Difference between revisions of "2010 AMC 10B Problems"
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Revision as of 11:15, 14 March 2020
2010 AMC 10B (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
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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
Problem 1
What is ?
Problem 2
Markala 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 3
A drawer contains red, green, blue, and white socks with at least 2 of each color. What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?
Problem 4
For a real number , define
to be the average of
and
. What is
?
Problem 5
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 6
A circle is centered at ,
is a diameter and
is a point on the circle with
.
What is the degree measure of
?
Problem 7
A triangle has side lengths ,
, and
. A rectangle has width
and area equal to the
area of the triangle. What is the perimeter of this rectangle?
Problem 8
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 9
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 10
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 11
A shopper plans to purchase an item that has a listed price greater than and can use any one of the three coupons. Coupon A gives
off the listed price, Coupon B gives
off the listed price, and Coupon C gives
off the amount by which the listed price exceeds
.
Let and
be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or Coupon C. What is
?
Problem 12
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 13
What is the sum of all the solutions of ?
Problem 14
The average of the numbers and
is
. What is
?
Problem 15
On a -question multiple choice math contest, students receive
points for a correct answer,
points for an answer left blank, and
point for an incorrect answer. Jesse’s total score on the contest was
. What is the maximum number of questions that Jesse could have answered correctly?
Problem 16
A square of side length and a circle of radius
share the same center. What is the area inside the circle, but outside the square?
Problem 17
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 18
Positive integers ,
, and
are randomly and independently selected with replacement from the set
. What is the probability that
is divisible by
?
Problem 19
A circle with center has area
. Triangle
is equilateral,
is a chord on the circle,
, and point
is outside
. What is the side length of
?
Problem 20
Two circles lie outside regular hexagon . The first is tangent to
, and the second is tangent to
. Both are tangent to lines
and
. What is the ratio of the area of the second circle to that of the first circle?
Problem 21
A palindrome between and
is chosen at random. What is the probability that it is divisible by
?
Problem 22
Seven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?
Problem 23
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 24
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 25
Let , and let
be a polynomial with integer coefficients such that
, and
.
What is the smallest possible value of ?