# Difference between revisions of "User:Rowechen"

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Here's the AIME compilation I will be doing: | Here's the AIME compilation I will be doing: | ||

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== Problem 2 == | == Problem 2 == | ||

− | + | A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls and once wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the probability each guest got one roll of each type is <math> \frac mn, </math> where <math> m </math> and <math> n </math> are relatively prime integers, find <math> m+n. </math> | |

− | [[ | + | [[2005 AIME II Problems/Problem 2|Solution]] |

+ | == Problem 4 == | ||

+ | Ana, Bob, and Cao bike at constant rates of <math>8.6</math> meters per second, <math>6.2</math> meters per second, and <math>5</math> meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point <math>D</math> on the south edge of the field. Cao arrives at point <math>D</math> at the same time that Ana and Bob arrive at <math>D</math> for the first time. The ratio of the field's length to the field's width to the distance from point <math>D</math> to the southeast corner of the field can be represented as <math>p : q : r</math>, where <math>p</math>, <math>q</math>, and <math>r</math> are positive integers with <math>p</math> and <math>q</math> relatively prime. Find <math>p+q+r</math>. | ||

− | == Problem | + | [[2012 AIME II Problems/Problem 4|Solution]] |

− | + | ==Problem 4== | |

+ | In the Cartesian plane let <math>A = (1,0)</math> and <math>B = \left( 2, 2\sqrt{3} \right)</math>. Equilateral triangle <math>ABC</math> is constructed so that <math>C</math> lies in the first quadrant. Let <math>P=(x,y)</math> be the center of <math>\triangle ABC</math>. Then <math>x \cdot y</math> can be written as <math>\tfrac{p\sqrt{q}}{r}</math>, where <math>p</math> and <math>r</math> are relatively prime positive integers and <math>q</math> is an integer that is not divisible by the square of any prime. Find <math>p+q+r</math>. | ||

− | [[ | + | [[2013 AIME II Problems/Problem 4|Solution]] |

− | == Problem | + | == Problem 7 == |

− | + | Given that <center><math>\frac 1{2!17!}+\frac 1{3!16!}+\frac 1{4!15!}+\frac 1{5!14!}+\frac 1{6!13!}+\frac 1{7!12!}+\frac 1{8!11!}+\frac 1{9!10!}=\frac N{1!18!}</math></center> find the greatest integer that is less than <math>\frac N{100}</math>. | |

− | [[ | + | [[2000 AIME II Problems/Problem 7|Solution]] |

== Problem 8 == | == Problem 8 == | ||

− | + | In trapezoid <math>ABCD</math>, leg <math>\overline{BC}</math> is perpendicular to bases <math>\overline{AB}</math> and <math>\overline{CD}</math>, and diagonals <math>\overline{AC}</math> and <math>\overline{BD}</math> are perpendicular. Given that <math>AB=\sqrt{11}</math> and <math>AD=\sqrt{1001}</math>, find <math>BC^2</math>. | |

− | [[ | + | [[2000 AIME II Problems/Problem 8|Solution]] |

== Problem 9 == | == Problem 9 == | ||

− | Given | + | Given that <math>z</math> is a complex number such that <math>z+\frac 1z=2\cos 3^\circ</math>, find the least integer that is greater than <math>z^{2000}+\frac 1{z^{2000}}</math>. |

− | [[ | + | [[2000 AIME II Problems/Problem 9|Solution]] |

== Problem 10 == | == Problem 10 == | ||

− | + | How many positive integer multiples of 1001 can be expressed in the form <math>10^{j} - 10^{i}</math>, where <math>i</math> and <math>j</math> are integers and <math>0\leq i < j \leq 99</math>? | |

− | [[2001 AIME | + | [[2001 AIME II Problems/Problem 10|Solution]] |

+ | == Problem 10 == | ||

+ | A circle of radius 1 is randomly placed in a 15-by-36 rectangle <math> ABCD </math> so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal <math> AC </math> is <math> m/n, </math> where <math> m </math> and <math> n </math> are relatively prime positive integers, find <math> m + n. </math> | ||

+ | [[2004 AIME I Problems/Problem 10|Solution]] | ||

+ | == Problem 10 == | ||

+ | Let <math> S </math> be the set of integers between 1 and <math> 2^{40} </math> whose binary expansions have exactly two 1's. If a number is chosen at random from <math> S, </math> the probability that it is divisible by 9 is <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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+ | [[2004 AIME II Problems/Problem 10|Solution]] | ||

== Problem 11 == | == Problem 11 == | ||

− | + | Define a <i>T-grid</i> to be a <math>3\times3</math> matrix which satisfies the following two properties: | |

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− | + | <OL> | |

− | + | <LI>Exactly five of the entries are <math>1</math>'s, and the remaining four entries are <math>0</math>'s.</LI> | |

− | + | <LI>Among the eight rows, columns, and long diagonals (the long diagonals are <math>\{a_{13},a_{22},a_{31}\}</math> and <math>\{a_{11},a_{22},a_{33}\})</math>, no more than one of the eight has all three entries equal.</LI></OL> | |

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− | + | Find the number of distinct <i>T-grids</i>. | |

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− | == Problem | + | [[2010 AIME II Problems/Problem 11|Solution]] |

− | + | == Problem 15 == | |

− | + | A long thin strip of paper is 1024 units in length, 1 unit in width, and is divided into 1024 unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a 512 by 1 strip of double thickness. Next, the right end of this strip is folded over to coincide with and lie on top of the left end, resulting in a 256 by 1 strip of quadruple thickness. This process is repeated 8 more times. After the last fold, the strip has become a stack of 1024 unit squares. How many of these squares lie below the square that was originally the 942nd square counting from the left? | |

