Difference between revisions of "2007 AIME II Problems"
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A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in <math>2007</math>. No character may appear in a [[sequence]] more times than it appears among the four letters in AIME or the four digits in <math>2007</math>. A set of plates in which each possible sequence appears exactly once contains <math>N</math> license plates. Find <math>\frac{N}{10}</math>. | A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in <math>2007</math>. No character may appear in a [[sequence]] more times than it appears among the four letters in AIME or the four digits in <math>2007</math>. A set of plates in which each possible sequence appears exactly once contains <math>N</math> license plates. Find <math>\frac{N}{10}</math>. | ||
| − | [ | + | [http://home.comcast.net/~wolfand/ Solution] |
== Problem 2 == | == Problem 2 == | ||
Find the number of ordered triples <math>(a,b,c)</math> where <math>a</math>, <math>b</math>, and <math>c</math> are positive [[integer]]s, <math>a</math> is a [[factor]] of <math>b</math>, <math>a</math> is a factor of <math>c</math>, and <math>a+b+c=100</math>. | Find the number of ordered triples <math>(a,b,c)</math> where <math>a</math>, <math>b</math>, and <math>c</math> are positive [[integer]]s, <math>a</math> is a [[factor]] of <math>b</math>, <math>a</math> is a factor of <math>c</math>, and <math>a+b+c=100</math>. | ||
| − | [ | + | [http://home.comcast.net/~wolfand/ Solution] |
== Problem 3 == | == Problem 3 == | ||
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<div style="text-align:center;">[[Image:2007 AIME II-3.png]]</div> | <div style="text-align:center;">[[Image:2007 AIME II-3.png]]</div> | ||
| − | [ | + | [http://home.comcast.net/~wolfand/ Solution] |
== Problem 4 == | == Problem 4 == | ||
The workers in a factory produce widgets and whoosits. For each product, production time is [[constant]] and identical for all workers, but not necessarily equal for the two products. In one hour, <math>100</math> workers can produce <math>300</math> widgets and <math>200</math> whoosits. In two hours, <math>60</math> workers can produce <math>240</math> widgets and <math>300</math> whoosits. In three hours, <math>50</math> workers can produce <math>150</math> widgets and <math>m</math> whoosits. Find <math>m</math>. | The workers in a factory produce widgets and whoosits. For each product, production time is [[constant]] and identical for all workers, but not necessarily equal for the two products. In one hour, <math>100</math> workers can produce <math>300</math> widgets and <math>200</math> whoosits. In two hours, <math>60</math> workers can produce <math>240</math> widgets and <math>300</math> whoosits. In three hours, <math>50</math> workers can produce <math>150</math> widgets and <math>m</math> whoosits. Find <math>m</math>. | ||
| − | [ | + | [http://home.comcast.net/~wolfand/ Solution] |
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== Problem 5 == | == Problem 5 == | ||
The [[graph]] of the [[equation]] <math>9x+223y=2007</math> is drawn on graph paper with each [[square]] representing one [[unit square|unit]] in each direction. How many of the <math>1</math> by <math>1</math> graph paper squares have interiors lying entirely below the graph and entirely in the first [[quadrant]]? | The [[graph]] of the [[equation]] <math>9x+223y=2007</math> is drawn on graph paper with each [[square]] representing one [[unit square|unit]] in each direction. How many of the <math>1</math> by <math>1</math> graph paper squares have interiors lying entirely below the graph and entirely in the first [[quadrant]]? | ||
| − | [ | + | [http://home.comcast.net/~wolfand/ Solution] |
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== Problem 6 == | == Problem 6 == | ||
An integer is called ''parity-monotonic'' if its decimal representation <math>a_{1}a_{2}a_{3}\cdots a_{k}</math> satisfies <math>a_{i}<a_{i+1}</math> if <math>a_{i}</math> is [[odd]], and <math>a_{i}>a_{i+1}</math> if <math>a_{i}</math> is [[even]]. How many four-digit parity-monotonic integers are there? | An integer is called ''parity-monotonic'' if its decimal representation <math>a_{1}a_{2}a_{3}\cdots a_{k}</math> satisfies <math>a_{i}<a_{i+1}</math> if <math>a_{i}</math> is [[odd]], and <math>a_{i}>a_{i+1}</math> if <math>a_{i}</math> is [[even]]. How many four-digit parity-monotonic integers are there? | ||
