their current edge. This allows more than <math>1</math> ant at a chosen edge and atleast <math>22</math> edges to be left empty at all times. What is the probability that after the bell has rung <math>2020</math> times, that no <math>2</math> ants are on the same edge?
 +
 +
 +
 +
 +
==Problem 20==
 +
 +
The number <math>6084</math> can be written as a sum of the cubes of a number of consecutive integers. This means it is possible to write <math>6084 = a_{1}^{3} + a_{2}^{3} + … + a_{n}^{3}</math> where <math>n</math> is a positive integer strictly greater than <math>1</math>. What is the sum of the digits of <math>n</math>?
 +
 +
 +
 +
==Problem 21==
 +
Let <math>Q(x) = x^{2020} + x^{2019} + x^{2018} + … + x^{2} + x + 1</math>, and let <math>R(x) = x^{4} + x^{3} + x^{2} + x + 1</math>. Let <math>P</math> be the product of the <math>kth</math> power roots of <math>Q(R(x))</math> with multiplicity. Given that the least integer <math>y</math> such that <math>2^{y} > P</math> is <math>101</math>, what is the product of the digits of <math>k</math>?
 +
 +
 +
 +
==Problem 22==
 +
 +
The remainder when <math>10^{10} + 10^{20} + 10^{30} + … + 10^{90} + 10^{100}</math> is divided by <math>10 + 10^{2} + 10^{3} + … + 10^{9} + 10^{10}</math> can be written as <math>- R</math> where <math>R</math> is a positive integer. (This is the negative remainder of the division). What is <math>R</math> squared?
 +
 +
 +
 +
==Problem 23==
 +
 +
Let <math>T</math> be a triangle on the complex plane with vertices at <math>(0, 0), (6, 0)</math>, and <math>(0, 5)</math>. Let <math>S</math> be a set of rigid transformations consistsing of rotataions <math>90, 180</math>, and <math>270</math> degrees, and reflections across the <math>x</math> and <math>y</math> axes. The polygon formed by connecting the <math>nth</math> roots of unity has <math>n</math> sides, and any combination of 5 transformations in <math>S</math> applied to <math>T</math> brings atleast <math>1</math> vertex of <math>T</math> on to the <math>n</math> sided polygon. What is <math>n</math>?
 +
 +
 +
 +
==Problem 24==
 +
 +
Let <math>k</math> be the least positive integer greater than <math>10000</math> such that
 +
 +
<math>\left\lfloor \frac{x^{9} + 1}{x - 1}\right\rfloor + \left\lfloor \frac{x^{8} + 1}{x - 1}\right\rfloor + \left\lfloor \frac{x^{7} + 1}{x - 1}\right\rfloor</math> 
 +
is divisible by <math>7</math>.
 +
 +
What is the sum of the digits of <math>k</math>? (Note: <math>\lfloor x \rfloor</math> denotes the greatest integer less than or equal to <math>x</math>. )
 +
 +
 +
 +
==Problem 25==
 +
Let there be multiple ordered pairs <math>(n, k)</math> where <math>n</math> and <math>k</math> are positive integerswhich satisfy
 +
 +
 +
<math>{n+k-1 \choose k-1}{n+k-1 \choose k+1} = 100</math>. How many such ordered pairs <math>(n, k)</math> are there?
 +
 +
 +
 +
 +
 +
 +
 +
[[AMC 10C 2020]]

Latest revision as of 17:33, 21 April 2020


Problem 1

What is the sum of the solutions to the equation $(x + 5)(x + 4) - (x + 5)(x - 6) = 0$ without multiplicity?


$\mathrm{(A) \ } -10\qquad \mathrm{(B) \ } -3\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 15$

Problem 2

How many increasing subsets of $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ contain no $2$ consecutive prime numbers?


Problem 3

A field is on the real $xy$ plane in the shape of a circle, centered at $(5, 6)$ with a a radius of $8$. The area that is in the field but above the line $y = x$ is planted. What fraction of the field is planted?


Problem 4

What is the numerical value of $1^{3} + 2^{3} + 3^{3} + … + 11^{3}$?


$\mathrm{(A) \ } -1000\qquad \mathrm{(B) \ } 1290\qquad \mathrm{(C) \ } 4356\qquad \mathrm{(D) \ } 7840\qquad \mathrm{(E) \ } 8764$


Problem 5

$10$ cows can consume $20$ kilograms of grass in $5$ days. How many more cows are required such that it takes all of the cows to consume $80$ kilograms of grass in $8$ days?


$\mathrm{(A) \ } 15\qquad \mathrm{(B) \ } 16\qquad \mathrm{(C) \ } 18\qquad \mathrm{(D) \ } 19\qquad \mathrm{(E) \ } 22$


Problem 6

$10$ candy canes and $9$ lollipops are to be distributed among $8$ children such that each child gets atleast $1$ candy. What is the probability that once the candies are distributed, no child has both types of candies?


Problem 7

Persons $A$ and $B$ can plough a field in $10$ days, persons $B$ and $C$ can plough the same field in $7$ days, and persons $A$ and $C$ can plough the same field in $15$ days. In how many days can all of them plough the field together?


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 $a + b$?


$\mathrm{(A) \ } 38\qquad \mathrm{(B) \ } 42\qquad \mathrm{(C) \ } 46\qquad \mathrm{(D) \ } 50\qquad \mathrm{(E) \ } 54$

Problem 9

On a summer evening stargazing, the probability of seeing a shooting star in any given hour on a sunny day is $\frac{3}{5}$ and the probability of seeing a shooting star on a rainy day is $\frac{1}{3}$. Both rainy and sunny days happen with equal chances. What is the probability of seeing a shooting star in the second $15$ minutes of an hour stargazing on a random night?


