Difference between revisions of "2007 AIME II Problems"
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== Problem 1 == | == 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 <math>2007</math>. No character may appear in a | + | 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 N license plates. Find N/10. |
− | [ | + | [[2007 AIME II Problems/Problem 1|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>. | ||
− | [ | + | [[2007 AIME II Problems/Problem 2|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> | ||
− | [ | + | [[2007 AIME II Problems/Problem 3|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>. | ||
− | [ | + | [[2007 AIME II Problems/Problem 4|Solution]] |
+ | |||
== 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]]? | ||
− | [ | + | [[2007 AIME II Problems/Problem 5|Solution]] |
+ | |||
== 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? |
+ | |||
+ | [[2007 AIME II Problems/Problem 6|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> | ||
− | [ | + | [[2007 AIME II Problems/Problem 7|Solution]] |
+ | |||
== 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. | ||
− | [ | + | [[2007 AIME II Problems/Problem 8|Solution]] |
+ | |||
== 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> | ||
− | [ | + | [[2007 AIME II Problems/Problem 9|Solution]] |
+ | |||
== 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 | + | Let <math>S</math> be a [[set]] with six [[element]]s. Let <math>\mathcal{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>\mathcal{P}</math>. The [[probability]] that <math>B</math> is contained in 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>) |
+ | |||
+ | [[2007 AIME II Problems/Problem 10|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> | ||
− | [ | + | [[2007 AIME II Problems/Problem 11|Solution]] |
+ | |||
== 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> | ||
− | [ | + | [[2007 AIME II Problems/Problem 12|Solution]] |
+ | |||
== 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>? | ||
− | [[ | + | <asy> |
+ | for (int i=0; i<12; ++i){ | ||
+ | for (int j=0; j<i; ++j){ | ||
+ | //dot((-j+i/2,-i)); | ||
+ | draw((-j+i/2,-i)--(-j+i/2+1,-i)--(-j+i/2+1,-i+1)--(-j+i/2,-i+1)--cycle); | ||
+ | } | ||
+ | } | ||
+ | </asy> | ||
+ | |||
+ | [[2007 AIME II Problems/Problem 13|Solution]] | ||
− | |||
== 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> | ||
− | [ | + | [[2007 AIME II Problems/Problem 14|Solution]] |
+ | |||
== 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> | ||
− | [ | + | [[2007 AIME II Problems/Problem 15|Solution]] |
+ | |||
+ | {{AIME box|year=2007|n=II|before=[[2007 AIME I]]|after=[[2008 AIME I]]}} | ||
+ | {{MAA Notice}} |
Latest revision as of 01:45, 6 December 2021
2007 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
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1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
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 N license plates. Find N/10.
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 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 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
2007 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2007 AIME I |
Followed by 2008 AIME I | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.