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Difference between revisions of "2004 AIME I Problems"

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== Problem 5 ==
 
== Problem 5 ==
−
Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta who did not attempt 300 points on the first day, had a positive integer score on each of the two days, and Beta's daily success rate (points scored divided by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was 300/500 = 3/5. The largest possible two-day success ratio that Beta could achieve is <math> m/n, </math> where <math> m </math> and <math> n </math> are relatively prime positive integers. What is <math> m+n </math>?
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Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of <math>500</math> points. Alpha scored <math>160</math> points out of <math>300</math> points attempted on the first day, and scored <math>140</math> points out of <math>200</math> points attempted on the second day. Beta who did not attempt <math>300</math> points on the first day, had a positive integer score on each of the two days, and Beta's daily success rate (points scored divided by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was <math>\frac{300}{500} = \frac{3}{5}</math>. The largest possible two-day success ratio that Beta could achieve is <math> m/n, </math> where <math> m </math> and <math> n </math> are relatively prime positive integers. What is <math> m+n </math>?
  
 
[[2004 AIME I Problems/Problem 5|Solution]]
 
[[2004 AIME I Problems/Problem 5|Solution]]
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* each of the <math> n </math> line segments intersects at least one of the other line segments at a point other than an endpoint,
 
* each of the <math> n </math> line segments intersects at least one of the other line segments at a point other than an endpoint,
 
* all of the angles at <math> P_1, P_2,\ldots, P_n </math> are congruent,
 
* all of the angles at <math> P_1, P_2,\ldots, P_n </math> are congruent,
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* all of the <math> n </math> line segments <math> P_2P_3,\ldots, P_nP_1 </math> are congruent, and
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* all of the <math> n </math> line segments <math>P_1P_2, P_2P_3,\ldots, P_nP_1 </math> are congruent, and
 
* the path <math> P_1P_2, P_2P_3,\ldots, P_nP_1 </math> turns counterclockwise at an angle of less than 180 degrees at each vertex.
 
* the path <math> P_1P_2, P_2P_3,\ldots, P_nP_1 </math> turns counterclockwise at an angle of less than 180 degrees at each vertex.
  
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For all positive integers <math>x</math>, let
 
For all positive integers <math>x</math>, let
 
<cmath>
 
<cmath>
−
f(x)=\begin{cases}1 &\mbox{if x = 1}\\ \frac x{10} &\mbox{if x is divisible by 10}\\ x+1 &\mbox{otherwise}\end{cases}
+
f(x)=\begin{cases}1 &\mbox{if }x = 1\\ \frac x{10} &\mbox{if }x\mbox{ is divisible by 10}\\ x+1 &\mbox{otherwise}\end{cases}
 
</cmath>
 
</cmath>
 
and define a sequence as follows: <math>x_1=x</math> and <math>x_{n+1}=f(x_n)</math> for all positive integers <math>n</math>. Let <math>d(x)</math> be the smallest <math>n</math> such that <math>x_n=1</math>. (For example, <math>d(100)=3</math> and <math>d(87)=7</math>.) Let <math>m</math> be the number of positive integers <math>x</math> such that <math>d(x)=20</math>. Find the sum of the distinct prime factors of <math>m</math>.
 
and define a sequence as follows: <math>x_1=x</math> and <math>x_{n+1}=f(x_n)</math> for all positive integers <math>n</math>. Let <math>d(x)</math> be the smallest <math>n</math> such that <math>x_n=1</math>. (For example, <math>d(100)=3</math> and <math>d(87)=7</math>.) Let <math>m</math> be the number of positive integers <math>x</math> such that <math>d(x)=20</math>. Find the sum of the distinct prime factors of <math>m</math>.
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== See Also ==
 
== See Also ==
 +
 +
{{AIME box|year = 2004|n=I|before=[[2003 AIME II Problems]]|after=[[2004 AIME II Problems]]}}
 +
 
* [[2004 AIME I]]
 
* [[2004 AIME I]]
 
* [[American Invitational Mathematics Examination]]
 
* [[American Invitational Mathematics Examination]]

Latest revision as of 12:34, 29 December 2021

2004 AIME I (Answer Key)
Printable version | AoPS Contest Collections • PDF

Instructions

  1. This is a 15-question, 3-hour examination. All answers are integers ranging from $000$ to $999$, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.
  2. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

Problem 1

The digits of a positive integer $n$ are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when $n$ is divided by 37?

Solution

Problem 2

Set $A$ consists of $m$ consecutive integers whose sum is $2m,$ and set $B$ consists of $2m$ consecutive integers whose sum is $m.$ The absolute value of the difference between the greatest element of $A$ and the greatest element of $B$ is 99. Find $m.$

Solution

Problem 3

A convex polyhedron $P$ has 26 vertices, 60 edges, and 36 faces, 24 of which are triangular, and 12 of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the same face. How many space diagonals does $P$ have?

