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Difference between revisions of "2007 AIME I Problems"

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{{AIME Problems|year=2007|n=I}}
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== Problem 1 ==
 
== Problem 1 ==
 
How many positive perfect squares less than <math>10^6</math> are multiples of 24?
 
How many positive perfect squares less than <math>10^6</math> are multiples of 24?
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== Problem 4 ==
 
== Problem 4 ==
Three planets orbit a star circularly in the same plane.  Each moves in the same direction and moves at constant speed.  Their periods are <math>60</math>,<math>84</math>, and <math>140</math>.  The three planets and the star are currently collinear.  What is the fewest number of years from now that they will all be collinear again?
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Three planets orbit a star circularly in the same plane.  Each moves in the same direction and moves at constant speed.  Their periods are <math>60</math>, <math>84</math>, and <math>140</math> years.  The three planets and the star are currently collinear.  What is the fewest number of years from now that they will all be collinear again?
  
 
[[2007 AIME I Problems/Problem 4|Solution]]
 
[[2007 AIME I Problems/Problem 4|Solution]]
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The formula for converting a Fahrenheit temperature <math>F</math> to the corresponding Celsius temperature <math>C</math> is <math>C = \frac{5}{9}(F-32).</math>  An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the nearest integer.
 
The formula for converting a Fahrenheit temperature <math>F</math> to the corresponding Celsius temperature <math>C</math> is <math>C = \frac{5}{9}(F-32).</math>  An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the nearest integer.
  
For how many integer Fahrenheit temperatures between 32 and 1000 inclusive does the original temperature equal the final temperature?
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For how many integer Fahrenheit temperatures between <math>32</math> and <math>1000</math> inclusive does the original temperature equal the final temperature?
  
 
[[2007 AIME I Problems/Problem 5|Solution]]
 
[[2007 AIME I Problems/Problem 5|Solution]]
  
 
== Problem 6 ==
 
== Problem 6 ==
A frog moves from 0 to 39 on an integral number line in the following way - on a given move, it jumps either to the next highest multiple of 3 or the next highest multiple of 13. Find the number of distinct possible paths the frog can take.
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A frog is placed at the [[origin]] on the [[number line]], and moves according to the following rule: in a given move, the frog advances to either the closest [[point]] with a greater [[integer]] [[coordinate]] that is a multiple of <math>3</math>, or to the closest point with a greater integer coordinate that is a multiple of <math>13</math>. A ''move sequence'' is a [[sequence]] of coordinates which correspond to valid moves, beginning with <math>0</math>, and ending with <math>39</math>. For example, <math>0,\ 3,\ 6,\ 13,\ 15,\ 26,\ 39</math> is a move sequence. How many move sequences are possible for the frog?
  
 
[[2007 AIME I Problems/Problem 6|Solution]]
 
[[2007 AIME I Problems/Problem 6|Solution]]
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== Problem 9 ==
 
== Problem 9 ==
In right triangle <math>ABC</math> with right angle <math>C</math>, <math>CA = 30</math> and <math>CB = 16</math>.  Its legs <math>CA</math> and <math>CB</math> are extended beyond <math>A</math> and <math>B</math>.  Points <math>O_1</math> and <math>O_2</math> lie in the exterior of the triangle and are the centers of two circles with equal radii.  The circle with center <math>O_1</math> is tangent to the hypotenuse and to the extension of leg <math>CA</math>, the circle with center <math>O_2</math> is tangent to the hypotenuse and to the extension of leg <math>CB</math>, and the circles are externally tangent to each other.  The length of the radius either circle can be expressed as <math>p/q</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers.  Find <math>p+q</math>.
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In right triangle <math>ABC</math> with right angle <math>C</math>, <math>CA = 30</math> and <math>CB = 16</math>.  Its legs <math>CA</math> and <math>CB</math> are extended beyond <math>A</math> and <math>B</math>.  Points <math>O_1</math> and <math>O_2</math> lie in the exterior of the triangle and are the centers of two circles with equal radii.  The circle with center <math>O_1</math> is tangent to the hypotenuse and to the extension of leg <math>CA</math>, the circle with center <math>O_2</math> is tangent to the hypotenuse and to the extension of leg <math>CB</math>, and the circles are externally tangent to each other.  The length of the radius of either circle can be expressed as <math>p/q</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers.  Find <math>p+q</math>.
 