| − | [ | + | [http://home.comcast.net/~wolfand/ Solution] |
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== Problem 7 == | == Problem 7 == | ||
Given a [[real number]] <math>x,</math> let <math>\lfloor x \rfloor</math> denote the [[floor function|greatest integer]] less than or equal to <math>x.</math> For a certain [[integer]] <math>k,</math> there are exactly <math>70</math> positive integers <math>n_{1}, n_{2}, \ldots, n_{70}</math> such that <math>k=\lfloor\sqrt[3]{n_{1}}\rfloor = \lfloor\sqrt[3]{n_{2}}\rfloor = \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor</math> and <math>k</math> divides <math>n_{i}</math> for all <math>i</math> such that <math>1 \leq i \leq 70.</math> | Given a [[real number]] <math>x,</math> let <math>\lfloor x \rfloor</math> denote the [[floor function|greatest integer]] less than or equal to <math>x.</math> For a certain [[integer]] <math>k,</math> there are exactly <math>70</math> positive integers <math>n_{1}, n_{2}, \ldots, n_{70}</math> such that <math>k=\lfloor\sqrt[3]{n_{1}}\rfloor = \lfloor\sqrt[3]{n_{2}}\rfloor = \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor</math> and <math>k</math> divides <math>n_{i}</math> for all <math>i</math> such that <math>1 \leq i \leq 70.</math> | ||
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Find the maximum value of <math>\frac{n_{i}}{k}</math> for <math>1\leq i \leq 70.</math> | Find the maximum value of <math>\frac{n_{i}}{k}</math> for <math>1\leq i \leq 70.</math> | ||
| − | [ | + | [http://home.comcast.net/~wolfand/ Solution] |
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== Problem 8 == | == Problem 8 == | ||
A [[rectangle|rectangular]] piece of paper measures 4 units by 5 units. Several [[line]]s are drawn [[parallel]] to the edges of the paper. A rectangle determined by the [[intersection]]s of some of these lines is called ''basic'' if | A [[rectangle|rectangular]] piece of paper measures 4 units by 5 units. Several [[line]]s are drawn [[parallel]] to the edges of the paper. A rectangle determined by the [[intersection]]s of some of these lines is called ''basic'' if | ||
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Given that the total length of all lines drawn is exactly 2007 units, let <math>N</math> be the maximum possible number of basic rectangles determined. Find the [[remainder]] when <math>N</math> is divided by 1000. | Given that the total length of all lines drawn is exactly 2007 units, let <math>N</math> be the maximum possible number of basic rectangles determined. Find the [[remainder]] when <math>N</math> is divided by 1000. | ||
| − | [ | + | [http://home.comcast.net/~wolfand/ Solution] |
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== Problem 9 == | == Problem 9 == | ||
[[Rectangle]] <math>ABCD</math> is given with <math>AB=63</math> and <math>BC=448.</math> Points <math>E</math> and <math>F</math> lie on <math>AD</math> and <math>BC</math> respectively, such that <math>AE=CF=84.</math> The [[inscribed circle]] of [[triangle]] <math>BEF</math> is [[tangent]] to <math>EF</math> at point <math>P,</math> and the inscribed circle of triangle <math>DEF</math> is tangent to <math>EF</math> at [[point]] <math>Q.</math> Find <math>PQ.</math> | [[Rectangle]] <math>ABCD</math> is given with <math>AB=63</math> and <math>BC=448.</math> Points <math>E</math> and <math>F</math> lie on <math>AD</math> and <math>BC</math> respectively, such that <math>AE=CF=84.</math> The [[inscribed circle]] of [[triangle]] <math>BEF</math> is [[tangent]] to <math>EF</math> at point <math>P,</math> and the inscribed circle of triangle <math>DEF</math> is tangent to <math>EF</math> at [[point]] <math>Q.</math> Find <math>PQ.</math> | ||
| − | [ | + | [http://home.comcast.net/~wolfand/ Solution] |
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== Problem 10 == | == Problem 10 == | ||
