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) = | + | 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> | + | 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? |
Line 17: | Line 17: | ||
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> | ||
+ | |||
==Problem 5== | ==Problem 5== | ||
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<math>10</math> cows can consume <math>20</math> kilograms of grass in <math>5</math> days. How many more cows are required such that it takes all of the cows to consume <math>80</math> kilograms of grass in <math>8</math> days? | <math>10</math> cows can consume <math>20</math> kilograms of grass in <math>5</math> days. How many more cows are required such that it takes all of the cows to consume <math>80</math> kilograms of grass in <math>8</math> days? | ||
+ | |||
+ | <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? | ||
+ | |||
+ | |||
+ | ==Problem 7== | ||
+ | |||
+ | 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? | ||
+ | |||
+ | |||
+ | ==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>? | ||
+ | |||
+ | |||
+ | <math>\mathrm{(A) \ } 38\qquad \mathrm{(B) \ } 42\qquad \mathrm{(C) \ } 46\qquad \mathrm{(D) \ } 50\qquad \mathrm{(E) \ } 54</math> | ||
+ | |||
+ | ==Problem 9== | ||
+ | |||
+ | On a summer evening stargazing, the probability of seeing a shooting star in any given hour | ||
+ | on a sunny day is <math>\frac{3}{5}</math> and the probability of seeing a shooting star on a rainy | ||
+ | day is <math>\frac{1}{3}</math>. Both rainy and sunny days happen with equal chances. What is the | ||
+ | probability of seeing a shooting star in the second <math>15</math> minutes of an hour stargazing on a | ||
+ | random night? | ||
+ | |||
+ | |||
+ | ==Problem 10== | ||
+ | |||
+ | 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 | ||
+ | |||
+ | <math>R(100) + R(99) + R(98) + R(97) + … + R(3) + R(2) + R(1) + R(0)</math>? | ||
+ | |||
+ | |||
+ | ==Problem 11== | ||
+ | |||
+ | 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? | ||
+ | |||
+ | |||
+ | ==Problem 12== | ||
+ | |||
+ | How many positive base<math> - 4</math> integers are divisible by <math>4</math> but the sum of their digits is not divisible by <math>4</math>? | ||
+ | |||
+ | |||
+ | ==Problem 13== | ||
+ | |||
+ | 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== | ||
+ | |||
+ | 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>? | ||
+ | |||
+ | |||
+ | ==Problem 15== | ||
+ | |||
+ | 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 18:33, 21 April 2020
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
Problem 1
What is the sum of the solutions to the equation without multiplicity?
Problem 2
How many increasing subsets of contain no
consecutive prime numbers?
Problem 3
A field is on the real plane in the shape of a circle, centered at
with a a radius of
. The area that is in the field but above the line
is planted. What fraction of the field is planted?
Problem 4
What is the numerical value of ?
Problem 5
cows can consume
kilograms of grass in
days. How many more cows are required such that it takes all of the cows to consume
kilograms of grass in
days?
Problem 6
candy canes and
lollipops are to be distributed among
children such that each child gets atleast
candy. What is the probability that once the candies are distributed, no child has both types of candies?
Problem 7
Persons and
can plough a field in
days, persons
and
can plough the same field in
days, and persons
and
can plough the same field in
days. In how many days can all of them plough the field together?
Problem 8
The real value of that satisfies the equation
can be written in the form
where
and
are integers. What is
?
Problem 9
On a summer evening stargazing, the probability of seeing a shooting star in any given hour
on a sunny day is and the probability of seeing a shooting star on a rainy
day is
. Both rainy and sunny days happen with equal chances. What is the
probability of seeing a shooting star in the second
minutes of an hour stargazing on a
random night?
Problem 10
Let denote the number of trailing
s in the numerical value of the expression
, for example,
since
which has
trailing zero. What is the sum
?
Problem 11
A line of hunters walk into a jungle where the distance between the first and last hunter is meters which maintains constant throughout their walk as the hunters walk at a constant speed of
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
meters. What was the speed of the butterfly?
Problem 12
How many positive base integers are divisible by
but the sum of their digits is not divisible by
?
Problem 13
The pentagon rolls on a straight line as each side of the pentagon touches the ground at
stage in the entire cycle. What is the length of the path that vertex
travels throughout
whole cycle?
Problem 14
Let be a polynomial with integral coefficients and
for all nonzero values of
. If
, what is the sum of the digits in the numerical value of
?
Problem 15
Let be
. (All the way till the number consisting of
zeroes starting with a
. What is the remainder when N is divided by
?
Problem 16
An urn consists of golden blocks and
silver blocks.
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
golden blocks has more than
silver blocks?
Problem 17
How many solutions does the trigonometric equation have in the interval
?
Problem 18
The triangle with
,
, and
is lifted up with an elevation angle of
. A pole is dropped from
perpendicular to the ground with an altitude of 6, at point
. Ropes are created to connect the points on the triangle to make segments
, and
. What is the volume of
?
Problem 19
A regular isocehedron(the polyhedron consisting of equilateral triangle faces) floats in empty space in which
ants are on
of the edges in the polyhedron, each edge chosen at random. (Note that there are a total of
edges with
edges meeting at each of the
vertices, as shown in the figure below). Each minute the following happens: When a bell rings(each minute), each of the
ants pick a random adjacent edge to crawl onto from their current edge. This allows more than
ant at a chosen edge and atleast
edges to be left empty at all times. What is the probability that after the bell has rung
times, that no
ants are on the same edge?
Problem 20
The number can be written as a sum of the cubes of a number of consecutive integers. This means it is possible to write
where
is a positive integer strictly greater than
. What is the sum of the digits of
?
Problem 21
Let , and let
. Let
be the product of the
power roots of
with multiplicity. Given that the least integer
such that
is
, what is the product of the digits of
?
Problem 22
The remainder when is divided by
can be written as
where
is a positive integer. (This is the negative remainder of the division). What is
squared?
Problem 23
Let be a triangle on the complex plane with vertices at
, and
. Let
be a set of rigid transformations consistsing of rotataions
, and
degrees, and reflections across the
and
axes. The polygon formed by connecting the
roots of unity has
sides, and any combination of 5 transformations in
applied to
brings atleast
vertex of
on to the
sided polygon. What is
?
Problem 24
Let be the least positive integer greater than
such that
is divisible by
.
What is the sum of the digits of ? (Note:
denotes the greatest integer less than or equal to
. )
Problem 25
Let there be multiple ordered pairs where
and
are positive integerswhich satisfy
. How many such ordered pairs
are there?