2024年4月2日发(作者：小盒老师数学试卷)

Problem 1

Call a -digit number geometric if it has distinct digits which, when read from

left to right, form a geometric sequence. Find the difference between the

largest and smallest geometric numbers.

Problem 2

There is a complex number with imaginary part

such that

Find .

Problem 3

A coin that comes up heads with probability and tails with probability

independently on each flip is flipped eight times. Suppose the

probability of three heads and five tails is equal to

heads and three tails. Let

integers. Find

Problem 4

In parallelogram

is on so that

.

, point is on so that and point

and

.

, where

of the probability of five

and a positive integer

and are relatively prime positive

. Let be the point of intersection of

. Find

Problem 5

Triangle has and . Points and are located on

and respectively so that , and is the angle bisector of

angle . Let be the point of intersection of and , and let be the

point on line

Problem 6

for which is the midpoint of . If , find .

How many positive integers less than are there such that the equation

denotes the greatest integer has a solution for ? (The notation

that is less than or equal to .)

Problem 7

The sequence satisfies and for

is an integer. Find .

. Let

be the least integer greater than for which

Problem 8

Let . Consider all possible positive differences of pairs

of elements of . Let be the sum of all of these differences. Find the

remainder when is divided by .

Problem 9

A game show offers a contestant three prizes A, B and C, each of which is

worth a whole number of dollars from $ to $ inclusive. The contestant

wins the prizes by correctly guessing the price of each prize in the order A, B,

C. As a hint, the digits of the three prices are given. On a particular day, the

digits given were . Find the total number of possible guesses for

all three prizes consistent with the hint.

Problem 10

The Annual Interplanetary Mathematics Examination (AIME) is written by a

committee of five Martians, five Venusians, and five Earthlings. At meetings,

committee members sit at a round table with chairs numbered from to in

clockwise order. Committee rules state that a Martian must occupy chair and

an Earthling must occupy chair , Furthermore, no Earthling can sit

immediately to the left of a Martian, no Martian can sit immediately to the left of

a Venusian, and no Venusian can sit immediately to the left of an Earthling.

The number of possible seating arrangements for the committee is

Find .

.

Problem 11

Consider the set of all triangles where is the origin and and are

distinct points in the plane with nonnegative integer coordinates such

that

a positive integer.

Problem 12

In right

altitude to

outside

. Find the number of such distinct triangles whose area is

with hypotenuse , , , and is the

. Let be the circle having as a diameter. Let be a point

such that and are both tangent to circle . The ratio of

to the length can be expressed in the form

.

, the perimeter of

where and are relatively prime positive integers. Find

Problem 13

The terms of the sequence

Problem 14

defined by for

.

are

positive integers. Find the minimum possible value of

For , define , where

.

. If and

, find the minimum possible value for

Problem 15

In triangle , , , and . Let be a point in the

interior of . Let and denote the incenters of triangles and

, respectively. The circumcircles of triangles and meet at

distinct points and . The maximum possible area of can be

expressed in the form , where , , and are positive integers and

is not divisible by the square of any prime. Find .

Problem 1

Of the students attending a school party, of the students are girls, and

of the students like to dance. After these students are joined by more

boy students, all of whom like to dance, the party is now girls. How many

students now at the party like to dance?

Solution

Problem 2

Square has sides of length units. Isosceles triangle

, and the area common to triangle and square

units. Find the length of the altitude to

Solution

Problem 3

Ed and Sue bike at equal and constant rates. Similarly, they jog at equal and

constant rates, and they swim at equal and constant rates. Ed covers

kilometers after biking for hours, jogging for hours, and swimming for

hours, while Sue covers kilometers after jogging for hours, swimming for

hours, and biking for hours. Their biking, jogging, and swimming rates are all

whole numbers of kilometers per hour. Find the sum of the squares of Ed\'s

biking, jogging, and swimming rates.