Let <math>S</math> be a [[set]] with six [[element]]s. Let <math>P</math> be the set of all [[subset]]s of <math>S.</math> Subsets <math>A</math> and <math>B</math> of <math>S</math>, not necessarily distinct, are chosen independently and at random from <math>P</math>. The [[probability]] that <math>B</math> is contained in at least one of <math>A</math> or <math>S-A</math> is <math>\frac{m}{n^{r}},</math> where <math>m</math>, <math>n</math>, and <math>r</math> are [[positive]] [[integer]]s, <math>n</math> is [[prime]], and <math>m</math> and <math>n</math> are [[relatively prime]]. Find <math>m+n+r.</math> (The set <math>S-A</math> is the set of all elements of <math>S</math> which are not in <math>A.</math>) | Let <math>S</math> be a [[set]] with six [[element]]s. Let <math>P</math> be the set of all [[subset]]s of <math>S.</math> Subsets <math>A</math> and <math>B</math> of <math>S</math>, not necessarily distinct, are chosen independently and at random from <math>P</math>. The [[probability]] that <math>B</math> is contained in at least one of <math>A</math> or <math>S-A</math> is <math>\frac{m}{n^{r}},</math> where <math>m</math>, <math>n</math>, and <math>r</math> are [[positive]] [[integer]]s, <math>n</math> is [[prime]], and <math>m</math> and <math>n</math> are [[relatively prime]]. Find <math>m+n+r.</math> (The set <math>S-A</math> is the set of all elements of <math>S</math> which are not in <math>A.</math>) | ||
| − | [ | + | [http://home.comcast.net/~wolfand/ Solution] |
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== Problem 11 == | == Problem 11 == | ||
Two long [[cylinder|cylindrical]] tubes of the same length but different [[diameter]]s lie [[parallel]] to each other on a [[plane|flat surface]]. The larger tube has [[radius]] <math>72</math> and rolls along the surface toward the smaller tube, which has radius <math>24</math>. It rolls over the smaller tube and continues rolling along the flat surface until it comes to rest on the same point of its [[circumference]] as it started, having made one complete revolution. If the smaller tube never moves, and the rolling occurs with no slipping, the larger tube ends up a [[distance]] <math>x</math> from where it starts. The distance <math>x</math> can be expressed in the form <math>a\pi+b\sqrt{c},</math> where <math>a,</math> <math>b,</math> and <math>c</math> are [[integer]]s and <math>c</math> is not divisible by the [[square]] of any [[prime]]. Find <math>a+b+c.</math> | Two long [[cylinder|cylindrical]] tubes of the same length but different [[diameter]]s lie [[parallel]] to each other on a [[plane|flat surface]]. The larger tube has [[radius]] <math>72</math> and rolls along the surface toward the smaller tube, which has radius <math>24</math>. It rolls over the smaller tube and continues rolling along the flat surface until it comes to rest on the same point of its [[circumference]] as it started, having made one complete revolution. If the smaller tube never moves, and the rolling occurs with no slipping, the larger tube ends up a [[distance]] <math>x</math> from where it starts. The distance <math>x</math> can be expressed in the form <math>a\pi+b\sqrt{c},</math> where <math>a,</math> <math>b,</math> and <math>c</math> are [[integer]]s and <math>c</math> is not divisible by the [[square]] of any [[prime]]. Find <math>a+b+c.</math> | ||
| − | [ | + | [http://home.comcast.net/~wolfand/ Solution] |
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== Problem 12 == | == Problem 12 == | ||
The increasing [[geometric sequence]] <math>x_{0},x_{1},x_{2},\ldots</math> consists entirely of [[integer|integral]] powers of <math>3.</math> Given that | The increasing [[geometric sequence]] <math>x_{0},x_{1},x_{2},\ldots</math> consists entirely of [[integer|integral]] powers of <math>3.</math> Given that | ||
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find <math>\log_{3}(x_{14}).</math> | find <math>\log_{3}(x_{14}).</math> | ||
| − | [ | + | [http://home.comcast.net/~wolfand/ Solution] |
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== Problem 13 == | == Problem 13 == | ||
