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Difference between revisions of "2003 AIME II Problems"

(→‎Problem 2)
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== Problem 3 ==
 
== Problem 3 ==
 +
Define a <math>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>.
  
 
[[2003 AIME II Problems/Problem 4|Solution]]
 
[[2003 AIME II Problems/Problem 4|Solution]]
  
 
== Problem 5 ==
 
== Problem 5 ==
 +
A cylindrical log has diameter <math>12</math> 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 <math>45^\circ</math> 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 <math>n\pi</math>, where n is a positive integer. Find <math>n</math>.
  
 
[[2003 AIME II Problems/Problem 5|Solution]]
 
[[2003 AIME II Problems/Problem 5|Solution]]
  
 
== Problem 6 ==
 
== Problem 6 ==
 +
In triangle <math>ABC,</math> <math>AB = 13,</math> <math>BC = 14,</math> <math>AC = 15,</math> and point <math>G</math> is the intersection of the medians. Points <math>A',</math> <math>B',</math> and <math>C',</math> are the images of <math>A,</math> <math>B,</math> and <math>C,</math> respectively, after a <math>180^\circ</math> rotation about <math>G.</math> What is the area if the union of the two regions enclosed by the triangles <math>ABC</math> and <math>A'B'C'?</math>
  
 
[[2003 AIME II Problems/Problem 6|Solution]]
 
[[2003 AIME II Problems/Problem 6|Solution]]
  
 
== Problem 7 ==
 
== Problem 7 ==
 +
Find the area of rhombus <math>ABCD</math> given that the radii of the circles circumscribed around triangles <math>ABD</math> and <math>ACD</math> are <math>12.5</math> and <math>25</math>, respectively.
  
 
[[2003 AIME II Problems/Problem 7|Solution]]
 
[[2003 AIME II Problems/Problem 7|Solution]]
  
 
== Problem 8 ==
 
== Problem 8 ==
 +
Find the eighth term of the sequence <math>1440,</math> <math>1716,</math> <math>1848,\ldots,</math> whose terms are formed by multiplying the corresponding terms of two arithmetic sequences.
  
 
[[2003 AIME II Problems/Problem 8|Solution]]
 
[[2003 AIME II Problems/Problem 8|Solution]]
  
 
== Problem 9 ==
 
== Problem 9 ==
 +
Consider the polynomials <math>P(x) = x^{6} - x^{5} - x^{3} - x^{2} - x</math> and <math>Q(x) = x^{4} - x^{3} - x^{2} - 1.</math> Given that <math>z_{1},z_{2},z_{3},</math> and <math>z_{4}</math> are the roots of <math>Q(x) = 0,</math> find <math>P(z_{1}) + P(z_{2}) + P(z_{3}) + P(z_{4}).</math>
  
 
[[2003 AIME II Problems/Problem 9|Solution]]
 
[[2003 AIME II Problems/Problem 9|Solution]]
  
 
== Problem 10 ==
 
== Problem 10 ==
 +
Two positive integers differ by <math>60.</math> 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?
  
 
[[2003 AIME II Problems/Problem 10|Solution]]
 
[[2003 AIME II Problems/Problem 10|Solution]]
  
 
== Problem 11 ==
 
== Problem 11 ==
 +
Triangle <math>ABC</math> is a right triangle with <math>AC = 7,</math> <math>BC = 24,</math> and right angle at <math>C.</math> Point <math>M</math> is the midpoint of <math>AB,</math> and <math>D</math> is on the same side of line <math>AB</math> as <math>C</math> so that <math>AD = BD = 15.</math> Given that the area of triangle <math>CDM</math> may be expressed as <math>\frac {m\sqrt {n}}{p},</math> where <math>m,</math> <math>n,</math> and <math>p</math> are positive integers, <math>m</math> and <math>p</math> are relatively prime, and <math>n</math> is not divisible by the square of any prime, find <math>m + n + p.</math>
  
 
[[2003 AIME II Problems/Problem 11|Solution]]
 
[[2003 AIME II Problems/Problem 11|Solution]]
  
 
== Problem 12 ==
 
== Problem 12 ==
 +
The members of a distinguished committee were choosing a president, and each member gave one vote to one of the <math>27</math> candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least <math>1</math> than the number of votes for that candidate. What is the smallest possible number of members of the committee?
  
