## 2011 AMC12B 真题

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## 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

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 4

In multiplying two positive integers and , Ron reversed the digits of the two-digit number . His erroneous product was 161. What is the correct value of the product of and ?

## Problem 5

Let be the second smallest positive integer that is divisible by every positive integer less than . What is the sum of the digits of ?

## Problem 6

Two tangents to a circle are drawn from a point . The points of contact and divide the circle into arcs with lengths in the ratio . What is the degree measure of ?

## Problem 7

Let and be two-digit positive integers with mean . What is the maximum value of the ratio ?

## Problem 8

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 width 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 9

Two real numbers are selected independently and at random from the interval . What is the probability that the product of those numbers is greater than zero?

## Problem 10

Rectangle has and . Point is chosen on side so that . What is the degree measure of ?

## Problem 11

A frog located at , with both and integers, makes successive jumps of length and always lands on points with integer coordinates. Suppose that the frog starts at and ends at . What is the smallest possible number of jumps the frog makes?

## Problem 12

A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?

## Problem 13

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 of ?

## Problem 14

A segment through the focus of a parabola with vertex is perpendicular to and intersects the parabola in points and . What is ?

## Problem 15

How many positive two-digit integers are factors of ?

## Problem 16

Rhombus has side length and . Region consists of all points inside of the rhombus that are closer to vertex than any of the other three vertices. What is the area of ?

## Problem 17

Let , and for integers . What is the sum of the digits of ?

## Problem 18

A pyramid has a square base with side of length 1 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 19

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 20

Triangle has , and . The points , and are the midpoints of , and respectively. Let be the intersection of the circumcircles of and . What is ?

## Problem 21

The arithmetic mean of two distinct positive integers and is a two-digit integer. The geometric mean of and is obtained by reversing the digits of the arithmetic mean. What is ?

## Problem 22

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 ?

## Problem 23

A bug travels in the coordinate plane, moving only along the lines that are parallel to the -axis or -axis. Let and . Consider all possible paths of the bug from to of length at most . How many points with integer coordinates lie on at least one of these paths?

## Problem 24

Let . What is the minimum perimeter among all the -sided polygons in the complex plane whose vertices are precisely the zeros of ?

## Problem 25

For every and integers with odd, denote by the integer closest to . For every odd integer , let be the probability that

for an integer randomly chosen from the interval . What is the minimum possible value of over the odd integers in the interval ?

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