2011 AMC 10B Problems
2011 AMC 10B (Answer Key) Printable version: | AoPS Resources • PDF | ||
Instructions
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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
Josanna's test scores to date are and . Her goal is to raise her test average at least points with her next test. What is the minimum test score she would need to accomplish this goal?
Problem 3
At a store, when a length is reported as inches that means the length is at least inches and at most inches. Suppose the dimensions of a rectangular tile are reported as inches by inches. In square inches, what is the minimum area for the rectangle?
Problem 4
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip, it turned out that LeRoy had paid dollars and Bernardo had paid dollars, where . How many dollars must LeRoy give to Bernardo so that they share the costs equally?
Problem 5
In multiplying two positive integers and , Ron reversed the digits of the two-digit number . His erroneous product was . What is the correct value of the product of and ?
Problem 6
On Halloween Casper ate of his candies and then gave candies to his brother. The next day he ate of his remaining candies and then gave candies to his sister. On the third day he ate his final candies. How many candies did Casper have at the beginning?
Problem 7
The sum of two angles of a triangle is of a right angle, and one of these two angles is larger than the other. What is the degree measure of the largest angle in the triangle?
Problem 8
At a certain beach if it is at least and sunny, then the beach will be crowded. On June 10 the beach was not crowded. What can be concluded about the weather conditions on June 10?
Problem 9
The area of is one third of the area of . Segment is perpendicular to segment . What is ?
Problem 10
Consider the set of numbers . The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer?
Problem 11
There are people in a room. What is the largest value of such that the statement "At least people in this room have birthdays falling in the same month" is always true?
Problem 12
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of meters, and it takes her seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?
Problem 13
Two real numbers are selected independently at random from the interval . What is the probability that the product of those numbers is greater than zero?
Problem 14
A rectangular parking lot has a diagonal of meters and an area of square meters. In meters, what is the perimeter of the parking lot?
Problem 15
Let denote the "averaged with" operation: . Which of the following distributive laws hold for all numbers and ?
Problem 16
A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is probability that the dart lands within the center square?
Problem 17
In the given circle, the diameter is parallel to , and is parallel to . The angles and are in the ratio . What is the degree measure of angle ?
Problem 18
Rectangle has and . Point is chosen on side so that . What is the degree measure of ?
Problem 19
What is the product of all the roots of the equation
Problem 20
Rhombus has side length and . Region consists of all points inside the rhombus that are closer to vertex than any of the other three vertices. What is the area of ?
Problem 21
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 for ?
Problem 22
A pyramid has a square base with sides of length 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 23
What is the hundreds digit of ?
Problem 24
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 25
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 ?
See also
2011 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by 2011 AMC 10A Problems |
Followed by 2012 AMC 10A Problems | |
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All AMC 10 Problems and Solutions |