A [[triangle|triangular]] [[array]] of [[square]]s has one square in the first row, two in the second, and in general, <math>k</math> squares in the <math>k</math>th row for <math>1 \leq k \leq 11.</math> With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in given diagram). In each square of the eleventh row, a <math>0</math> or a <math>1</math> is placed. Numbers are then placed into the other squares, with the entry for each square being the sum of the entries in the two squares below it. For how many initial distributions of <math>0</math>'s and <math>1</math>'s in the bottom row is the number in the top square a [[multiple]] of <math>3</math>? | A [[triangle|triangular]] [[array]] of [[square]]s has one square in the first row, two in the second, and in general, <math>k</math> squares in the <math>k</math>th row for <math>1 \leq k \leq 11.</math> With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in given diagram). In each square of the eleventh row, a <math>0</math> or a <math>1</math> is placed. Numbers are then placed into the other squares, with the entry for each square being the sum of the entries in the two squares below it. For how many initial distributions of <math>0</math>'s and <math>1</math>'s in the bottom row is the number in the top square a [[multiple]] of <math>3</math>? | ||
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[[Image:2007 AIME II-13.png]] | [[Image:2007 AIME II-13.png]] | ||
| − | [ | + | [http://home.comcast.net/~wolfand/ Solution] |
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== Problem 14 == | == Problem 14 == | ||
Let <math>f(x)</math> be a [[polynomial]] with real [[coefficient]]s such that <math>f(0) = 1,</math> <math>f(2)+f(3)=125,</math> and for all <math>x</math>, <math>f(x)f(2x^{2})=f(2x^{3}+x).</math> Find <math>f(5).</math> | Let <math>f(x)</math> be a [[polynomial]] with real [[coefficient]]s such that <math>f(0) = 1,</math> <math>f(2)+f(3)=125,</math> and for all <math>x</math>, <math>f(x)f(2x^{2})=f(2x^{3}+x).</math> Find <math>f(5).</math> | ||
| − | [ | + | [http://home.comcast.net/~wolfand/ Solution] |
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== Problem 15 == | == Problem 15 == | ||
Four [[circle]]s <math>\omega,</math> <math>\omega_{A},</math> <math>\omega_{B},</math> and <math>\omega_{C}</math> with the same [[radius]] are drawn in the interior of [[triangle]] <math>ABC</math> such that <math>\omega_{A}</math> is [[tangent]] to sides <math>AB</math> and <math>AC</math>, <math>\omega_{B}</math> to <math>BC</math> and <math>BA</math>, <math>\omega_{C}</math> to <math>CA</math> and <math>CB</math>, and <math>\omega</math> is [[externally tangent]] to <math>\omega_{A},</math> <math>\omega_{B},</math> and <math>\omega_{C}</math>. If the sides of triangle <math>ABC</math> are <math>13,</math> <math>14,</math> and <math>15,</math> the radius of <math>\omega</math> can be represented in the form <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are [[relatively prime]] positive integers. Find <math>m+n.</math> | Four [[circle]]s <math>\omega,</math> <math>\omega_{A},</math> <math>\omega_{B},</math> and <math>\omega_{C}</math> with the same [[radius]] are drawn in the interior of [[triangle]] <math>ABC</math> such that <math>\omega_{A}</math> is [[tangent]] to sides <math>AB</math> and <math>AC</math>, <math>\omega_{B}</math> to <math>BC</math> and <math>BA</math>, <math>\omega_{C}</math> to <math>CA</math> and <math>CB</math>, and <math>\omega</math> is [[externally tangent]] to <math>\omega_{A},</math> <math>\omega_{B},</math> and <math>\omega_{C}</math>. If the sides of triangle <math>ABC</math> are <math>13,</math> <math>14,</math> and <math>15,</math> the radius of <math>\omega</math> can be represented in the form <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are [[relatively prime]] positive integers. Find <math>m+n.</math> | ||
| − | [ | + | [http://home.comcast.net/~wolfand/ Solution] |
Revision as of 13:44, 9 May 2011
| 2007 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
|
Instructions
| ||
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
to 
, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.Contents
[hide]Problem 1
A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in 
. No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in . A set of plates in which each possible sequence appears exactly once contains 
license plates. Find 
.
