AMC美国高中数学竞赛难题精选

F

组题

A contains four liters of a solution that is 45% acid. Jar B contains five liters of

a solution that is 48% acid. Jar C contains one liter of a solution that is

From

jar

C,

acid.

liters

of

the

solution

is

added

to

jar

A,

and

the

remainder

of

the

and

are relatively prime positive integers, find

solution in jar C is added to jar B. At the end both jar A and jar B contain solutions

that are 50% acid. Given that

.

Answer:

Solution:Omited.

Resource: 2011 AIME I Problems1

2.

Let

be

the

line

with

slope

that

contains

the

point

,

and

let

. The original

the

-axis. In

is on the

be the line perpendicular to line

that contains the point

coordinate axes are erased, and line

is made the

-axis and line

the new coordinate system, point

positive

-axis. The point

coordinates

with coordinates

is on the positive

-axis, and point

.

in the original system has

in the new coordinate system. Find

Answer:

Solution:Omited.

Resource: 2011 AIME I Problems3

3.

Suppose that a parabola has vertex

where

and

and equation

,

is an integer

. The minimum possible value of

can be

written

in

the

form

,

where

and

are

relatively

prime

positive

integers.

Find

.

Answer:

Solution:Omited.

Resource: 2011 AIME I Problems6

4.

Suppose

is in the interval

and

. Find

.

Answer:

Solution:Omited.

Resource: 2011 AIME I Problems9

5.

For some integer

,

, and

. Find

Answer:

Solution:Omited.

, the polynomial

.

has the three integer roots

Resource: 2011 AIME I Problems15

6.

The sum of the first 2011 terms of a

geometric sequence

is 200. The sum of the

first 4022 terms is 380. Find the sum of the first 6033 terms.

Answer:

the first

Solution:Omited.

Resource: 2011 AIME II Problems/Problem 5

7.

Gary purchased a large beverage, but only drank

terms is

.

of it, where

and

are

relatively prime

positive integers. If he had purchased half as much and drunk twice

as much, he would have wasted only

Answer:

Solution:Omited.

Resource: 2011 AIME II Problems/Problem 1

8.

On

square

as much beverage. Find

.

, point

lies on side

and point

lies on side

.

, so that

. Find the area of the square

Answer:

810

Solution:Omited.

Resource: 2011 AIME II Problems/Problem 2

9.

The

degree

measures

of

the

angles

in

a

convex

18-sided

polygon

form

an

increasing

arithmetic sequence

with integer values. Find the degree measure of the

smallest

angle

.

Answer:

The first term is then

Solution:

The average angle in an 18-gon is

. In an arithmetic sequence the

average

is

the

same

as

the

median,

so

the

middle

two

terms

of

the

sequence

average

to

.

Thus

for

some

positive

(the

sequence

is

increasing

and

thus

,

the

middle

two

terms

are

and

.

the last term of the sequence is

, which must be

,

so

the

only

non-constant)

integer

Since the step is

less

than

,

since

the

polygon

is

convex.

This

gives

suitable positive integer

is 1.

Resource: 2011 AIME II Problems/Problem 3

10.

In

triangle

,

.

The

angle

bisector

of

. Let

intersects

at

point

of

, and point

and

is the midpoint of

to

be the point of the intersection

, where

. The ratio of

can be expressed in the form

.

and

are relatively prime positive integers. Find

Answer:

Solution:Omited.

Resource: 2011 AIME II Problems/Problem 4

.

11.

The sum of the first 2011 terms of a

geometric sequence

is 200. The sum of the

first 4022 terms is 380. Find the sum of the first 6033 terms

Answer:

542

.

Solution:Omited.

Resource: 2011 AIME II Problems/Problem 5

12.

Let

. A real number

is chosen at random from the interval

. The probability that

is equal to

, where

,

,

,

, and

are positive integers. Find

.

Answer:

850

Solution:Omited.

Resource: 2011 AIME II Problems/Problem15

13.

Maya lists all the positive divisors of

. She then randomly selects two

distinct divisors from this list. Let

be the

probability

that exactly one of the

selected divisors is a

perfect square

. The probability

can be expressed in the

form

, where

and

are

relatively prime

positive integers. Find

.

Answer:

107

Solution:Omited.

Resource: 2010 AIME I Problems/Problem1

14.

Suppose that

and

. The quantity

can be expressed as a

.

rational number

, where

and

are relatively prime positive integers. Find

Answer:

529

Solution:Omited.

Resource: 2010 AIME I Problems/Problem3

15.

Positive integers

,

,

, and

satisfy

,

,

and

Answer:

501

Solution:Omited.

. Find the number of possible values of

.

Resource: 2010 AIME I Problems/Problem5

16.

Rectangle

and a

semicircle

with diameter

are coplanar and have

nonoverlapping interiors. Let

denote the region enclosed by the semicircle and

the rectangle. Line

meets the semicircle, segment

, and segment

at

distinct points

,

, and

, respectively. Line

divides region

into two regions

with areas in the ratio

. Suppose that

,

, and

.

Then

can be represented as

, where

and

are positive integers and

is not divisible by the square of any prime. Find

.

Answer:

069

Solution:Omited.

Resource: 2010 AIME I Problems/Problem13

17.

In

with

,

, and

, let

be a point on

such that the

incircles

of

and

have equal

radii

. Let

and

be

. Find

.

positive

relatively prime

integers such that

Answer:

045