Difference between revisions of "2013 AMC 12B Problems"
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− | [[2013 AMC | + | [[2013 AMC 10B Problems/Problem 25|Solution]] |
==Problem 24== | ==Problem 24== |
Revision as of 11:22, 7 April 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
- 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 to one more than the last number said by the other person. Jo starts by saying
, so Blair follows by saying
. Jo then says
, 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
. 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
Alex has red tokens and
blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token, and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?
Problem 11
Two bees start at the same spot and fly at the same rate in the following directions. Bee travels
foot north, then
foot east, then
foot upwards, and then continues to repeat this pattern. Bee
travels
foot south, then
foot west, and then continues to repeat this pattern. In what directions are the bees traveling when they are exactly
feet away from each other?
east,
west
north,
south
north,
west
up,
south
up,
west
Problem 12
Cities ,
,
,
, and
are connected by roads
,
,
,
,
,
, and
. How many different routes are there from
to
that use each road exactly once? (Such a route will necessarily visit some cities more than once.)
Problem 13
The internal angles of quadrilateral form an arithmetic progression. Triangles
and
are similar with
and
. Moreover, the angles in each of these two triangles also form an arithemetic progression. In degrees, what is the largest possible sum of the two largest angles of
?
Problem 14
Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is . What is the smallest possible value of
?
Problem 15
the number is expressed in the form
![$2013 = \frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$](http://latex.artofproblemsolving.com/6/6/9/6690a904c827dd9d3643929e24f32e7eea1074e5.png)
where and
are positive integers and
is as small as possible. What is
?
Problem 16
Let be an equiangular convex pentagon of perimeter
. The pairwise intersections of the lines that extend the sides of the pentagon determine a five-pointed star polygon. Let
be the perimeter of this star. What is the difference between the maximum and the minimum possible values of
.
Problem 17
Let and
be real numbers such that
What is the difference between the maximum and minimum possible values of ?
Problem 18
Barbara and Jenna play the following game, in which they take turns. A number of coins lie on a table. When it is Barbara’s turn, she must remove or
coins, unless only one coin remains, in which case she loses her turn. What it is Jenna’s turn, she must remove
or
coins. A coin flip determines who goes first. Whoever removes the last coin wins the game. Assume both players use their best strategy. Who will win when the game starts with
coins and when the game starts with
coins?
Barbara will win with
coins and Jenna will win with
coins.
Jenna will win with
coins, and whoever goes first will win with
coins.
Barbara will win with
coins, and whoever goes second will win with
coins.
Jenna will win with
coins, and Barbara will win with
coins.
Whoever goes first will win with
coins, and whoever goes second will win with
coins.
Problem 19
In triangle ,
,
, and
. Distinct points
,
, and
lie on segments
,
, and
, respectively, such that
,
, and
. The length of segment
can be written as
, where
and
are relatively prime positive integers. What is
?
Problem 20
For , points
and
are the vertices of a trapezoid. What is
?
Problem 21
Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point (0,0) and the directrix lines have the form with a and b integers such that
and
. No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?
Problem 22
Let and
be integers. Suppose that the product of the solutions for
of the equation
is the smallest possible integer. What is
?
Problem 23
Bernardo chooses a three-digit positive integer and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer
. For example, if
, Bernardo writes the numbers 10,444 and 3,245, and LeRoy obtains the sum
. For how many choices of
are the two rightmost digits of
, in order, the same as those of
?
Problem 24
Let be a triangle where
is the midpoint of
, and
is the angle bisector of
with
on
. Let
be the intersection of the median
and the bisector
. In addition
is equilateral with
. What is
?
Problem 25
Let be the set of polynomials of the form
where
are integers and
has distinct roots of the form
with
and
integers. How many polynomials are in
?
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 |