Difference between revisions of "2005 AIME I Problems"
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+ | {{AIME Problems|year=2005|n=I}} | ||
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== Problem 1 == | == Problem 1 == | ||
− | Six circles form a ring | + | Six congruent circles form a ring with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle <math> C </math> with radius 30. Let <math> K </math> be the area of the region inside circle <math> C </math> and outside of the six circles in the ring. Find <math> \lfloor K \rfloor. </math> |
[[2005 AIME I Problems/Problem 1|Solution]] | [[2005 AIME I Problems/Problem 1|Solution]] | ||
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== Problem 3 == | == Problem 3 == | ||
− | How many positive integers have exactly three proper divisors, each of which is less than 50? | + | How many positive integers have exactly three proper divisors (positive integral divisors excluding itself), each of which is less than 50? |
[[2005 AIME I Problems/Problem 3|Solution]] | [[2005 AIME I Problems/Problem 3|Solution]] | ||
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== Problem 5 == | == Problem 5 == | ||
− | Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible | + | Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the 8 coins. |
[[2005 AIME I Problems/Problem 5|Solution]] | [[2005 AIME I Problems/Problem 5|Solution]] | ||
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== Problem 9 == | == Problem 9 == | ||
− | Twenty seven unit cubes are painted orange on a set of four faces so that two | + | Twenty-seven unit cubes are painted orange on a set of four faces so that the two unpainted faces share an edge. The 27 cubes are then randomly arranged to form a <math> 3\times 3 \times 3 </math> cube. Given that the probability that the entire surface of the larger cube is orange is <math> \frac{p^a}{q^br^c}, </math> where <math> p,q, </math> and <math> r </math> are distinct primes and <math> a,b, </math> and <math> c </math> are positive integers, find <math> a+b+c+p+q+r. </math> |
[[2005 AIME I Problems/Problem 9|Solution]] | [[2005 AIME I Problems/Problem 9|Solution]] | ||
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== Problem 14 == | == Problem 14 == | ||
− | Consider the points <math> A(0,12), B(10,9), C(8,0), and D(-4,7). </math> There is a unique square <math> S </math> such that each of the four points is on a different side of <math> S. </math> Let <math> K </math> be the area of <math> S. </math> Find the remainder when <math> 10K </math> is divided by 1000. | + | Consider the points <math> A(0,12), B(10,9), C(8,0), </math> and <math> D(-4,7). </math> There is a unique square <math> S </math> such that each of the four points is on a different side of <math> S. </math> Let <math> K </math> be the area of <math> S. </math> Find the remainder when <math> 10K </math> is divided by 1000. |
[[2005 AIME I Problems/Problem 14|Solution]] | [[2005 AIME I Problems/Problem 14|Solution]] | ||
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== See Also == | == See Also == | ||
+ | |||
+ | {{AIME box|year = 2005|n=I|before=[[2004 AIME II Problems]]|after=[[2005 AIME II Problems]]}} | ||
+ | |||
* [[American Invitational Mathematics Examination]] | * [[American Invitational Mathematics Examination]] | ||
* [[AIME Problems and Solutions]] | * [[AIME Problems and Solutions]] | ||
* [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=50 2005 AIME I Math Jam Transcript] | * [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=50 2005 AIME I Math Jam Transcript] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
+ | {{MAA Notice}} |
Latest revision as of 05:06, 20 February 2019
2005 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
Six congruent circles form a ring with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle with radius 30. Let
be the area of the region inside circle
and outside of the six circles in the ring. Find
Problem 2
For each positive integer let
denote the increasing arithmetic sequence of integers whose first term is 1 and whose common difference is
For example,
is the sequence
For how many values of
does
contain the term 2005?
Problem 3
How many positive integers have exactly three proper divisors (positive integral divisors excluding itself), each of which is less than 50?
Problem 4
The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are 5 members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. Find the maximum number of members this band can have.
Problem 5
Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the 8 coins.
Problem 6
Let be the product of the nonreal roots of
Find
Problem 7
In quadrilateral and
Given that
where
and
are positive integers, find
Problem 8
The equation has three real roots. Given that their sum is
where
and
are relatively prime positive integers, find
Problem 9
Twenty-seven unit cubes are painted orange on a set of four faces so that the two unpainted faces share an edge. The 27 cubes are then randomly arranged to form a cube. Given that the probability that the entire surface of the larger cube is orange is
where
and
are distinct primes and
and
are positive integers, find
Problem 10
Triangle lies in the Cartesian Plane and has an area of 70. The coordinates of
and
are
and
respectively, and the coordinates of
are
The line containing the median to side
has slope
Find the largest possible value of
Problem 11
A semicircle with diameter is contained in a square whose sides have length 8. Given the maximum value of
is
find
Problem 12
For positive integers let
denote the number of positive integer divisors of
including 1 and
For example,
and
Define
by
Let
denote the number of positive integers
with
odd, and let
denote the number of positive integers
with
even. Find
Problem 13
A particle moves in the Cartesian Plane according to the following rules:
- From any lattice point
the particle may only move to
or
- There are no right angle turns in the particle's path.
How many different paths can the particle take from to
?
Problem 14
Consider the points and
There is a unique square
such that each of the four points is on a different side of
Let
be the area of
Find the remainder when
is divided by 1000.
Problem 15
Triangle has
The incircle of the triangle evenly trisects the median
If the area of the triangle is
where
and
are integers and
is not divisible by the square of a prime, find
See Also
2005 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2004 AIME II Problems |
Followed by 2005 AIME II 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
- 2005 AIME I Math Jam Transcript
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.