2003 AIME II Problems
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| 2003 AIME II (Answer Key) Printable version: AoPS Resources • PDF | MathLinks Resources • PDF | ||
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Instructions
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| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
Contents |
Problem 1
The product 
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 .
Problem 2
Let be the greastest integer multiple of 8, no two whose digits are the same. What is the remainder when is divided by 1000.
Problem 3
Define a 
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 is never immediately followed by , is never immediately followed by , and is never immediately followed by . How many seven-letter good words are there?
Problem 4
In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is 
, where 
and 
are relatively prime positive integers. Find 
.
Problem 5
A cylindrical log has diameter 
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 .
Problem 6
In triangle 
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 
Problem 7
Find the area of rhombus 
given that the radii of the circles circumscribed around triangles 
and 
are 
and 
, respectively.
Problem 8
Find the eighth term of the sequence 
whose terms are formed by multiplying the corresponding terms of two arithmetic sequences.
Problem 9
Consider the polynomials 
and 
Given that 
and 
are the roots of 
find 
Problem 10
Two positive integers differ by 
The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers?
Problem 11
Triangle is a right triangle with 
and right angle at 
Point 
is the midpoint of 
and 
is on the same side of line 
as so that 
Given that the area of triangle 
may be expressed as 
where 
and 
are positive integers, and are relatively prime, and is not divisible by the square of any prime, find 
Problem 12
The members of a distinguished committee were choosing a president, and each member gave one vote to one of the 
candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least 
than the number of votes for that candidate. What is the smallest possible number of members of the committee?
Problem 13
A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is 
where and are relatively prime positive integers, find 
Problem 14
Let 
and 
be points on the coordinate plane. Let 
be a convex equilateral hexagon such that 
and the y-coordinates of its vertices are distinct elements of the set 
The area of the hexagon can be written in the form 
where and are positive integers and n is not divisible by the square of any prime. Find
Problem 15
Let
Let 
be the distinct zeros of 
and let 
for 
where 
and 
and 
are real numbers. Let
where and are integers and is not divisible by the square of any prime. Find
