2003 AMC12A(美国数学竞赛)

2003 AMC 12A Problems

Problem 1

What is the difference between the sum of the first

and the sum of the first

odd counting numbers?

Solution

even 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?

Solution

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?

Solution

Problem 4

It takes Mary

her only

minutes to walk uphill

km from her home to school, but it takes

minutes to walk from school to her home along the same route. What is

her average speed, in km/hr, for the round trip?

Solution

Problem 5

The sum of the two 5-digit numbers

is

?

Solution

and

is

. What

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

Solution

Problem 7

How many non-congruent triangles with perimeter

have integer side lengths?

Solution

Problem 8

What is the probability that a randomly drawn positive factor of

Solution

is less than

?

Problem 9

A set

of points in the

and the line

. If

-plane is symmetric about the orgin, both coordinate axes,

is in

, what is the smallest number of points in

?

Solution

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?

Solution

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

.

Solution

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?

Solution

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?

Solution

Problem 14

Points

that

,

and

,

lie in the plane of the square

, and

such

has an

are equilateral triangles. If

.

area of 16, find the area of

Solution

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.

Solution

Problem 16

A point P is chosen at random in the interior of equilateral triangle

the probability that

has a greater area than each of

Solution

. What is

and

?

Problem 17

Square

has sides of length

, and

is the midpoint of

. A circle with

radius

and center

points

and

intersects a circle with radius

and center

at

?

. What is the distance from

to

Solution

Problem 18

Let

be a

-digit number, and let

and

be the quotient and the remainder,

respectively, when

is divided by

by

?

Solution

. For how many values of

is

divisible

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

the following describes the graph of

Solution

and

?

, respectively. Which 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?

Solution

Problem 21

The graph of the polynomial

has five distinct

-intercepts, one of which is at

coefficients cannot be zero?

Solution

. Which of the following

Problem 22

Objects

and

move simultaneously in the coordinate plane via a sequence of

steps, each of length one. Object

starts at

right or up, both equally likely. Object

starts at

and each of its steps is either

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?

Solution

Problem 23

How many perfect squares are divisors of the product

?

Solution

Problem 24

If

what is the largest possible value of

Solution

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

are the same set?

Solution

答案：

Problem 1

Solution

Solution 1

The first

The first

even counting numbers are

odd counting numbers are

.

.

Thus, the problem is asking for the value

of

.

Solution 2

Using the sum of an

arithmetic progression

formula, we can write this

as

.

Solution 3

The formula for the sum of the first

even numbers, is

for even).

Sum of first

odd numbers, is

Knowing this, plug

for

,

.

, (O standing for odd).

, (E standing

Problem 2

Solution

Since T-shirts cost

dollars more than a pair of socks, T-shirts

cost

dollars.

Since each member needs

pairs of socks and

T-shirts, the total cost

for

member is

Since

dollars.

was the cost per member, the

dollars was the cost for the club, and

number of members in the League is

Problem 3

Solution

The volume of the original box is

The volume of each cube that is removed is

Since there are

corners on the box,

cubes are removed.

So the total volume removed is

Therefore, the desired percentage is

.

Problem 4

Solution

Solution 1

Since she walked

km to school and

km back home, her total distance

is

km.

minutes walking to school and

minutes =

minutes walking back home,

Since she spent

her total time is

hours.

Therefore her average speed in km/hr is

.

Solution 2

The average speed of two speeds that travel the same distance is the

harmonic

mean

of the speeds, or

to school is

(for speeds

and

). Mary's speed going

. Plugging the numbers

, and her speed coming back is

in, we get that the average speed is

.

Problem 5

Solution

Since

,

Therefore,

, and

are digits,

,

,

.

.

Problem 6

Solution

Examining statement C:

when

, but statement C says that it does for all

.

Therefore the statement that is not true is

Problem 7