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 | + | 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, | ||
− | * all of the <math> n </math> line segments <math> P_2P_3,\ldots, P_nP_1 </math> are congruent, and | + | * 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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== Problem 12 == | == Problem 12 == | ||
− | Let <math> S </math> be the set of ordered pairs <math> (x, y) </math> such that <math> 0 < x \le 1, 0<y\le 1, </math> and <math> \left | + | Let <math> S </math> be the set of ordered pairs <math> (x, y) </math> such that <math> 0 < x \le 1, 0<y\le 1, </math> and <math> \left \lfloor{\log_2{\left(\frac 1x\right)}}\right \rfloor </math> and <math> \left \lfloor{\log_5{\left(\frac 1y\right)}}\right \rfloor </math> are both even. Given that the area of the graph of <math> S </math> is <math> m/n, </math> where <math> m </math> and <math> n </math> are relatively prime positive integers, find <math> m+n. </math> The notation <math> \left \lfloor{z}\right \rfloor</math> denotes the greatest integer that is less than or equal to <math> z. </math> |
[[2004 AIME I Problems/Problem 12|Solution]] | [[2004 AIME I Problems/Problem 12|Solution]] | ||
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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 | + | 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 13:34, 29 December 2021
2004 AIME I (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 digits of a positive integer are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when
is divided by 37?
Problem 2
Set consists of
consecutive integers whose sum is
and set
consists of
consecutive integers whose sum is
The absolute value of the difference between the greatest element of
and the greatest element of
is 99. Find
Problem 3
A convex polyhedron 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
have?
Problem 4
A square has sides of length 2. Set 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
enclose a region whose area to the nearest hundredth is
Find
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 points. Alpha scored
points out of
points attempted on the first day, and scored
points out of
points attempted on the second day. Beta who did not attempt
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
. The largest possible two-day success ratio that Beta could achieve is
where
and
are relatively prime positive integers. What is
?
Problem 6
An integer is called snakelike if its decimal representation satisfies
if
is odd and
if
is even. How many snakelike integers between 1000 and 9999 have four distinct digits?
Problem 7
Let be the coefficient of
in the expansion of the product
Find
Problem 8
Define a regular -pointed star to be the union of
line segments
such that
- the points
are coplanar and no three of them are collinear,
- each of the
line segments intersects at least one of the other line segments at a point other than an endpoint,
- all of the angles at
are congruent,
- all of the
line segments
are congruent, and
- the path
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?
Problem 9
Let be a triangle with sides 3, 4, and 5, and
be a 6-by-7 rectangle. A segment is drawn to divide triangle
into a triangle
and a trapezoid
and another segment is drawn to divide rectangle
into a triangle
and a trapezoid
such that
is similar to
and
is similar to
The minimum value of the area of
can be written in the form
where
and
are relatively prime positive integers. Find
Problem 10
A circle of radius 1 is randomly placed in a 15-by-36 rectangle so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal
is
where
and
are relatively prime positive integers, find
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 and a frustum-shaped solid
in such a way that the ratio between the areas of the painted surfaces of
and
and the ratio between the volumes of
and
are both equal to
Given that
where
and
are relatively prime positive integers, find
Problem 12
Let be the set of ordered pairs
such that
and
and
are both even. Given that the area of the graph of
is
where
and
are relatively prime positive integers, find
The notation
denotes the greatest integer that is less than or equal to
Problem 13
The polynomial has 34 complex roots of the form
with
and
Given that
where
and
are relatively prime positive integers, find
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 feet, where
and
are positive integers, and
is prime. Find
Problem 15
For all positive integers , let
and define a sequence as follows:
and
for all positive integers
. Let
be the smallest
such that
. (For example,
and
.) Let
be the number of positive integers
such that
. Find the sum of the distinct prime factors of
.
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 |
- 2004 AIME I
- 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.