Solution

Problem 4

There exist unique positive integers and that satisfy the equation

. Find

Solution

Problem 5

A right circular cone has base radius and height . The cone lies on its side

on a flat table. As the cone rolls on the surface of the table without slipping, the

point where the cone\'s base meets the table traces a circular arc centered at

the point where the vertex touches the table. The cone first returns to its

original position on the table after making complete rotations. The value of

can be written in the form , where and are positive integers and

is not divisible by the square of any prime. Find .

.

in .

has base

is square

Problem 6

A triangular array of numbers has a first row consisting of the odd integers

in increasing order. Each row below the first has one fewer entry

than the row above it, and the bottom row has a single entry. Each entry in any

row after the top row equals the sum of the two entries diagonally above it in

the row immediately above it. How many entries in the array are multiples of

?

Problem 7

Let be the set of all integers such that . For example,

is the set . How many of the sets

do not contain a perfect square?

Problem 8

Find the positive integer such that

Problem 9

Ten identical crates each of dimensions ft ft ft. The first crate is placed

flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top

of the previous crate, and the orientation of each crate is chosen at random.

Let be the probability that the stack of crates is exactly

.

ft tall, where

and are relatively prime positive integers. Find

Problem 10

Let be an isosceles trapezoid with

is . The diagonals have length

whose angle at the

, and point is at longer base

distances and

foot of the altitude from

from vertices and , respectively. Let be the

to . The distance can be expressed in the

form

, where and are positive integers and is not divisible by the

. square of any prime. Find

Problem 11

Consider sequences that consist entirely of \'s and \'s and that have the

property that every run of consecutive \'s has even length, and every run of

consecutive \'s has odd length. Examples of such sequences are , ,

and , while is not such a sequence. How many such

sequences have length 14?

Problem 12

On a long straight stretch of one-way single-lane highway, cars all travel at the

same speed and all obey the safety rule: the distance form the back of the car

ahead to the front of the car behind is exactly one car length for each 15

kilometers per hour of speed or fraction thereof (Thus the front of a car

traveling 52 kilometers per hour will be four car lengths behind the back of the

car in front of it.) A photoelectric eye by the side of the road counts the number

of cars that pass in one hour. Assuming that each car is 4 meters long and that

the cars can travel at any speed, let be the maximum whole number of cars

that can pass the photoelectric eye in one hour. Find the quotient when

divided by 10.

Problem 13

Let

.

Suppose that

is

.

There is a point for which for all such polynomials, where

. Find , , and are positive integers, and are relatively prime, and

.

Solution

Problem 14

Let be a diameter of circle . Extend through to . Point lies on

so that line is tangent to . Point is the foot of the perpendicular from

to line . Suppose , and let denote the maximum possible

length of segment

Solution

Problem 15

A square piece of paper has sides of length . From each corner a wedge is

cut in the following manner: at each corner, the two cuts for the wedge each

start at distance from the corner, and they meet on the diagonal at an

angle of (see the figure below). The paper is then folded up along the lines

joining the vertices of adjacent cuts. When the two edges of a cut meet, they

are taped together. The result is a paper tray whose sides are not at right

angles to the base. The height of the tray, that is, the perpendicular distance

between the plane of the base and the plane formed by the upper edges, can

be written in the form , where and are positive integers, ,

and is not divisible by the th power of any prime. Find .

. Find .

Problem 1

How many positive perfect squares less than

Problem 2

A 100 foot long moving walkway moves at a constant rate of 6 feet per second.

Al steps onto the start of the walkway and stands. Bob steps onto the start of

the walkway two seconds later and strolls forward along the walkway at a

constant rate of 4 feet per second. Two seconds after that, Cy reaches the

start of the walkway and walks briskly forward beside the walkway at a

constant rate of 8 feet per second. At a certain time, one of these three

persons is exactly halfway between the other two. At that time, find the

distance in feet between the start of the walkway and the middle person.

Problem 3

The complex number is equal to , where is a positive real number

and . Given that the imaginary parts of and are the same, what is

equal to?

Problem 4

Three planets orbit a star circularly in the same plane. Each moves in the

same direction and moves at constant speed. Their periods are , , and

. The three planets and the star are currently collinear. What is the fewest

number of years from now that they will all be collinear again?