 
[[2003 AIME II Problems/Problem 12|Solution]]
 
[[2003 AIME II Problems/Problem 12|Solution]]
  
 
== Problem 13 ==
 
== 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 <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 13|Solution]]
 
[[2003 AIME II Problems/Problem 13|Solution]]
  
 
== Problem 14 ==
 
== Problem 14 ==
 +
Let <math>A = (0,0)</math> and <math>B = (b,2)</math> be points on the coordinate plane. Let <math>ABCDEF</math> be a convex equilateral hexagon such that <math>\angle FAB = 120^\circ,</math> <math>\overline{AB}\parallel \overline{DE},</math> <math>\overline{BC}\parallel \overline{EF,}</math> <math>\overline{CD}\parallel \overline{FA},</math> and the y-coordinates of its vertices are distinct elements of the set <math>\{0,2,4,6,8,10\}.</math> The area of the hexagon can be written in the form <math>m\sqrt {n},</math> where <math>m</math> and <math>n</math> are positive integers and n is not divisible by the square of any prime. Find <math>m + n.</math>
  
 
[[2003 AIME II Problems/Problem 14|Solution]]
 
[[2003 AIME II Problems/Problem 14|Solution]]
  
 
== Problem 15 ==
 
== Problem 15 ==
 +
Let
 +
<center><math>P(x) = 24x^{24} + \sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).</math></center>
 +
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>i = \sqrt { - 1},</math> and <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>
 +
where <math>m,</math> <math>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]]

Revision as of 13:31, 21 November 2007

Problem 1

The product $N$ 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 $N$.

Solution

Problem 2

Let $N$ be the greastest integer multiple of 8, no two whose digits are the same. What is the remainder when $N$ is divided by 1000.

Solution

Problem 3

Define a $good~word$ as a sequence of letters that consists only of the letters $A$, $B$, and $C$ - some of these letters may not appear in the sequence - and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, and $C$ is never immediately followed by $A$. How many seven-letter good words are there?

Solution

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 $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 5

A cylindrical log has diameter $12$ 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 $45^\circ$ 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 $n\pi$, where n is a positive integer. Find $n$.

Solution

Problem 6

In triangle $ABC,$ $AB = 13,$ $BC = 14,$ $AC = 15,$ and point $G$ is the intersection of the medians. Points $A',$ $B',$ and $C',$ are the images of $A,$ $B,$ and $C,$ respectively, after a $180^\circ$ rotation about $G.$ What is the area if the union of the two regions enclosed by the triangles $ABC$ and $A'B'C'?$

Solution

Problem 7

Find the area of rhombus $ABCD$ given that the radii of the circles circumscribed around triangles $ABD$ and $ACD$ are $12.5$ and $25$, respectively.

Solution

Problem 8

Find the eighth term of the sequence $1440,$ $1716,$ $1848,\ldots,$ whose terms are formed by multiplying the corresponding terms of two arithmetic sequences.

Solution

Problem 9

Consider the polynomials $P(x) = x^{6} - x^{5} - x^{3} - x^{2} - x$ and $Q(x) = x^{4} - x^{3} - x^{2} - 1.$ Given that $z_{1},z_{2},z_{3},$ and $z_{4}$ are the roots of $Q(x) = 0,$ find $P(z_{1}) + P(z_{2}) + P(z_{3}) + P(z_{4}).$

Solution

Problem 10

Two positive integers differ by $60.$ 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?

Solution

Problem 11

Triangle $ABC$ is a right triangle with $AC = 7,$ $BC = 24,$ and right angle at $C.$ Point $M$ is the midpoint of $AB,$ and $D$ is on the same side of line $AB$ as $C$ so that $AD = BD = 15.$ Given that the area of triangle $CDM$ may be expressed as $\frac {m\sqrt {n}}{p},$ where $m,$ $n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime, find $m + n + p.$

Solution

Problem 12

The members of a distinguished committee were choosing a president, and each member gave one vote to one of the $27$ candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least $1$ than the number of votes for that candidate. What is the smallest possible number of members of the committee?

Solution

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 $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m + n.$

Solution

Problem 14

Let $A = (0,0)$ and $B = (b,2)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\angle FAB = 120^\circ,$ $\overline{AB}\parallel \overline{DE},$ $\overline{BC}\parallel \overline{EF,}$ $\overline{CD}\parallel \overline{FA},$ and the y-coordinates of its vertices are distinct elements of the set $\{0,2,4,6,8,10\}.$ The area of the hexagon can be written in the form $m\sqrt {n},$ where $m$ and $n$ are positive integers and n is not divisible by the square of any prime. Find $m + n.$

Solution

Problem 15

Let

$P(x) = 24x^{24} + \sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).$

Let $z_{1},z_{2},\ldots,z_{r}$ be the distinct zeros of $P(x),$ and let $z_{K}^{2} = a_{k} + b_{k}i$ for $k = 1,2,\ldots,r,$ where $i = \sqrt { - 1},$ and $a_{k}$ and $b_{k}$ are real numbers. Let

$\sum_{k = 1}^{r}|b_{k}| = m + n\sqrt {p},$

where $m,$ $n,$ and $p$ are integers and $p$ is not divisible by the square of any prime. Find $m + n + p.$

Solution

See also

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