# 2003 AMC12B 真题及答案详细解析

## Problem 1

Which of the following is the same as

## Problem 2

Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs 1 dollar more than a pink pill, and Al's pills cost a total of 546 dollars for the two weeks. How much does one green pill cost?

## Problem 8

Let  denote the sum of the digits of the positive integer . For example,  and  For how many two-digit values of  is

## Problem 9

Let  be a linear function for which  What is

## Problem 10

Several figures can be made by attaching two equilateral triangles to the regular pentagon ABCDE in two of the five positions shown. How many non-congruent figures can be constructed in this way?

## Problem 11

Cassandra sets her watch to the correct time at noon. At the actual time of 1:00 PM, she notices that her watch reads 12:57 and 36 seconds. Assuming that her watch loses time at a constant rate, what will be the actual time when her watch first reads 10:00 PM?

## Problem 12

What is the largest integer that is a divisor of  for all positive even integers ?

## Problem 13

An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies  of the volume of the frozen ice cream. What is the ratio of the cone’s height to its radius?

## Problem 14

In rectangle ABCD, AB = 5 and BC = 3. Points F and G are on CD so that DF = 1 and GC = 2. Lines AF and BG intersect at E. Find the area of

## Problem 15

A regular octagon  has an area of one square unit. What is the area of the rectangle ?

## Problem 16

Three semicircles of radius 1 are constructed on diameter AB of a semicircle of radius 2. The centers of the small semicircles divide AB into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles?

## Problem 17

If  and , what is ?

## Problem 18

Let  and  be positive integers such that  The minimum possible value of  has a prime factorization  What is ?

## Problem 19

Let  be the set of permutations of the sequence  for which the first term is not . A permutation is chosen randomly from . The probability that the second term is , in lowest terms, is . What is ?

## Problem 20

Part of the graph of  is shown. What is ?

## Problem 21

An object moves  cm in a straight line from  to , turns at an angle , measured in radians and chosen at random from the interval , and moves  cm in a straight line to . What is the probability that ?

## Problem 22

Let  be a rhombus with  and . Let  be a point on , and let  and  be the feet of the perpendiculars from to  and , respectively. Which of the following is closest to the minimum possible value of ?

## Problem 23

The number of -intercepts on the graph of  in the interval  is closest to

## Problem 24

Positive integers  and  are chosen so that , and the system of equations

and
has exactly one solution. What is the minimum value of ?

## Problem 25

Three points are chosen randomly and independently on a circle. What is the probability that all three pairwise distance between the points are less than the radius of the circle?

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