2005年AIME I 真题：
Problem 1
Six congruent circles form a ring 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 
For how many values of 
does contain the term 2005?
Problem 3
How many positive integers have exactly three proper divisors (positive integral divisors excluding itself), 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 distinguishable arrangements of the 8 coins.
Problem 6
Let 
be the product of the nonreal roots of 
Find 
Problem 7
In quadrilateral 
and 
Given that 
where 
and 
are positive integers, find 
Problem 8
The equation 
has three real roots. Given that their sum is 
where 
and 
are relatively prime positive integers, find 
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, 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 
is greater than 
, the probability of obtaining the face opposite is less than 
. Given that the probability of obtaining face 
where 
has sides of length 1. Points 
and 
and 
respectively, so that 
is equilateral. A square with vertex 
has sides that are parallel to those of 
The length of a side of this smaller square is 
where 
and 
are positive integers and 
is not divisible by the square of any prime. Find 
such that 
and neither 
nor ![[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 
are positive integers and 
is a right angle, diagonal 
is perpendicular to 
and 
Find the perimeter of 
be a 90-element subset of 
and let 
be the sum of the elements of 
Find the number of possible values of 
of the original integer.
be the number of consecutive 0's at the right end of the decimal representation of the product 
Find the remainder when 
can be written as 
where 
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 
. Find the ratio of shaded region 
to the area 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 
. No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in 
where 
.
, and points 
and 
. Find 
.
workers can produce 
widgets and 
whoosits. In two hours, 
workers can produce 
widgets and 
workers can produce 
widgets and 
is drawn on graph paper with each square representing one unit in each direction. How many of the 
by 
satisfies 
if 
is odd, and 
if 
let 
denote the greatest integer less than or equal to 
For a certain integer 
positive integers 
such that ![$k=\lfloor\sqrt[3]{n_{1}}\rfloor = \lfloor\sqrt[3]{n_{2}}\rfloor = \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor$](https://latex.artofproblemsolving.com/a/2/a/a2a31bac0a2bc6d9943fbdf0ef0b1a3bc933895e.png)
and 
for all 
such that 
for 
are multiples of 24?
is equal to 
, where 
. Given that the imaginary parts of 
and 
are the same, what is 
, and 
years. 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?
An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the nearest integer.
and 
inclusive does the original temperature equal the final temperature?
, or to the closest point with a greater integer coordinate that is a multiple of 
, and ending with 
. For example, 
is a move sequence. How many move sequences are possible for the frog?
is the greatest integer less than or equal to 
is the least integer greater than or equal to 
is cubic. What is the largest value of 
and 
are both factors of 
with right angle 
and 
. Its legs 
and 
are extended beyond 
and 
lie in the exterior of the triangle and are the centers of two circles with equal radii. The circle with center 
, where 
.
, where the additions and subtractions alternate in pairs. Find the remainder when 
cm by 
cm. Ten slices are cut from the cheese. Each slice has a width of 
and 
(
) with each 
such that![\[a_13^{n_1} + a_23^{n_2} + \cdots + a_r3^{n_r} = 2008.\]](https://latex.artofproblemsolving.com/8/5/8/8583fcf8b5e36851cf3d1546038070e8b70e6061.png)
Find 
.
, let 
and 
. Let 
, 
, and 
and 
, respectively. Find the length 
.
is defined by![\[a_0 = 1,a_1 = 1, \text{ and } a_n = a_{n - 1} + \frac {a_{n - 1}^2}{a_{n - 2}}\text{ for }n\ge2.\]](https://latex.artofproblemsolving.com/4/0/8/408d131150595befc5ae2662573b99ec545e5819.png)
The sequence 
is defined by![\[b_0 = 1,b_1 = 3, \text{ and } b_n = b_{n - 1} + \frac {b_{n - 1}^2}{b_{n - 2}}\text{ for }n\ge2.\]](https://latex.artofproblemsolving.com/e/3/a/e3a90aa84fa76712b0de3ce0ce81cdef9c1d3e8d.png)
Find 
.
