Difference between revisions of "2013 AMC 12B Problems"
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==Problem 9== | ==Problem 9== | ||
− | + | What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides <math>12!</math> ? | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12 </math> |
[[2013 AMC 12B Problems/Problem 9|Solution]] | [[2013 AMC 12B Problems/Problem 9|Solution]] |
Revision as of 09:51, 22 February 2013
2013 AMC 12B (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
[hide]- 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
On a particular January day, the high temperature in Lincoln, Nebraska, was degrees higher than the low temperature, and the average of the high and low temperatures was
. In degrees, what was the low temperature in Lincoln that day?
Problem 2
Mr. Green measures his rectangular garden by walking two of the sides and finds that it is steps by
steps. Each of Mr. Green's steps is
feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden?
Problem 3
When counting from to
,
is the
number counted. When counting backwards from
to
,
is the
number counted. What is
?
Problem 4
Ray's car averages miles per gallon of gasoline, and Tom's car averages
miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rate of miles per gallon of gasoline?
Problem 5
The average age of fifth-graders is
. The average age of
of their parents is
. What is the average age of all of these parents and fifth-graders?
Problem 6
Real numbers and
satisfy the equation
. What is
?
Problem 7
Jo and Blair take turns counting from 1 to one more than the last number said by the other person. Jo starts by saing "1", so Blair follows by saying "1, 2". Jo then says "1, 2, 3", and so on. What is the number said?
Problem 8
Line has equation
and goes through
. Line
has equation
and meets line
at point
. Line
has positive slope, goes through point
, and meets
at point
. The area of
is 3. What is the slope of
?
Problem 9
What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides ?
Problem 10
Rectangle has
and
. Point
is chosen on side
so that
. What is the degree measure of
?
Problem 11
A frog located at , with both
and
integers, makes successive jumps of length
and always lands on points with integer coordinates. Suppose that the frog starts at
and ends at
. What is the smallest possible number of jumps the frog makes?
Problem 12
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?
Problem 13
Brian writes down four integers whose sum is
. The pairwise positive differences of these numbers are
and
. What is the sum of the possible values of
?
Problem 14
A segment through the focus of a parabola with vertex
is perpendicular to
and intersects the parabola in points
and
. What is
?
Problem 15
How many positive two-digit integers are factors of ?
Problem 16
Rhombus has side length
and
. Region
consists of all points inside of the rhombus that are closer to vertex
than any of the other three vertices. What is the area of
?
Problem 17
Let , and
for integers
. What is the sum of the digits of
?
Problem 18
A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
Problem 19
A lattice point in an -coordinate system is any point
where both
and
are integers. The graph of
passes through no lattice point with
for all
such that
. What is the maximum possible value of
?
Problem 20
Triangle has
, and
. The points
, and
are the midpoints of
, and
respectively. Let
be the intersection of the circumcircles of
and
. What is
?
Problem 21
The arithmetic mean of two distinct positive integers and
is a two-digit integer. The geometric mean of
and
is obtained by reversing the digits of the arithmetic mean. What is
?
Problem 22
Let be a triangle with sides
, and
. For
, if
and
, and
are the points of tangency of the incircle of
to the sides
, and
, respectively, then
is a triangle with side lengths
, and
, if it exists. What is the perimeter of the last triangle in the sequence
?
Problem 23
A bug travels in the coordinate plane, moving only along the lines that are parallel to the -axis or
-axis. Let
and
. Consider all possible paths of the bug from
to
of length at most
. How many points with integer coordinates lie on at least one of these paths?
Problem 24
Let . What is the minimum perimeter among all the
-sided polygons in the complex plane whose vertices are precisely the zeros of
?
Problem 25
For every and
integers with
odd, denote by
the integer closest to
. For every odd integer
, let
be the probability that
for an integer randomly chosen from the interval
. What is the minimum possible value of
over the odd integers
in the interval
?
See also
2013 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by 2013 AMC 12A Problems |
Followed by 2014 AMC 12A 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 |