2000年AIME I 真题：
Problem 1
Find the least positive integer 
such that no matter how 
is expressed as the product of any two positive integers, at least one of these two integers contains the digit 
.
Problem 2
Let 
and 
be integers satisfying 
. Let 
, let 
be the reflection of 
across the line 
, let 
be the reflection of across the y-axis, let 
be the reflection of across the x-axis, and let 
be the reflection of across the y-axis. The area of pentagon 
is 
. Find 
.
Problem 3
In the expansion of 
where 
and 
are relatively prime positive integers, the coefficients of 
and 
are equal. Find 
.
Problem 4
The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle.
![[asy]defaultpen(linewidth(0.7)); draw((0,0)--(69,0)--(69,61)--(0,61)--(0,0));draw((36,0)--(36,36)--(0,36)); draw((36,33)--(69,33));draw((41,33)--(41,61));draw((25,36)--(25,61)); draw((34,36)--(34,45)--(25,45)); draw((36,36)--(36,38)--(34,38)); draw((36,38)--(41,38)); draw((34,45)--(41,45));[/asy]](https://latex.artofproblemsolving.com/b/4/b/b4bba1a273e9631a49fa74039b89e66ad4433573.png)
Problem 5
Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is 
One marble is taken out of each box randomly. The probability that both marbles are black is 
and the probability that both marbles are white is 
where 
and are relatively prime positive integers. What is 
?
Problem 6
For how many ordered pairs 
of integers is it true that 
and that the arithmetic mean of 
and 
is exactly 
more than the geometric mean of and ?
Problem 7
Suppose that 
and 
are three positive numbers that satisfy the equations 
and 
Then 
where and are relatively prime positive integers. Find .
Problem 8
A container in the shape of a right circular cone is 12 inches tall and its base has a 5-inch radius. The liquid that is sealed inside is 9 inches deep when the cone is held with its point down and its base horizontal. When the liquid is held with its point up and its base horizontal, the height of the liquid is ![$m - n\sqrt [3]{p},$](https://latex.artofproblemsolving.com/1/5/8/1586f2bca69882cdcb09b653822a61d6a5931db7.png)
where 
and 
are positive integers and is not divisible by the cube of any prime number. Find 
.
Problem 9
The system of equations
has two solutions 
and 
. Find 
.
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be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of 
be the largest number of students who could study both languages. Find 
.
find the value of 
.
. The lines whose equations are 
and 
contain points 
and 
, respectively, such that 
is the midpoint of 
. The length of 
equals 
, where 
if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets 
of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of 
is inscribed in a circle. Square 
has vertices 
on 
and vertices 
and 
on the circle. The ratio of the area of square 
. Find 
.
be a right triangle with 
, 
, and 
. Let 
be the inscribed circle. Construct 
with 
on 
and 
on 
, such that 
with 
on 
on 
be the inscribed circle of 
and 
the inscribed circle of 
. The distance between the centers of 
. What is 
has the properties that 
for all positive real values of 
for 
. Find the smallest 
.
of distinct real numbers has the following properties: the mean of 
is 
less than the mean of 
is 
more than the mean of 
, given that there are no multiple roots.
, angles 
degrees and 
degrees, respectively. The bisector of angle 
at 
. The area of triangle 
, where 
are positive integers, and 
.
. One vertex of the triangle is 
, one altitude is contained in the y-axis, and the length of each side is 
, where 
.
, where 
, 
and 
. Points 
and 
, respectively, such that 
is parallel to 
, where 
if the digits of the base-7 representation of 
is a 7-10 double because its base-7 representation is 
. What is the largest 7-10 double?
, 
and 
. Point 
. Let 
, 
, and 
, where 
, and 
are positive and satisfy 
and 
. The ratio of the area of triangle 
to the area of triangle 
and 
, inclusive; 
. How many distinct values of 
, 
, and 
. What is the surface area of the cube?
, where 
is the square of an integer. Find 
.
unit on a side are used to outline a garden by placing the blocks edge to edge with 
.
, then the area of the garden enclosed by the path, not including the path itself, is 
square units, where 
.
where 
is not a divisor of 
.
.