, and 
be the three roots of the equation![\[8x^3 + 1001x + 2008 = 0.\]](https://latex.artofproblemsolving.com/d/d/3/dd319820bf0b62935d0d87d0751f25f499d310b6.png)
Find 
.
. Find the smallest positive integer ![\[2[\cos(a)\sin(a) + \cos(4a)\sin(2a) + \cos(9a)\sin(3a) + \cdots + \cos(n^2a)\sin(na)]\]](https://latex.artofproblemsolving.com/1/3/0/130591afb17e8bdcc9d7334e4fcc26d57a8f65bd.png)
is an integer.
. Define a move for the particle as a counterclockwise rotation of 
radians about the origin followed by a translation of 
-direction. Given that the particle's position after 
, find the greatest integer less than or equal to 
.
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?
has sides of length 
has base 
, and the area common to triangle 
square units. Find the length of the altitude to 
.
kilometers after biking for 
hours, jogging for 
hours, while Sue covers 
kilometers after jogging for 
that satisfy the equation 
. Find 
.
. 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 
.
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 
?
be the set of all integers 
. For example, 
is the set 
. How many of the sets 
do not contain a perfect square?![\[\arctan\frac {1}{3} + \arctan\frac {1}{4} + \arctan\frac {1}{5} + \arctan\frac {1}{n} = \frac {\pi}{4}.\]](https://latex.artofproblemsolving.com/4/7/6/476f930a87b85c9e1e2e2b56aa2160dcc516e3a0.png)
be the probability that the stack of crates is exactly 
ft tall, where 
ounces of blue paint, 
ounces of red paint, and 
ounces of white paint. Bill painted four equally sized stripes on a wall, making a blue stripe, a red stripe, a white stripe, and a pink stripe. Pink is a mixture of red and white, not necessarily in equal amounts. When Bill finished, he had equal amounts of blue, red, and white paint left. Find the total number of ounces of paint Bill had left.
, 
, and 
. Find![\[a^{(\log_3 7)^2} + b^{(\log_7 11)^2} + c^{(\log_{11} 25)^2}.\]](https://latex.artofproblemsolving.com/6/8/0/680c5fe965e912d7a3cbf4cf823fe104d86d3124.png)
. Let 
and line 
are perpendicular, find the greatest integer less than 
.
grapes. The total number of grapes eaten in the contest was 
. Find the smallest possible value of 
, where 
.![[asy] unitsize(3mm); defaultpen(linewidth(.8pt)); dotfactor=4; pair A=(0,0), D=8*dir(330), C=8*dir(210), B=7*dir(90); pair Ep=(0,4-27/5); pair[] dotted={A,B,C,D,Ep}; draw(Circle(A,10)); draw(Circle(B,3)); draw(Circle(C,2)); draw(Circle(D,2)); draw(Circle(Ep,27/5)); dot(dotted); label("$E$",Ep,E); label("$A$",A,W); label("$B$",B,W); label("$C$",C,W); label("$D$",D,E); [/asy]](https://latex.artofproblemsolving.com/2/c/9/2c9d04c4ba8f11721bf58a4058fa3c4a849369c4.png)
to be 
for 
for 
is expressed as a fraction in lowest terms, its denominator is 
with 
.
, and let 
. Find the remainder when 
is divided by 
![\[\frac {z}{z + n} = 4i.\]](https://latex.artofproblemsolving.com/8/d/e/8deda249b1855d6825b6496e58411d8b23717c6d.png)
and tails with probability 
independently on each flip is flipped eight times. Suppose the probability of three heads and five tails is equal to 
of the probability of five heads and three tails. Let 
, where 
so that 
and point 
. Let 
. Find 
.
and 
. Points 
are located on 
, and 
is the angle bisector of angle 
and 
for which 
. If 
, find 
.
has a solution for 
satisfies 
and 
for 
. Let 
. Consider all possible positive differences of pairs of elements of 
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.
