2002年AIME II 真题：
Problem 1
Given that 
and 
are both integers between 
and 
, inclusive; is the number formed by reversing the digits of ; and 
. How many distinct values of 
are possible?
Problem 2
Three vertices of a cube are 
, 
, and 
. What is the surface area of the cube?
Problem 3
It is given that 
, where 
, 
, and 
are positive integers that form an increasing geometric sequence and 
is the square of an integer. Find 
.
Problem 4
Patio blocks that are hexagons 
unit on a side are used to outline a garden by placing the blocks edge to edge with 
on each side. The diagram indicates the path of blocks around the garden when 
.
If 
, then the area of the garden enclosed by the path, not including the path itself, is 
square units, where 
is a positive integer. Find the remainder when is divided by 
.
Problem 5
Find the sum of all positive integers 
where and are non-negative integers, for which 
is not a divisor of 
.
Problem 6
Find the integer that is closest to 
.
Problem 7
It is known that, for all positive integers 
,
. such that 
is a multiple of 
.
Problem 8
Find the least positive integer for which the equation 
has no integer solutions for . (The notation 
means the greatest integer less than or equal to .)
Problem 9
Let 
be the set 
. Let be the number of sets of two non-empty disjoint subsets of . (Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when is divided by .
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, where 
.
, where 
and 
are positive integers. Find 
.![[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 
.
with 
,
?
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 
th picket. Call the positive integer 
when the triple 
of positive integers results in every picket being painted exactly once. Find the sum of all the paintable integers.
of three positive integers is 6 times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of 
as a sequence of letters that consists only of the letters 
, 
, and 
- some of these letters may not appear in the sequence - and in which 
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 
is the intersection of the medians. Points 
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 
given that the radii of the circles circumscribed around 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 
with 
and 
Point 
is not on 
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 positive integers, 
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 
is on 
and 
is on 
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 
has 26 vertices, 60 edges, and 36 faces, 24 of which are triangular, and 12 of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the same face. How many space diagonals does 
is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set 
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 
be the area of the region inside circle 
let 
denote the increasing arithmetic sequence of integers whose first term is 1 and whose common difference is 
is the sequence 
For how many values of 
Find 
and 
Given that 
where 
has three real roots. Given that their sum is 
, all six sides are congruent, 
and 
are right angles, and 
and 
are congruent. The area of the hexagonal region is 
Find 
, 
, and 
, where 
.
be a permutation of 
for which
Find the number of such permutations.
, and 
, the probability of obtaining face 
, the probability of obtaining the face opposite is less than 
. Given that the probability of obtaining face 
where 
and 
respectively, so that 
is equilateral. A square with vertex 
The length of a side of this smaller square is 
where 
and 
and neither ![[asy] size(50); pair A,B; A=(0,0); B=(2,0); pair C=rotate(60,A)*B; pair D, E, F; D = (1,0); E=rotate(60,A)*D; F=rotate(60,C)*E; draw(C--A--B--cycle); draw(D--E--F--cycle); [/asy]](https://latex.artofproblemsolving.com/1/1/d/11d5abd7384d03c423da478807ee2c6f2a5dfada.png)
and 
have their centers at 
, 
, and 
, and have radii 1, 2, and 4, respectively. Line 
is a common internal tangent to 
and 
and has a positive slope, and line 
is a common internal tangent to 
and that 
where 
and 
is a right angle, diagonal 
and 
Find the perimeter of 
be a 90-element subset of 
and let 
Find the number of possible values of 
of the original integer.
Find the remainder when 
can be written as 
where 
where 
are distinct digits. Find the sum of the elements of 
. Find the ratio of shaded region ![[asy] size(6cm); defaultpen(linewidth(0.7)+fontsize(10)); for(int i=0; i<4; i=i+1) { fill((2*i,0)--(2*i+1,0)--(2*i+1,6)--(2*i,6)--cycle, mediumgray); } pair A=(1/3,4), B=A+7.5*dir(-17), C=A+7*dir(10); draw(B--A--C); fill((7.3,0)--(7.8,0)--(7.8,6)--(7.3,6)--cycle, white); clip(B--A--C--cycle); for(int i=0; i<9; i=i+1) { draw((i,1)--(i,6)); } label("$\mathcal{A}$", A+0.2*dir(-17), S); label("$\mathcal{B}$", A+2.3*dir(-17), S); label("$\mathcal{C}$", A+4.4*dir(-17), S); label("$\mathcal{D}$", A+6.5*dir(-17), S); [/asy]](https://latex.artofproblemsolving.com/1/9/8/1985e47141f84873a1d3fffdb0d43fb205c5f8d8.png)
and 
, as shown. Rhombuses 
and 
Let 
![[asy] // TheMathGuyd size(8cm); pair A=(0,0), B=(4.2,0), C=(5.85,-1.6), D=(4.2,-3.2), EE=(0,-3.2), F=(-1.65,-1.6), G=(0.45,-1.6), H=(3.75,-1.6), I=(2.1,0), J=(2.1,-3.2), K=(2.1,-1.6); draw(A--B--C--D--EE--F--cycle); draw(F--G--(2.1,0)); draw(C--H--(2.1,0)); draw(G--(2.1,-3.2)); draw(H--(2.1,-3.2)); label("$\mathcal{T}$",(2.1,-1.6)); label("$\mathcal{P}$",(0,-1),NE); label("$\mathcal{Q}$",(4.2,-1),NW); label("$\mathcal{R}$",(0,-2.2),SE); label("$\mathcal{S}$",(4.2,-2.2),SW); [/asy]](https://latex.artofproblemsolving.com/6/a/0/6a063c7eb18cc2a02ebddd2f3216e2c02cb09c89.png)
is geometric with 
and common ratio 
where 
find the number of possible ordered pairs 
