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==Problem 14== | ==Problem 14== | ||
What is the value of<cmath>\left(\sum_{k=1}^{20} \log_{5^k} 3^{k^2}\right)\cdot\left(\sum_{k=1}^{100} \log_{9^k} 25^k\right)?</cmath> | What is the value of<cmath>\left(\sum_{k=1}^{20} \log_{5^k} 3^{k^2}\right)\cdot\left(\sum_{k=1}^{100} \log_{9^k} 25^k\right)?</cmath> | ||
− | <math>\textbf{(A) }21 \qquad \textbf{(B) }100\log_5 3 \qquad \textbf{(C) }200\log_3 5 \qquad \textbf{(D) }2,200\qquad \textbf{(E) }21,000</math> | + | <math>\textbf{(A) }21 \qquad \textbf{(B) }100\log_5 3 \qquad \textbf{(C) }200\log_3 5 \qquad \textbf{(D) }2{,}200\qquad \textbf{(E) }21{,}000</math> |
[[2021 AMC 12A Problems/Problem 14|Solution]] | [[2021 AMC 12A Problems/Problem 14|Solution]] |
Revision as of 09:17, 15 August 2021
2021 AMC 12A (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
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
- 26 See also
Problem 1
What is the value of
Problem 2
Under what conditions does hold, where
and
are real numbers?
It is never true.
It is true if and only if
.
It is true if and only if
.
It is true if and only if
and
.
It is always true.
Problem 3
The sum of two natural numbers is . One of the two numbers is divisible by
. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?
Problem 4
Tom has a collection of snakes,
of which are purple and
of which are happy. He observes that
- all of his happy snakes can add,
- none of his purple snakes can subtract, and
- all of his snakes that can't subtract also can't add.
Which of these conclusions can be drawn about Tom's snakes?
Purple snakes can add.
Purple snakes are happy.
Snakes that can add are purple.
Happy snakes are not purple.
Happy snakes can't subtract.
Problem 5
When a student multiplied the number by the repeating decimal,
where
and
are digits, he did not notice the notation and just multiplied
times
Later he found that his answer is
less than the correct answer. What is the
-digit number
Problem 6
A deck of cards has only red cards and black cards. The probability of a randomly chosen card being red is . When
black cards are added to the deck, the probability of choosing red becomes
. How many cards were in the deck originally?
Problem 7
What is the least possible value of for all real numbers
and
Problem 8
A sequence of numbers is defined by and
for
. What are the parities (evenness or oddness) of the triple of numbers
, where
denotes even and
denotes odd?
Problem 9
Which of the following is equivalent to
Problem 10
Two right circular cones with vertices facing down as shown in the figure below contains the same amount of liquid. The radii of the tops of the liquid surfaces are cm and
cm. Into each cone is dropped a spherical marble of radius
cm, which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of the rise of the liquid level in the narrow cone to the rise of the liquid level in the wide cone?
Problem 11
A laser is placed at the point . The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the
-axis, then hit and bounce off the
-axis, then hit the point
. What is the total distance the beam will travel along this path?
Problem 12
All the roots of the polynomial are positive integers, possibly repeated. What is the value of
?
Problem 13
Of the following complex numbers , which one has the property that
has the greatest real part?
Problem 14
What is the value of
Problem 15
A choir direction must select a group of singers from among his tenors and
basses. The only
requirements are that the difference between the numbers of tenors and basses must be a multiple
of
, and the group must have at least one singer. Let
be the number of different groups that could be
selected. What is the remainder when
is divided by
?
Problem 16
In the following list of numbers, the integer appears
times in the list for
.
What is the median of the numbers in this list?
Problem 17
Trapezoid has
, and
. Let
be the intersection of the diagonals
and
, and let
be the midpoint of
. Given that
, the length of
can be written in the form
, where
and
are positive integers and
is not divisible by the square of any prime. What is
?
Problem 18
Let be a function defined on the set of positive rational numbers with the property that
for all positive rational numbers
and
. Furthermore, suppose that
also has the property that
for every prime number
. For which of the following numbers
is
?
Problem 19
How many solutions does the equation have in the closed interval
?
Problem 20
Suppose that on a parabola with vertex and a focus
there exists a point
such that
and
. What is the sum of all possible values of the length
Problem 21
The five solutions to the equation may be written in the form
for
where
and
are real. Let
be the unique ellipse that passes through the points
and
. The eccentricity of
can be written in the form
where
and
are relatively prime positive integers. What is
? (Recall that the eccentricity of an ellipse
is the ratio
, where
is the length of the major axis of
and
is the is the distance between its two foci.)
Problem 22
Suppose that the roots of the polynomial are
and
, where angles are in radians. What is
?
Problem 23
Frieda the frog begins a sequence of hops on a grid of squares, moving one square on each hop and choosing at random the direction of each hop up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite edge. For example if Frieda begins in the center square and makes two hops "up", the first hop would place her in the top row middle square, and the second hop would cause Frieda to jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the center square, makes at most four hops at random, and stops hopping if she lands on a corner square. What is the probability that she reaches a corner square on one of the four hops?
Problem 24
Semicircle has diameter
of length
. Circle
lies tangent to
at a point
and intersects
at points
and
. If
and
, then the area of
equals
, where
and
are relatively prime positive integers, and
is a positive integer not divisible by the square of any prime. What is
?
Problem 25
Let denote the number of positive integers that divide
, including
and
. For example,
and
. (This function is known as the divisor function.) Let
There is a unique positive integer
such that
for all positive integers
. What is the sum of the digits of
See also
2021 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by 2020 AMC 12B Problems |
Followed by 2021 AMC 12B Problems |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.