Difference between revisions of "2003 AIME II Problems"
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== Problem 2 == | == Problem 2 == | ||
− | Let <math>N</math> be the greatest integer multiple of 8, whose digits are all different. What is the remainder when <math>N</math> is divided by 1000 | + | Let <math>N</math> be the greatest integer multiple of 8, whose digits are all different. What is the remainder when <math>N</math> is divided by 1000? |
[[2003 AIME II Problems/Problem 2|Solution]] | [[2003 AIME II Problems/Problem 2|Solution]] | ||
== 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]] | ||
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== Problem 15 == | == Problem 15 == | ||
− | Let | + | Let <math>P(x) = x+2x^{2}+3x^{3} \dots 24x^{24}+23x^{25}+22x^{26} \dots x^{47}</math>. 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 |
− | |||
− | 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 | ||
<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, | + | 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> |
[[2003 AIME II Problems/Problem 15|Solution]] | [[2003 AIME II Problems/Problem 15|Solution]] | ||
== See also == | == See also == | ||
+ | |||
+ | {{AIME box|year = 2003|n=II|before=[[2003 AIME I Problems]]|after=[[2004 AIME I Problems]]}} | ||
+ | |||
* [[American Invitational Mathematics Examination]] | * [[American Invitational Mathematics Examination]] | ||
* [[AIME Problems and Solutions]] | * [[AIME Problems and Solutions]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 12:27, 22 June 2021
2003 AIME II (Answer Key) | AoPS Contest Collections | ||
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 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
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.