 
[[Image:AIME I 2007-9.png]]
 
  
 
[[2007 AIME I Problems/Problem 9|Solution]]
 
[[2007 AIME I Problems/Problem 9|Solution]]
  
 
== Problem 10 ==
 
== Problem 10 ==
In a 6 x 4 grid (6 rows, 4 columns), 12 of the 24 squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column.  Let <math>N</math> be the number of shadings with this property.  Find the remainder when <math>N</math> is divided by 1000.
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In a <math>6 \times 4</math> grid (<math>6</math> rows, <math>4</math> columns), <math>12</math> of the <math>24</math> squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column.  Let <math>N</math> be the number of shadings with this property.  Find the remainder when <math>N</math> is divided by <math>1000</math>.
  
 
[[Image:AIME I 2007-10.png]]
 
[[Image:AIME I 2007-10.png]]
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== Problem 11 ==
 
== Problem 11 ==
For each positive integer <math>p</math>, let <math>b(p)</math> denote the unique positive integer <math>k</math> such that <math>|k-\sqrt{p}| < \frac{1}{2}</math>.  For example, <math>b(6) = 2</math> and <math>b(23) = 5</math>.  If <math>S = \Sigma_{p=1}^{2007} b(p),</math> find the remainder when <math>S</math> is divided by 1000.
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For each positive integer <math>p</math>, let <math>b(p)</math> denote the unique positive integer <math>k</math> such that <math>|k-\sqrt{p}| < \frac{1}{2}</math>.  For example, <math>b(6) = 2</math> and <math>b(23) = 5</math>.  If <math>S = \sum_{p=1}^{2007} b(p),</math> find the remainder when <math>S</math> is divided by 1000.
  
 
[[2007 AIME I Problems/Problem 11|Solution]]
 
[[2007 AIME I Problems/Problem 11|Solution]]
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== Problem 12 ==
 
== Problem 12 ==
 
In isosceles triangle <math>ABC</math>, <math>A</math> is located at the origin and <math>B</math> is located at (20,0).  Point <math>C</math> is in the first quadrant with <math>AC = BC</math> and angle <math>BAC = 75^{\circ}</math>.  If triangle <math>ABC</math> is rotated counterclockwise about point <math>A</math> until the image of <math>C</math> lies on the positive <math>y</math>-axis, the area of the region common to the original and the rotated triangle is in the form <math>p\sqrt{2} + q\sqrt{3} + r\sqrt{6} + s</math>, where <math>p,q,r,s</math> are integers.  Find <math>(p-q+r-s)/2</math>.
 
In isosceles triangle <math>ABC</math>, <math>A</math> is located at the origin and <math>B</math> is located at (20,0).  Point <math>C</math> is in the first quadrant with <math>AC = BC</math> and angle <math>BAC = 75^{\circ}</math>.  If triangle <math>ABC</math> is rotated counterclockwise about point <math>A</math> until the image of <math>C</math> lies on the positive <math>y</math>-axis, the area of the region common to the original and the rotated triangle is in the form <math>p\sqrt{2} + q\sqrt{3} + r\sqrt{6} + s</math>, where <math>p,q,r,s</math> are integers.  Find <math>(p-q+r-s)/2</math>.
 
[[Image:AIME I 2007-12.png]]
 
  
 
[[2007 AIME I Problems/Problem 12|Solution]]
 
[[2007 AIME I Problems/Problem 12|Solution]]
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== Problem 13 ==
 
== Problem 13 ==
 
A square pyramid with base <math>ABCD</math> and vertex <math>E</math> has eight edges of length 4.  A plane passes through the midpoints of <math>AE</math>, <math>BC</math>, and <math>CD</math>.  The plane's intersection with the pyramid has an area that can be expressed as <math>\sqrt{p}</math>.  Find <math>p</math>.
 
A square pyramid with base <math>ABCD</math> and vertex <math>E</math> has eight edges of length 4.  A plane passes through the midpoints of <math>AE</math>, <math>BC</math>, and <math>CD</math>.  The plane's intersection with the pyramid has an area that can be expressed as <math>\sqrt{p}</math>.  Find <math>p</math>.
 
[[Image:AIME I 2007-13.png]]
 
  
 
[[2007 AIME I Problems/Problem 13|Solution]]
 
[[2007 AIME I Problems/Problem 13|Solution]]
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* [[AIME Problems and Solutions]]
 
* [[AIME Problems and Solutions]]
 
* [[Mathematics competition resources]]
 
* [[Mathematics competition resources]]
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{{MAA Notice}}

Latest revision as of 22:22, 28 September 2020

2007 AIME I (Answer Key)
Printable version | AoPS Contest CollectionsPDF

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

How many positive perfect squares less than $10^6$ are multiples of 24?

Solution

Problem 2

A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds after that, Cy reaches the start of the walkway and walks briskly forward beside the walkway at a constant rate of 8 feet per second. At a certain time, one of these three persons is exactly halfway between the other two. At that time, find the distance in feet between the start of the walkway and the middle person.