Problem 2
Find the number of ordered triples 
where 
, 
, and 
are positive integers, is a factor of , is a factor of , and 
.
Problem 3
Square 
has side length 
, and points 
and 
are exterior to the square such that 
and 
. Find 
.
Problem 4
The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, 
workers can produce 
widgets and 
whoosits. In two hours, 
workers can produce 
widgets and whoosits. In three hours, 
workers can produce 
widgets and 
whoosits. Find .
Problem 5
The graph of the equation 
is drawn on graph paper with each square representing one unit in each direction. How many of the 
by graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant?
Problem 6
An integer is called parity-monotonic if its decimal representation 
satisfies 
if 
is odd, and 
if is even. How many four-digit parity-monotonic integers are there?
Problem 7
Given a real number 
let 
denote the greatest integer less than or equal to 
For a certain integer 
there are exactly 
positive integers 
such that ![$k=\lfloor\sqrt[3]{n_{1}}\rfloor = \lfloor\sqrt[3]{n_{2}}\rfloor = \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor$](http://latex.artofproblemsolving.com/a/2/a/a2a31bac0a2bc6d9943fbdf0ef0b1a3bc933895e.png)
and 
divides 
for all 
such that 
Find the maximum value of 
for 
Problem 8
A rectangular piece of paper measures 4 units by 5 units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called basic if
- (i) all four sides of the rectangle are segments of drawn line segments, and
- (ii) no segments of drawn lines lie inside the rectangle.
Given that the total length of all lines drawn is exactly 2007 units, let be the maximum possible number of basic rectangles determined. Find the remainder when is divided by 1000.
Problem 9
Rectangle is given with 
and 
Points and lie on 
and 
respectively, such that 
The inscribed circle of triangle 
is tangent to 
at point 
and the inscribed circle of triangle 
is tangent to at point 
Find 
Problem 10
Let 
be a set with six elements. Let 
be the set of all subsets of 
Subsets 
and 
of , not necessarily distinct, are chosen independently and at random from . The probability that is contained in at least one of or 
is 
where , 
, and 
are positive integers, is prime, and and are relatively prime. Find 
(The set is the set of all elements of which are not in 
)
Problem 11
Two long cylindrical tubes of the same length but different diameters lie parallel to each other on a flat surface. The larger tube has radius 
and rolls along the surface toward the smaller tube, which has radius 
. It rolls over the smaller tube and continues rolling along the flat surface until it comes to rest on the same point of its circumference as it started, having made one complete revolution. If the smaller tube never moves, and the rolling occurs with no slipping, the larger tube ends up a distance 
from where it starts. The distance can be expressed in the form 
where 
and are integers and is not divisible by the square of any prime. Find 
Problem 12
The increasing geometric sequence 
consists entirely of integral powers of 
Given that
and 
find 
Problem 13
A triangular array of squares has one square in the first row, two in the second, and in general, squares in the th row for 
With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in given diagram). In each square of the eleventh row, a 
or a is placed. Numbers are then placed into the other squares, with the entry for each square being the sum of the entries in the two squares below it. For how many initial distributions of 's and 's in the bottom row is the number in the top square a multiple of 
?
Problem 14
Let 
be a polynomial with real coefficients such that 
and for all , 
Find 
Problem 15
Four circles 
and 
with the same radius are drawn in the interior of triangle 
such that 
is tangent to sides 
and 
, 
to and 
, to 
and 
, and 
is externally tangent to and . If the sides of triangle are 
and 
the radius of can be represented in the form 
, where and are relatively prime positive integers. Find 
