2003 AMC 12A 真题与答案及免费中文视频详解(翰林独家)
2003 AMC 12A Problems & Solutions
2003 AMC 12A 真题与答案及中文视频详解(翰林独家)
Problem 1
What is the difference between the sum of the first 
even counting numbers and the sum of the first odd counting numbers?
Problem 2
Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $2366, how many members are in the League?
Problem 3
A solid box is 
cm by 
cm by 
cm. A new solid is formed by removing a cube 
cm on a side from each corner of this box. What percent of the original volume is removed?
Problem 4
It takes Mary 
minutes to walk uphill 
km from her home to school, but it takes her only minutes to walk from school to her home along the same route. What is her average speed, in km/hr, for the round trip?
Problem 5
The sum of the two 5-digit numbers 
and 
is 
. What is 
?
Problem 6
Define 
to be 
for all real numbers 
and 
. Which of the following statements is not true?
for all and
for all and
for all
for all
if 
Problem 7
How many non-congruent triangles with perimeter 
have integer side lengths?
Problem 8
What is the probability that a randomly drawn positive factor of 
is less than ?
Problem 9
A set 
of points in the 
-plane is symmetric about the origin, both coordinate axes, and the line 
. If 
is in , what is the smallest number of points in ?
Problem 10
Al, Bert, and Carl are the winners of a school drawing for a pile of Halloween candy, which they are to divide in a ratio of 
, respectively. Due to some confusion they come at different times to claim their prizes, and each assumes he is the first to arrive. If each takes what he believes to be the correct share of candy, what fraction of the candy goes unclaimed?
Problem 11
A square and an equilateral triangle have the same perimeter. Let 
be the area of the circle circumscribed about the square and 
the area of the circle circumscribed around the triangle. Find 
.
Problem 12
Sally has five red cards numbered through 
and four blue cards numbered through 
. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards?
Problem 13
The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing?
Problem 14
Points 
and 
lie in the plane of the square 
such that 
, 
, 
, and 
are equilateral triangles. If has an area of 16, find the area of 
.
Problem 15
A semicircle of diameter sits at the top of a semicircle of diameter 
, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune.
Problem 16
A point P is chosen at random in the interior of equilateral triangle 
. What is the probability that 
has a greater area than each of 
and 
?
Problem 17
Square has sides of length 
, and 
is the midpoint of 
. A circle with radius and center intersects a circle with radius and center at points 
and 
. What is the distance from to 
?
Problem 18
Let 
be a -digit number, and let 
and 
be the quotient and the remainder, respectively, when is divided by 
. For how many values of is 
divisible by 
?
Problem 19
A parabola with equation 
is reflected about the -axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of 
and 
, respectively. Which of the following describes the graph of 
?
Problem 20
How many -letter arrangements of A's, B's, and C's have no A's in the first letters, no B's in the next letters, and no C's in the last letters?
Problem 21
The graph of the polynomial
has five distinct -intercepts, one of which is at 
. Which of the following coefficients cannot be zero?
Problem 22
Objects and move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object starts at and each of its steps is either right or up, both equally likely. Object starts at 
and each of its steps is either to the left or down, both equally likely. Which of the following is closest to the probability that the objects meet?
Problem 23
How many perfect squares are divisors of the product 
?
Problem 24
If 
what is the largest possible value of 
Problem 25
Let 
. For how many real values of 
is there at least one positive value of 
for which the domain of 
and the range of are the same set?
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