Problem 5

The formula for converting a Fahrenheit temperature

Celsius temperature is

to the corresponding

are multiples of 24?

An integer Fahrenheit temperature

is converted to Celsius, rounded to the nearest integer, converted back to

Fahrenheit, and again rounded to the nearest integer.

For how many integer Fahrenheit temperatures between 32 and 1000

inclusive does the original temperature equal the final temperature?

Problem 6

A frog is placed at the origin on the number line, and moves according to the

following rule: in a given move, the frog advances to either the closest point

with a greater integer coordinate that is a multiple of 3, or to the closest point

with a greater integer coordinate that is a multiple of 13. A move sequence is a

sequence of coordinates which correspond to valid moves, beginning with 0,

and ending with 39. For example, is a move sequence.

How many move sequences are possible for the frog?

Problem 7

Let

Find the remainder when

than or equal to , and

Problem 8

The polynomial

polynomials

are both factors of

Problem 9

?

is divided by 1000. ( is the greatest integer less

is the least integer greater than or equal to .)

is cubic. What is the largest value of for which the

and

In right triangle with right angle ,

and are extended beyond and . Points

and

and

. Its legs

lie in the exterior of

the triangle and are the centers of two circles with equal radii. The circle with

center is tangent to the hypotenuse and to the extension of leg , the

circle with center is tangent to the hypotenuse and to the extension of leg

, and the circles are externally tangent to each other. The length of the

radius of either circle can be expressed as

prime positive integers. Find .

, where and are relatively

Problem 10

In a 6 x 4 grid (6 rows, 4 columns), 12 of the 24 squares are to be shaded so

that there are two shaded squares in each row and three shaded squares in

each column. Let be the number of shadings with this property. Find the

remainder when is divided by 1000.

Problem 11

For each positive integer , let denote the unique positive integer such

that . For example, and . If find

the remainder when is divided by 1000.

Problem 12

In isosceles triangle , is located at the origin and

Point is in the first quadrant with and angle

is located at (20,0).

. If

triangle is rotated counterclockwise about point until the image of

lies on the positive -axis, the area of the region common to the original and

the rotated triangle is in the form

integers. Find

Problem 13

A square pyramid with base and vertex

A plane passes through the midpoints of ,

has eight edges of length 4.

, and . The plane\'s

. Find

.

, where are

intersection with the pyramid has an area that can be expressed as

.

Problem 14

A sequence is defined over non-negative integral indexes in the following way:

, .

Find the greatest integer that does not exceed

Problem 15

Let

and

be an equilateral triangle, and let

, respectively, with and

are

and be points on sides

. Point lies on side

is . The two

, where and are

such that angle . The area of triangle

possible values of the length of side

rational, and is an integer not divisible by the square of a prime. Find .

Problem 1

In quadrilateral is a right angle, diagonal

and

Problem 2

Let set be a 90-element subset of and let be the sum of

is perpendicular to

Find the perimeter of

the elements of

Problem 3

Find the number of possible values of

Find the least positive integer such that when its leftmost digit is deleted, the

resulting integer is

Problem 4

Let be the number of consecutive 0\'s at the right end of the decimal

representation of the product Find the remainder when

divided by 1000.

Problem 5

The number

where

Problem 6

Let be the set of real numbers that can be represented as repeating

decimals of the form where are distinct digits. Find the sum of the

elements of

can be written as

and are positive integers. Find

of the original integer.

is

Problem 7

An angle is drawn on a set of equally spaced parallel lines as shown. The ratio

of the area of shaded region to the area of shaded region is 11/5. Find the

ratio of shaded region to the area of shaded region

Problem 8

Hexagon is divided into five rhombuses, and as

shown. Rhombuses and are congruent, and each has area

Let be the area of rhombus Given that is a positive integer, find the

number of possible values for

Problem 9

The sequence is geometric with and common ratio where

and are positive integers. Given that

find the number of possible ordered

pairs

Problem 10

Eight circles of diameter 1 are packed in the first quadrant of the coordinte

plane as shown. Let region be the union of the eight circular regions. Line

with slope 3, divides into two regions of equal area. Line \'s equation can be

expressed in the form where and are positive integers

whose greatest common divisor is 1. Find

Problem 11

A collection of 8 cubes consists of one cube with edge-length for each

integer A tower is to be built using all 8 cubes according to the

rules:

Any cube may be the bottom cube in the tower.