Solution

Problem 4

A square has sides of length 2. Set $S$ is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $S$ enclose a region whose area to the nearest hundredth is $k.$ Find $100k.$

Solution

Problem 5

Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of $500$ points. Alpha scored $160$ points out of $300$ points attempted on the first day, and scored $140$ points out of $200$ points attempted on the second day. Beta who did not attempt $300$ points on the first day, had a positive integer score on each of the two days, and Beta's daily success rate (points scored divided by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was $\frac{300}{500} = \frac{3}{5}$. The largest possible two-day success ratio that Beta could achieve is $m/n,$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

Solution

Problem 6

An integer is called snakelike if its decimal representation $a_1a_2a_3\cdots a_k$ satisfies $a_i<a_{i+1}$ if $i$ is odd and $a_i>a_{i+1}$ if $i$ is even. How many snakelike integers between 1000 and 9999 have four distinct digits?

Solution

Problem 7

Let $C$ be the coefficient of $x^2$ in the expansion of the product $(1 - x)(1 + 2x)(1 - 3x)\cdots(1 + 14x)(1 - 15x).$ Find $|C|.$

Solution

Problem 8

Define a regular $n$-pointed star to be the union of $n$ line segments $P_1P_2, P_2P_3,\ldots, P_nP_1$ such that

  • the points $P_1, P_2,\ldots, P_n$ are coplanar and no three of them are collinear,
  • each of the $n$ line segments intersects at least one of the other line segments at a point other than an endpoint,
  • all of the angles at $P_1, P_2,\ldots, P_n$ are congruent,
  • all of the $n$ line segments $P_1P_2, P_2P_3,\ldots, P_nP_1$ are congruent, and
  • the path $P_1P_2, P_2P_3,\ldots, P_nP_1$ turns counterclockwise at an angle of less than 180 degrees at each vertex.

There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed stars are similar, but there are two non-similar regular 7-pointed stars. How many non-similar regular 1000-pointed stars are there?

Solution

Problem 9

Let $ABC$ be a triangle with sides 3, 4, and 5, and $DEFG$ be a 6-by-7 rectangle. A segment is drawn to divide triangle $ABC$ into a triangle $U_1$ and a trapezoid $V_1$ and another segment is drawn to divide rectangle $DEFG$ into a triangle $U_2$ and a trapezoid $V_2$ such that $U_1$ is similar to $U_2$ and $V_1$ is similar to $V_2.$ The minimum value of the area of $U_1$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Solution

Problem 10

A circle of radius 1 is randomly placed in a 15-by-36 rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m + n.$

Solution

Problem 11

A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum-shaped solid $F,$ in such a way that the ratio between the areas of the painted surfaces of $C$ and $F$ and the ratio between the volumes of $C$ and $F$ are both equal to $k.$ Given that $k=m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

Solution

Problem 12

Let $S$ be the set of ordered pairs $(x, y)$ such that $0 < x \le 1, 0<y\le 1,$ and $\left \lfloor{\log_2{\left(\frac 1x\right)}}\right \rfloor$ and $\left \lfloor{\log_5{\left(\frac 1y\right)}}\right \rfloor$ are both even. Given that the area of the graph of $S$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$ The notation $\left \lfloor{z}\right \rfloor$ denotes the greatest integer that is less than or equal to $z.$

Solution

Problem 13

The polynomial $P(x)=(1+x+x^2+\cdots+x^{17})^2-x^{17}$ has 34 complex roots of the form $z_k = r_k[\cos(2\pi a_k)+i\sin(2\pi a_k)], k=1, 2, 3,\ldots, 34,$ with $0 < a_1 \le a_2 \le a_3 \le \cdots \le a_{34} < 1$ and $r_k>0.$ Given that $a_1 + a_2 + a_3 + a_4 + a_5 = m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

Solution

Problem 14

A unicorn is tethered by a 20-foot silver rope to the base of a magician's cylindrical tower whose radius is 8 feet. The rope is attached to the tower at ground level and to the unicorn at a height of 4 feet. The unicorn has pulled the rope taut, the end of the rope is 4 feet from the nearest point on the tower, and the length of the rope that is touching the tower is $\frac{a-\sqrt{b}}c$ feet, where $a, b,$ and $c$ are positive integers, and $c$ is prime. Find $a+b+c.$

Solution

Problem 15

For all positive integers $x$, let \[f(x)=\begin{cases}1 &\mbox{if }x = 1\\ \frac x{10} &\mbox{if }x\mbox{ is divisible by 10}\\ x+1 &\mbox{otherwise}\end{cases}\] and define a sequence as follows: $x_1=x$ and $x_{n+1}=f(x_n)$ for all positive integers $n$. Let $d(x)$ be the smallest $n$ such that $x_n=1$. (For example, $d(100)=3$ and $d(87)=7$.) Let $m$ be the number of positive integers $x$ such that $d(x)=20$. Find the sum of the distinct prime factors of $m$.

Solution

See Also

2004 AIME I (Problems • Answer Key • Resources)
Preceded by
2003 AIME II Problems 		Followed by
2004 AIME II Problems
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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