Problem 10

Let $R(x)$ denote the number of trailing $0$s in the numerical value of the expression $x!$, for example, $R(5) = 1$ since $5! = 120$ which has $1$ trailing zero. What is the sum

$R(100) + R(99) + R(98) + R(97) + … + R(3) + R(2) + R(1) + R(0)$?


Problem 11

A line of hunters walk into a jungle where the distance between the first and last hunter is $125$ meters which maintains constant throughout their walk as the hunters walk at a constant speed of $5$ meters per second. A butterfly starts from the front of their line and flies to the back as they come forward and then turns and comes back as soon as it reaches the back of the line. When the butterfly is back at the front of the line, the hunter finds out that the butterfly has travelled a distance of $625$ meters. What was the speed of the butterfly?


Problem 12

How many positive base$- 4$ integers are divisible by $4$ but the sum of their digits is not divisible by $4$?


Problem 13

The pentagon $ABCDE$ rolls on a straight line as each side of the pentagon touches the ground at $1$ stage in the entire cycle. What is the length of the path that vertex $C$ travels throughout $1$ whole cycle?


Problem 14

Let $P(x)$ be a polynomial with integral coefficients and $S(x) = \frac{P^{2}(x)}{x(2 - x)}$ for all nonzero values of $x$. If $P(2) = P(3) = 8$, what is the sum of the digits in the numerical value of $P(100)$?


Problem 15

Let $N$ be $10^{100^{1000^{10000…}}}$. (All the way till the number consisting of $100$ zeroes starting with a $1$. What is the remainder when N is divided by $629$?


Problem 16

An urn consists of $7$ golden blocks and $8$ silver blocks. $3$ pirates find the urn and randomly split the blocks equally. In how many ways can the pirates split the blocks such that no pirate who has more than $2$ golden blocks has more than $2$ silver blocks?


Problem 17

How many solutions does the trigonometric equation $tan(cos(x)) = cos((x\pi^{2} - sin(x))$ have in the interval $[-\pi, \pi]$?


Problem 18

The triangle $ABC$ with $\overline {AB} = 8$, $\overline {BC} = 15$, and $\overline {AC} = 17$ is lifted up with an elevation angle of $60^\circ$. A pole is dropped from $A$ perpendicular to the ground with an altitude of 6, at point $D$. Ropes are created to connect the points on the triangle to make segments $\overline {BD}$, and $\overline {CD}$. What is the volume of $ABCD$?


Problem 19

A regular isocehedron(the polyhedron consisting of $20$ equilateral triangle faces) floats in empty space in which $8$ ants are on $8$ of the edges in the polyhedron, each edge chosen at random. (Note that there are a total of $30$ edges with $5$ edges meeting at each of the $12$ vertices, as shown in the figure below). Each minute the following happens: When a bell rings(each minute), each of the $8$ ants pick a random adjacent edge to crawl onto from their current edge. This allows more than $1$ ant at a chosen edge and atleast $22$ edges to be left empty at all times. What is the probability that after the bell has rung $2020$ times, that no $2$ ants are on the same edge?



Problem 20

The number $6084$ can be written as a sum of the cubes of a number of consecutive integers. This means it is possible to write $6084 = a_{1}^{3} + a_{2}^{3} + … + a_{n}^{3}$ where $n$ is a positive integer strictly greater than $1$. What is the sum of the digits of $n$?


Problem 21

Let $Q(x) = x^{2020} + x^{2019} + x^{2018} + … + x^{2} + x + 1$, and let $R(x) = x^{4} + x^{3} + x^{2} + x + 1$. Let $P$ be the product of the $kth$ power roots of $Q(R(x))$ with multiplicity. Given that the least integer $y$ such that $2^{y} > P$ is $101$, what is the product of the digits of $k$?


Problem 22

The remainder when $10^{10} + 10^{20} + 10^{30} + … + 10^{90} + 10^{100}$ is divided by $10 + 10^{2} + 10^{3} + … + 10^{9} + 10^{10}$ can be written as $- R$ where $R$ is a positive integer. (This is the negative remainder of the division). What is $R$ squared?


Problem 23

Let $T$ be a triangle on the complex plane with vertices at $(0, 0), (6, 0)$, and $(0, 5)$. Let $S$ be a set of rigid transformations consistsing of rotataions $90, 180$, and $270$ degrees, and reflections across the $x$ and $y$ axes. The polygon formed by connecting the $nth$ roots of unity has $n$ sides, and any combination of 5 transformations in $S$ applied to $T$ brings atleast $1$ vertex of $T$ on to the $n$ sided polygon. What is $n$?


Problem 24

Let $k$ be the least positive integer greater than $10000$ such that

$\left\lfloor \frac{x^{9} + 1}{x - 1}\right\rfloor + \left\lfloor \frac{x^{8} + 1}{x - 1}\right\rfloor + \left\lfloor \frac{x^{7} + 1}{x - 1}\right\rfloor$ is divisible by $7$.

What is the sum of the digits of $k$? (Note: $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. )


Problem 25

Let there be multiple ordered pairs $(n, k)$ where $n$ and $k$ are positive integerswhich satisfy


${n+k-1 \choose k-1}{n+k-1 \choose k+1} = 100$. How many such ordered pairs $(n, k)$ are there?




AMC 10C 2020

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