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− | [[ | + | [[2004 AIME II Problems/Problem 15|Solution]] |

== Problem 14 == | == Problem 14 == | ||

− | + | 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]] |

− | == Problem | + | == Problem 15 == |

− | + | Let <math> w_1 </math> and <math> w_2 </math> denote the circles <math> x^2+y^2+10x-24y-87=0 </math> and <math> x^2 +y^2-10x-24y+153=0, </math> respectively. Let <math> m </math> be the smallest positive value of <math> a </math> for which the line <math> y=ax </math> contains the center of a circle that is externally tangent to <math> w_2 </math> and internally tangent to <math> w_1. </math> Given that <math> m^2=\frac pq, </math> where <math> p </math> and <math> q </math> are relatively prime integers, find <math> p+q. </math> | |

− | [[ | + | [[2005 AIME II Problems/Problem 15|Solution]] |

− | == Problem | + | == Problem 15 == |

− | + | In <math>\triangle{ABC}</math> with <math>AB = 12</math>, <math>BC = 13</math>, and <math>AC = 15</math>, let <math>M</math> be a point on <math>\overline{AC}</math> such that the incircles of <math>\triangle{ABM}</math> and <math>\triangle{BCM}</math> have equal radii. Let <math>p</math> and <math>q</math> be positive relatively prime integers such that <math>\frac {AM}{CM} = \frac {p}{q}</math>. Find <math>p + q</math>. | |

− | [[ | + | [[2010 AIME I Problems/Problem 15|Solution]] |

− | == Problem | + | == Problem 15 == |

− | + | In triangle <math>ABC</math>, <math>AC=13</math>, <math>BC=14</math>, and <math>AB=15</math>. Points <math>M</math> and <math>D</math> lie on <math>AC</math> with <math>AM=MC</math> and <math>\angle ABD = \angle DBC</math>. Points <math>N</math> and <math>E</math> lie on <math>AB</math> with <math>AN=NB</math> and <math>\angle ACE = \angle ECB</math>. Let <math>P</math> be the point, other than <math>A</math>, of intersection of the circumcircles of <math>\triangle AMN</math> and <math>\triangle ADE</math>. Ray <math>AP</math> meets <math>BC</math> at <math>Q</math>. The ratio <math>\frac{BQ}{CQ}</math> can be written 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>. | |

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− | == | ||

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− | [[ | + | [[2010 AIME II Problems/Problem 15|Solution]] |

## Revision as of 09:28, 28 May 2020

Hey how did you get to this page? If you aren't me then I have to say hello. If you are me then I must be pretty conceited to waste my time looking at my own page. If you aren't me, seriously, how did you get to this page? This is pretty cool. Well, nice meeting you! I'm going to stop wasting my time typing this up and do some math. Gtg. Bye.

Here's the AIME compilation I will be doing:

## Contents

## Problem 2

A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls and once wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the probability each guest got one roll of each type is where and are relatively prime integers, find

## Problem 4

Ana, Bob, and Cao bike at constant rates of meters per second, meters per second, and meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point on the south edge of the field. Cao arrives at point at the same time that Ana and Bob arrive at for the first time. The ratio of the field's length to the field's width to the distance from point to the southeast corner of the field can be represented as , where , , and are positive integers with and relatively prime. Find .

## Problem 4

In the Cartesian plane let and . Equilateral triangle is constructed so that lies in the first quadrant. Let be the center of . Then can be written as , where and are relatively prime positive integers and is an integer that is not divisible by the square of any prime. Find .

## Problem 7

Given that

find the greatest integer that is less than .

## Problem 8

In trapezoid , leg is perpendicular to bases and , and diagonals and are perpendicular. Given that and , find .

## Problem 9

Given that is a complex number such that , find the least integer that is greater than .

## Problem 10

How many positive integer multiples of 1001 can be expressed in the form , where and are integers and ?

## Problem 10

A circle of radius 1 is randomly placed in a 15-by-36 rectangle so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal is where and are relatively prime positive integers, find

## Problem 10

Let be the set of integers between 1 and whose binary expansions have exactly two 1's. If a number is chosen at random from the probability that it is divisible by 9 is where and are relatively prime positive integers. Find

## Problem 11

Define a *T-grid* to be a matrix which satisfies the following two properties:

- Exactly five of the entries are 's, and the remaining four entries are 's.
- Among the eight rows, columns, and long diagonals (the long diagonals are and , no more than one of the eight has all three entries equal.

Find the number of distinct *T-grids*.

## Problem 15

A long thin strip of paper is 1024 units in length, 1 unit in width, and is divided into 1024 unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a 512 by 1 strip of double thickness. Next, the right end of this strip is folded over to coincide with and lie on top of the left end, resulting in a 256 by 1 strip of quadruple thickness. This process is repeated 8 more times. After the last fold, the strip has become a stack of 1024 unit squares. How many of these squares lie below the square that was originally the 942nd square counting from the left?

## 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

Let and denote the circles and respectively. Let be the smallest positive value of for which the line contains the center of a circle that is externally tangent to and internally tangent to Given that where and are relatively prime integers, find

## Problem 15

In with , , and , let be a point on such that the incircles of and have equal radii. Let and be positive relatively prime integers such that . Find .

## Problem 15

In triangle , , , and . Points and lie on with and . Points and lie on with and . Let be the point, other than , of intersection of the circumcircles of and . Ray meets at . The ratio can be written in the form , where and are relatively prime positive integers. Find .