Solution

Problem 3

The complex number $z$ is equal to $9+bi$, where $b$ is a positive real number and $i^{2}=-1$. Given that the imaginary parts of $z^{2}$ and $z^{3}$ are the same, what is $b$ equal to?

Solution

Problem 4

Three planets orbit a star circularly in the same plane. Each moves in the same direction and moves at constant speed. Their periods are $60$, $84$, and $140$ years. The three planets and the star are currently collinear. What is the fewest number of years from now that they will all be collinear again?

Solution

Problem 5

The formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ is $C = \frac{5}{9}(F-32).$ An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the nearest integer.

For how many integer Fahrenheit temperatures between $32$ and $1000$ inclusive does the original temperature equal the final temperature?

Solution

Problem 6

A frog is placed at the origin on the number line, and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of $3$, or to the closest point with a greater integer coordinate that is a multiple of $13$. A move sequence is a sequence of coordinates which correspond to valid moves, beginning with $0$, and ending with $39$. For example, $0,\ 3,\ 6,\ 13,\ 15,\ 26,\ 39$ is a move sequence. How many move sequences are possible for the frog?

Solution

Problem 7

Let $N = \sum_{k = 1}^{1000} k ( \lceil \log_{\sqrt{2}} k \rceil  - \lfloor \log_{\sqrt{2}} k \rfloor )$

Find the remainder when $N$ is divided by 1000. ($\lfloor{k}\rfloor$ is the greatest integer less than or equal to $k$, and $\lceil{k}\rceil$ is the least integer greater than or equal to $k$.)

Solution

Problem 8

The polynomial $P(x)$ is cubic. What is the largest value of $k$ for which the polynomials $Q_1(x) = x^2 + (k-29)x - k$ and $Q_2(x) = 2x^2+ (2k-43)x + k$ are both factors of $P(x)$?

Solution

Problem 9

In right triangle $ABC$ with right angle $C$, $CA = 30$ and $CB = 16$. Its legs $CA$ and $CB$ are extended beyond $A$ and $B$. Points $O_1$ and $O_2$ lie in the exterior of the triangle and are the centers of two circles with equal radii. The circle with center $O_1$ is tangent to the hypotenuse and to the extension of leg $CA$, the circle with center $O_2$ is tangent to the hypotenuse and to the extension of leg $CB$, and the circles are externally tangent to each other. The length of the radius of either circle can be expressed as $p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Solution

Problem 10

In a $6 \times 4$ grid ($6$ rows, $4$ columns), $12$ of the $24$ squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let $N$ be the number of shadings with this property. Find the remainder when $N$ is divided by $1000$.

AIME I 2007-10.png

Solution

Problem 11

For each positive integer $p$, let $b(p)$ denote the unique positive integer $k$ such that $|k-\sqrt{p}| < \frac{1}{2}$. For example, $b(6) = 2$ and $b(23) = 5$. If $S = \sum_{p=1}^{2007} b(p),$ find the remainder when $S$ is divided by 1000.

Solution

Problem 12

In isosceles triangle $ABC$, $A$ is located at the origin and $B$ is located at (20,0). Point $C$ is in the first quadrant with $AC = BC$ and angle $BAC = 75^{\circ}$. If triangle $ABC$ is rotated counterclockwise about point $A$ until the image of $C$ lies on the positive $y$-axis, the area of the region common to the original and the rotated triangle is in the form $p\sqrt{2} + q\sqrt{3} + r\sqrt{6} + s$, where $p,q,r,s$ are integers. Find $(p-q+r-s)/2$.

Solution

Problem 13

A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length 4. A plane passes through the midpoints of $AE$, $BC$, and $CD$. The plane's intersection with the pyramid has an area that can be expressed as $\sqrt{p}$. Find $p$.

Solution

Problem 14

A sequence is defined over non-negative integral indexes in the following way: $a_{0}=a_{1}=3$, $a_{n+1}a_{n-1}=a_{n}^{2}+2007$.

Find the greatest integer that does not exceed $\frac{a_{2006}^{2}+a_{2007}^{2}}{a_{2006}a_{2007}}$

Solution

Problem 15

Let $ABC$ be an equilateral triangle, and let $D$ and $F$ be points on sides $BC$ and $AB$, respectively, with $FA = 5$ and $CD = 2$. Point $E$ lies on side $CA$ such that angle $DEF = 60^{\circ}$. The area of triangle $DEF$ is $14\sqrt{3}$. The two possible values of the length of side $AB$ are $p \pm q \sqrt{r}$, where $p$ and $q$ are rational, and $r$ is an integer not divisible by the square of a prime. Find $r$.

Solution

See also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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