Difference between revisions of "AMC 12C 2020 Problems"
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− | + | ==Problem 1 | |
− | + | A plane flies at a speed of <math>590</math> miles/hour. How many miles in two hours | |
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==Problem 3== | ==Problem 3== | ||
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==Problem 4== | ==Problem 4== | ||
− | <math> | + | 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> | <math>\textbf{(A)}\ 20 \qquad\textbf{(B)}\ \ 21 \qquad\textbf{(C)}\ \ 22 \qquad\textbf{(D)}\ \ 23 \qquad\textbf{(E)}\ 24</math> | ||
+ | ==Problem 5== | ||
− | == | + | 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== | |
+ | How many increasing(lower to higher numbered) subsets of <math>\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}</math> contain no <math>2</math> consecutive prime numbers? | ||
− | <math>\ | + | ==Problem 7== |
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+ | 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== | ||
− | + | 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>? | |
− | + | ==Problem 9== | |
+ | 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>? | ||
− | ==Problem | + | ==Problem 10== |
− | + | 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>? Source: JHMMC 2019 |
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+ | <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== | |
+ | 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== | |
− | <math> | + | 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== | ||
− | <math> | + | Let <math>K</math> be the set of solutions to the equation <math>(x + i)^{10} = 1</math> on the complex plane, where <math>i = \sqrt -1</math>. <math>2</math> points from <math>K</math> are chosen, such that a circle <math>\Omega</math> passes through both points. What is the least possible area of <math>\Omega</math>? |
+ | ==Problem 15== | ||
− | = | + | Let <math>N = 10^{10^{100…^{10000…(100 zeroes)}}}</math>. What is the remainder when <math>N</math> is divided by <math>629</math>? |
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− | ==Problem | + | ==Problem 16== |
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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>? | ||
− | + | ==Problem 17== | |
+ | 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} = 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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+ | <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== | ||
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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== | ||
+ | What is the maximum value of <math>\sum_{k = 1}^{6}(2^{x} + 3^{x})</math> as <math>x</math> varies through all real numbers to the nearest integer? | ||
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− | + | <math>\textbf{(A)}\ -3\qquad\textbf{(B)}\ -2\qquad\textbf{(C)}\ -1\qquad\textbf{(D)}\ 0\qquad\textbf{(E)}\ 1</math> | |
+ | ==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>? | ||
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− | + | ==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>? | ||
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− | + | ==Problem 23== | |
− | + | ==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> | + | <math>a_{n + 1} = \frac{2a_n-1}{1 - a_n^2}</math> |
+ | What is <math>\lfloor \ a_{2020}\rfloor</math>? | ||
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− | + | ==Problem 25== | |
− | <math> | + | 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 15:02, 27 November 2022
1 plus 1
==Problem 1
A plane flies at a speed of miles/hour. How many miles in two hours
Contents
- 1 Problem 3
- 2 Problem 4
- 3 Problem 5
- 4 Problem 6
- 5 Problem 7
- 6 Problem 8
- 7 Problem 9
- 8 Problem 10
- 9 Problem 11
- 10 Problem 12
- 11 Problem 13
- 12 Problem 14
- 13 Problem 15
- 14 Problem 16
- 15 Problem 17
- 16 Problem 18
- 17 Problem 19
- 18 Problem 20
- 19 Problem 21
- 20 Problem 22
- 21 Problem 23
- 22 Problem 24
- 23 Problem 25
Problem 3
In a bag are marbles consisting of
blue marbles and
red marbles. If each marble is pulled out
at a time, what is the probability that the
marble pulled out red?
Problem 4
A spaceship flies in space at a speed of miles/hour and the spaceship is paid
dollars for each
miles traveled. It’s only expense is fuel in which it pays
dollars per gallon, while going at a rate of
hours per gallon. Traveling
miles, how much money would the spaceship have gained?
Problem 5
Let be a function satisfying
for all real numbers
and
. Let
What is
?
Problem 6
How many increasing(lower to higher numbered) subsets of contain no
consecutive prime numbers?
Problem 7
A Regular Octagon has an area of . What is the sum of the lengths of the diagonals of the octagon?
Problem 8
What is the value of ?
Problem 9
Let denote the sum of the even digits of a positive integer and let
denote the sum of the odd digits of a positive integer. For some positive integer
,
=
. What is the product of the digits of the least possible such
?
Problem 10
In how many ways can candy canes and
lollipops be split between
children if each child must receive atleast
candy but no child receives both types?
Problem 11
Let be an isosceles trapezoid with
being parallel to
and
,
, and
. If
is the intersection of
and
, and
is the circumcenter of
, what is the length of
? Source: JHMMC 2019
Problem 12
For some positive integer , let
satisfy the equation
.
What is the sum of the digits of
?
Problem 13
An alien walks horizontally on the real number line starting at the origin. On each move, the alien can walk or
numbers the right or left of it. What is the expected distance from the alien to the origin after
moves?
Problem 14
Let be the set of solutions to the equation
on the complex plane, where
.
points from
are chosen, such that a circle
passes through both points. What is the least possible area of
?
Problem 15
Let . What is the remainder when
is divided by
?
Problem 16
Let and
be the vertex and focus of the Parabola
respectively. For a point
lying on the directrix of
, and a point
lying on
,
and Quadrilateral
is cyclic. If
has integral side lengths, what is the minimum possible area of
?
Problem 17
Let denote the
nonzero digit from the right in the base -
expansion of
, for example,
. What is the sum of the digits of
?
Problem 18
lays flat on the ground and has side lengths
, and
. Vertex
is then lifted up creating an elevation angle with the triangle and the ground of
. A wooden pole is dropped from
perpendicular to the ground, making an altitude of a
Dimensional figure. Ropes are connected from the foot of the pole,
, to form
other segments,
and
. What is the volume of
?
Problem 19
Let be a cubic polynomial with integral coefficients and roots
,
, and
. What is the least possible sum of the coefficients of
?
Problem 20
What is the maximum value of as
varies through all real numbers to the nearest integer?
Problem 21
Let denote the greatest integer less than or equal to
. How many positive integers
, satisfy the equation
?
Problem 22
A convex hexagon is inscribed in a circle.
.
. The measure of
can be written as
where
and
are relatively prime positive integers. What is
?
Problem 23
Problem 24
A sequence is defined as
,
, and for all
,
What is ?
Problem 25
Let and let
. Let
be the sum of the
power of the roots of
. It is given that the least positive integer
, such that
is
. What is
?