Difference between revisions of "2007 AIME I Problems"
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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? | + | 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? | + | 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 | + | 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>. | + | 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>. |
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[[2007 AIME I Problems/Problem 9|Solution]] | [[2007 AIME I Problems/Problem 9|Solution]] | ||
== Problem 10 == | == Problem 10 == | ||
− | In a 6 | + | 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 = \ | + | 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>. | ||
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[[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>. | ||
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[[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]] | ||
+ | {{MAA Notice}} |
Latest revision as of 22:22, 28 September 2020
2007 AIME I (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
How many positive perfect squares less than are multiples of 24?
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.
Problem 3
The complex number is equal to , where is a positive real number and . Given that the imaginary parts of and are the same, what is equal to?
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 , , and 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?
Problem 5
The formula for converting a Fahrenheit temperature to the corresponding Celsius temperature is 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 and inclusive does the original temperature equal the final temperature?
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 , or to the closest point with a greater integer coordinate that is a multiple of . A move sequence is a sequence of coordinates which correspond to valid moves, beginning with , and ending with . For example, is a move sequence. How many move sequences are possible for the frog?
Problem 7
Let
Find the remainder when is divided by 1000. ( is the greatest integer less than or equal to , and is the least integer greater than or equal to .)
Problem 8
The polynomial is cubic. What is the largest value of for which the polynomials and are both factors of ?
Problem 9
In right triangle with right angle , and . Its legs and are extended beyond and . Points and lie in the exterior of the triangle and are the centers of two circles with equal radii. The circle with center is tangent to the hypotenuse and to the extension of leg , the circle with center is tangent to the hypotenuse and to the extension of leg , and the circles are externally tangent to each other. The length of the radius of either circle can be expressed as , where and are relatively prime positive integers. Find .
Problem 10
In a grid ( rows, columns), of the squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let be the number of shadings with this property. Find the remainder when is divided by .
Problem 11
For each positive integer , let denote the unique positive integer such that . For example, and . If find the remainder when is divided by 1000.
Problem 12
In isosceles triangle , is located at the origin and is located at (20,0). Point is in the first quadrant with and angle . If triangle is rotated counterclockwise about point until the image of lies on the positive -axis, the area of the region common to the original and the rotated triangle is in the form , where are integers. Find .
Problem 13
A square pyramid with base and vertex has eight edges of length 4. A plane passes through the midpoints of , , and . The plane's intersection with the pyramid has an area that can be expressed as . Find .
Problem 14
A sequence is defined over non-negative integral indexes in the following way: , .
Find the greatest integer that does not exceed
Problem 15
Let be an equilateral triangle, and let and be points on sides and , respectively, with and . Point lies on side such that angle . The area of triangle is . The two possible values of the length of side are , where and are rational, and is an integer not divisible by the square of a prime. Find .
See also
- 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.