2010 AMC 12B Problems
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
Makarla attended two meetings during her -hour work day. The first meeting took
minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?
Problem 2
A big is formed as shown. What is its area?
![[asy] unitsize(4mm); defaultpen(linewidth(.8pt)); draw((0,0)--(5,0)--(5,2)--(2,2)--(2,8)--(0,8)--cycle); label("8",(0,4),W); label("5",(5/2,0),S); label("2",(5,1),E); label("2",(1,8),N); [/asy]](http://latex.artofproblemsolving.com/1/2/2/122a4290575eda1b63a0b302c47384ced6bb393e.png)
Problem 3
A ticket to a school play cost dollars, where
is a whole number. A group of 9th graders buys tickets costing a total of $
, and a group of 10th graders buys tickets costing a total of $
. How many values for
are possible?
Problem 4
A month with days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?
Problem 5
Lucky Larry's teacher asked him to substitute numbers for ,
,
,
, and
in the expression
and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The numbers Larry substituted for
,
,
, and
were
,
,
, and
, respectively. What number did Larry substitute for
?
Problem 6
At the beginning of the school year, of all students in Mr. Wells' math class answered "Yes" to the question "Do you love math", and
answered "No." At the end of the school year,
answered "Yes" and
answered "No." Altogether,
of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of
?
Problem 7
Shelby drives her scooter at a speed of miles per hour if it is not raining, and
miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of
miles in
minutes. How many minutes did she drive in the rain?
Problem 8
Every high school in the city of Euclid sent a team of students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed
th and
th, respectively. How many schools are in the city?
Problem 9
Let be the smallest positive integer such that
is divisible by
,
is a perfect cube, and
is a perfect square. What is the number of digits of
?
Problem 10
The average of the numbers and
is
. What is
?
Problem 11
A palindrome between and
is chosen at random. What is the probability that it is divisible by
?
Problem 12
For what value of does
Problem 13
In ,
and
. What is
?
Problem 14
Let ,
,
,
, and
be positive integers with
and let
be the largest of the sums
,
,
and
. What is the smallest possible value of
?
Problem 15
For how many ordered triples of nonnegative integers less than
are there exactly two distinct elements in the set
, where
?
Problem 16
Positive integers ,
, and
are randomly and independently selected with replacement from the set
. What is the probability that
is divisible by
?
Problem 17
The entries in a array include all the digits from
through
, arranged so that the entries in every row and column are in increasing order. How many such arrays are there?
Problem 18
A frog makes jumps, each exactly
meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog's final position is no more than
meter from its starting position?
Problem 19
A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than points. What was the total number of points scored by the two teams in the first half?
Problem 20
A geometric sequence has
,
, and
for some real number
. For what value of
does
?
Problem 21
Let , and let
be a polynomial with integer coefficients such that
, and
.
What is the smallest possible value of ?
Problem 22
Let be a cyclic quadrilateral. The side lengths of
are distinct integers less than
such that
. What is the largest possible value of
?
Problem 23
Monic quadratic polynomials and
have the property that
has zeros at
and
, and
has zeros at
and
. What is the sum of the minimum values of
and
?
Problem 24
The set of real numbers for which
is the union of intervals of the form . What is the sum of the lengths of these intervals?
Problem 25
For every integer , let
be the largest power of the largest prime that divides
. For example
. What is the largest integer
such that
divides
?
See also
2010 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by 2010 AMC 12A Problems |
Followed by 2011 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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.