2010 AMC 10B Problems
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
- 26 See also
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 ?
See also
2010 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by 2010 AMC 10A Problems |
Followed by 2011 AMC 10A 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 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.