The cube immediately on top of a cube with

edge-length must have edge-length at most

Let be the number of different towers than can be constructed. What is the

remainder when is divided by 1000?

Problem 12

Find the sum of the values of such that

where is measured in degrees and

Problem 13

For each even positive integer

divides For example,

let

and

denote the greatest power of 2 that

For each positive integer

let

perfect square.

Problem 14

Find the greatest integer less than 1000 such that is a

A tripod has three legs each of length 5 feet. When the tripod is set up, the

angle between any pair of legs is equal to the angle between any other pair,

and the top of the tripod is 4 feet from the ground In setting up the tripod, the

lower 1 foot of one leg breaks off. Let be the height in feet of the top of the

tripod from the ground when the broken tripod is set up. Then can be written

in the form where and are positive integers and is not divisible by

(The notation

)

denotes the greatest the square of any prime. Find

integer that is less than or equal to

Problem 15

Given that a sequence satisfies and for all integers

find the minimum possible value of

Problem 1

Six circles form a ring with with each circle externally tangent to two circles

adjacent to it. All circles are internally tangent to a circle with radius 30. Let

be the area of the region inside circle and outside of the six circles in the

ring. Find

Problem 2

For each positive integer let denote the increasing arithmetic sequence of

integers whose first term is 1 and whose common difference is For example,

is the sequence

term 2005?

Problem 3

How many positive integers have exactly three proper divisors, each of which

is less than 50?

Problem 4

The director of a marching band wishes to place the members into a formation

that includes all of them and has no unfilled positions. If they are arranged in a

square formation, there are 5 members left over. The director realizes that if he

arranges the group in a formation with 7 more rows than columns, there are no

members left over. Find the maximum number of members this band can

have.

Problem 5

Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins.

Each coin has an engraving of one face on one side, but not on the other. He

wants to stack the eight coins on a table into a single stack so that no two

adjacent coins are face to face. Find the number of possible distunguishable

arrangements of the 8 coins.

For how many values of does contain the

Problem 6

Let

Problem 7

In quadrilateral

Given that

integers, find

Problem 8

The equation

their sum is

Problem 9

Twenty seven unit cubes are painted orange on a set of four faces so that two

non-painted faces share an edge. The 27 cubes are randomly arranged to

form a cube. Given the probability of the entire surface area of the

where and are distinct primes and

and

where

has three real roots. Given that

and are relatively prime positive integers, find

and

where and are positive

be the product of the nonreal roots of Find

larger cube is orange is

are positive integers, find

Problem 10

Triangle lies in the Cartesian Plane and has an area of 70. The

and are and respectively, and the

has

coordinates of

coordinates of are

slope

The line containing the median to side

Find the largest possible value of

Problem 11

A semicircle with diameter is contained in a square whose sides have length

8. Given the maximum value of is find

Problem 12

For positive integers

of including 1 and

let denote the number of positive integer divisors

and Define by For example,

Let denote the number of positive integers

with

with

Problem 13

A particle moves in the Cartesian Plane according to the following rules:

1. From any lattice point

only move to

the particle may

or

odd, and let denote the number of positive integers

even. Find

2. There are no right angle turns in the particle\'s

path.

How many different paths can the particle take from

Problem 14

Consider the points and There is a unique

Let be

to ?

square such that each of the four points is on a different side of

the area of Find the remainder when is divided by 1000.

Problem 15

Triangle

median

has The incircle of the triangle evenly trisects the

If the area of the triangle is where and are integers and

is not divisible by the square of a prime, find


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