Difference between revisions of "2003 AIME II Problems"
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== Problem 3 == | == Problem 3 == | ||
− | Define a <math>good | + | Define a <math>\text{good word}</math> as a sequence of letters that consists only of the letters <math>A</math>, <math>B</math>, and <math>C</math> - some of these letters may not appear in the sequence - and in which <math>A</math> is never immediately followed by <math>B</math>, <math>B</math> is never immediately followed by <math>C</math>, and <math>C</math> is never immediately followed by <math>A</math>. How many seven-letter good words are there? |
[[2003 AIME II Problems/Problem 3|Solution]] | [[2003 AIME II Problems/Problem 3|Solution]] | ||
== Problem 4 == | == 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 <math>m/n</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | + | 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 <math>m/n</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. |
[[2003 AIME II Problems/Problem 4|Solution]] | [[2003 AIME II Problems/Problem 4|Solution]] | ||
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== Problem 15 == | == Problem 15 == | ||
− | Let < | + | Let <cmath>P(x) = 24x^{24} + \sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).</cmath> Let <math>z_{1},z_{2},\ldots,z_{r}</math> be the distinct zeros of <math>P(x),</math> and let <math>z_{k}^{2} = a_{k} + b_{k}i</math> for <math>k = 1,2,\ldots,r,</math> where <math>a_{k}</math> and <math>b_{k}</math> are real numbers. Let |
<center><math>\sum_{k = 1}^{r}|b_{k}| = m + n\sqrt {p},</math></center> | <center><math>\sum_{k = 1}^{r}|b_{k}| = m + n\sqrt {p},</math></center> | ||
where <math>m, n,</math> and <math>p</math> are integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p.</math> | where <math>m, n,</math> and <math>p</math> are integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p.</math> |
Latest revision as of 09:02, 11 July 2023
2003 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
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 greatest integer multiple of 8, whose digits are all different. 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
are real numbers. Let
![$\sum_{k = 1}^{r}|b_{k}| = m + n\sqrt {p},$](http://latex.artofproblemsolving.com/b/6/9/b6959c0d9b67d1a2d6914af2b95338ccf226924b.png)
where and
are integers and
is not divisible by the square of any prime. Find
See also
2003 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2003 AIME I Problems |
Followed by 2004 AIME I Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.