,
.
is a multiple of 
.
has no integer solutions for 
means the greatest integer less than or equal to 
. Let 
, where 
.![[asy] size(250);real x=sqrt(3); int i; draw(origin--(14,0)--(14,2+2x)--(0,2+2x)--cycle); for(i=0; i<7; i=i+1) { draw(Circle((2*i+1,1), 1)^^Circle((2*i+1,1+2x), 1)); } for(i=0; i<6; i=i+1) { draw(Circle((2*i+2,1+x), 1)); } [/asy]](https://latex.artofproblemsolving.com/9/5/4/954424e16552d40041fb782328143ad217cdfdf9.png)
be Dick's present age. How many ordered pairs of positive integers 
are possible?
for 
. Given that 
, for positive integers 
, find 
be the vertices of a regular dodecagon. How many distinct squares in the plane of the dodecagon have at least two vertices in the set 
?
are 
and 
. Find 
.
,
?
is a nondecreasing sequence of positive integers
for all 
th picket; Tanya starts with the second picket and paints every 
th picket; and Ulysses starts with the third picket and paints every 
when the triple 
of positive integers results in every picket being painted exactly once. Find the sum of all the paintable integers.
as a sequence of letters that consists only of the letters 
inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms a 
angle with the plane of the first cut. The intersection of these two planes has exactly one point in common with the log. The number of cubic inches in the wedge can be expressed as 
, where n is a positive integer. Find 
and point 
and 
are the images of 
and 
respectively, after a 
rotation about 
What is the area of the union of the two regions enclosed by the triangles 
and 
are 
and 
, respectively.
whose terms are formed by multiplying the corresponding terms of two arithmetic sequences.
and 
Given that 
and 
are the roots of 
find 
are drawn in a plane. The interior of the circle of radius 1 is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio of the total area of the green regions to the area of the circle of radius 100 can be expressed as 
where 
Susan makes a list as follows: for each two-element subset of 
she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list.
and that 
find 
where 
and 
where 
and 
Point 
and 
and 
are integers. Let 
be the sum of all possible perimeters of 
Find 
Find the sum of the four terms.
inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there?
where 
and 
are relatively prime, and neither 
is divided by 1000.
what is the least possible total for the number of bananas?
on edge 
The crease is 
where 
and 
The dimensions 
and 
are given. The perimeter of rectangle ![[asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(25,0), C=(25,70/3), D=(0,70/3), E=(8,0), F=(22,70/3), Bp=reflect(E,F)*B, Cp=reflect(E,F)*C; draw(F--D--A--E); draw(E--B--C--F, linetype("4 4")); filldraw(E--F--Cp--Bp--cycle, white, black); pair point=( 12.5, 35/3 ); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$B^\prime$", Bp, dir(point--Bp)); label("$C^\prime$", Cp, dir(point--Cp));[/asy]](https://latex.artofproblemsolving.com/8/3/2/8320d3082101e1e3a6a8d6b283794ffbdc9cf7b5.png)
are divisible by exactly 2004 positive integers?
and 
is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic progression, and, in general, for all 
the terms 
are in geometric progression, and the terms 
and 
are in arithmetic progression. Let 
be the greatest term in this sequence that is less than 1000. Find 
and set 
consecutive integers whose sum is 
The absolute value of the difference between the greatest element of 
Find 
points. Alpha scored 
points out of 
points attempted on the first day, and scored 
points out of 
. The largest possible two-day success ratio that Beta could achieve is 
satisfies 
if 
is odd and 
if 
in the expansion of the product 
Find 
such that
are coplanar and no three of them are collinear,
The number of possible sets of 6 cards that can be drawn from the deck is 6 times the number of possible sets of 3 cards that can be drawn. Find 
where 
where 
of integers such that 
and 
cards are numbered consecutively from 1 through 
The remaining cards form pile 
The cards are then restacked by taking cards alternately from the tops of pile 
becomes the bottom card of the new stack, card number 1 is on top of this card, and so on, until piles ![$x=\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)}.$](https://latex.artofproblemsolving.com/6/f/7/6f71d4755273a86295f916ae662a281e75af55c6.png)
Find 
and 
are externally tangent, and they are both internally tangent to circle 
The radii of 
is also a common external tangent of 
Given that the length of the chord is 
where 
and 
true for all real 
