# Difference between revisions of "User:Rowechen"

Line 4: | Line 4: | ||

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

− | == Problem | + | == Problem 2 == |

− | + | Find the number of [[ordered pair]]s <math>(x,y)</math> of positive integers that satisfy <math>x \le 2y \le 60</math> and <math>y \le 2x \le 60</math>. | |

− | [[ | + | [[1998 AIME Problems/Problem 2|Solution]] |

== Problem 5 == | == Problem 5 == | ||

− | + | Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is <math> \frac{m + n \pi}{p}, </math> where <math> m, n, </math> and <math> p </math> are positive integers, and <math> n </math> and <math> p </math> are relatively prime, find <math> m + n + p. </math> | |

− | [[ | + | [[2003 AIME I Problems/Problem 5|Solution]] |

− | + | == Problem 6 == | |

− | == Problem | + | The cards in a stack of <math> 2n </math> cards are numbered consecutively from 1 through <math> 2n </math> from top to bottom. The top <math> n </math> cards are removed, kept in order, and form pile <math> A. </math> The remaining cards form pile <math> B. </math> The cards are then restacked by taking cards alternately from the tops of pile <math> B </math> and <math> A, </math> respectively. In this process, card number <math> (n+1) </math> becomes the bottom card of the new stack, card number 1 is on top of this card, and so on, until piles <math> A </math> and <math> B </math> are exhausted. If, after the restacking process, at least one card from each pile occupies the same position that it occupied in the original stack, the stack is named magical. Find the number of cards in the magical stack in which card number 131 retains its original position. |

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

+ | [[2005 AIME II Problems/Problem 6|Solution]] | ||

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

− | + | How many different <math>4\times 4</math> arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0 and the sum of the entries in each column is 0? | |

− | |||

− | |||

+ | [[1997 AIME Problems/Problem 8|Solution]] | ||

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

− | + | Given a nonnegative real number <math>x</math>, let <math>\langle x\rangle</math> denote the fractional part of <math>x</math>; that is, <math>\langle x\rangle=x-\lfloor x\rfloor</math>, where <math>\lfloor x\rfloor</math> denotes the greatest integer less than or equal to <math>x</math>. Suppose that <math>a</math> is positive, <math>\langle a^{-1}\rangle=\langle a^2\rangle</math>, and <math>2<a^2<3</math>. Find the value of <math>a^{12}-144a^{-1}</math>. | |

− | [[ | + | [[1997 AIME Problems/Problem 9|Solution]] |

+ | == Problem 10 == | ||

+ | Let <math>S</math> be the set of points whose coordinates <math>x,</math> <math>y,</math> and <math>z</math> are integers that satisfy <math>0\le x\le2,</math> <math>0\le y\le3,</math> and <math>0\le z\le4.</math> Two distinct points are randomly chosen from <math>S.</math> The probability that the midpoint of the segment they determine also belongs to <math>S</math> is <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n.</math> | ||

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

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

− | + | In a rectangular array of points, with 5 rows and <math>N</math> columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through <math>N,</math> the second row is numbered <math>N + 1</math> through <math>2N,</math> and so forth. Five points, <math>P_1, P_2, P_3, P_4,</math> and <math>P_5,</math> are selected so that each <math>P_i</math> is in row <math>i.</math> Let <math>x_i</math> be the number associated with <math>P_i.</math> Now renumber the array consecutively from top to bottom, beginning with the first column. Let <math>y_i</math> be the number associated with <math>P_i</math> after the renumbering. It is found that <math>x_1 = y_2,</math> <math>x_2 = y_1,</math> <math>x_3 = y_4,</math> <math>x_4 = y_5,</math> and <math>x_5 = y_3.</math> Find the smallest possible value of <math>N.</math> | |

− | [[ | + | [[2001 AIME I Problems/Problem 11|Solution]] |

− | == Problem | + | == Problem 12 == |

− | + | A sphere is inscribed in the tetrahedron whose vertices are <math>A = (6,0,0), B = (0,4,0), C = (0,0,2),</math> and <math>D = (0,0,0).</math> The radius of the sphere is <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n.</math> | |

− | [[ | + | [[2001 AIME I Problems/Problem 12|Solution]] |

+ | == Problem 9 == | ||

+ | The system of equations | ||

+ | <cmath>\begin{eqnarray*}\log_{10}(2000xy) - (\log_{10}x)(\log_{10}y) & = & 4 \\ | ||

+ | \log_{10}(2yz) - (\log_{10}y)(\log_{10}z) & = & 1 \\ | ||

+ | \log_{10}(zx) - (\log_{10}z)(\log_{10}x) & = & 0 \\ | ||

+ | \end{eqnarray*}</cmath> | ||

− | + | has two solutions <math>(x_{1},y_{1},z_{1})</math> and <math>(x_{2},y_{2},z_{2})</math>. Find <math>y_{1} + y_{2}</math>. | |

− | |||

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

− | == Problem | + | == Problem 12 == |

− | Given | + | Given a function <math>f</math> for which |

+ | <center><math>f(x) = f(398 - x) = f(2158 - x) = f(3214 - x)</math></center> | ||

+ | holds for all real <math>x,</math> what is the largest number of different values that can appear in the list <math>f(0),f(1),f(2),\ldots,f(999)</math>? | ||

− | [[ | + | [[2000 AIME I Problems/Problem 12|Solution]] |

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

− | + | There are <math>2n</math> complex numbers that satisfy both <math>z^{28} - z^{8} - 1 = 0</math> and <math>|z| = 1</math>. These numbers have the form <math>z_{m} = \cos\theta_{m} + i\sin\theta_{m}</math>, where <math>0\leq\theta_{1} < \theta_{2} < \ldots < \theta_{2n} < 360</math> and angles are measured in degrees. Find the value of <math>\theta_{2} + \theta_{4} + \ldots + \theta_{2n}</math>. | |

+ | |||

+ | [[2001 AIME II Problems/Problem 14|Solution]] | ||

+ | == Problem 13 == | ||

+ | In triangle <math>ABC</math> the medians <math>\overline{AD}</math> and <math>\overline{CE}</math> have lengths 18 and 27, respectively, and <math>AB = 24</math>. Extend <math>\overline{CE}</math> to intersect the circumcircle of <math>ABC</math> at <math>F</math>. The area of triangle <math>AFB</math> is <math>m\sqrt {n}</math>, where <math>m</math> and <math>n</math> are positive integers and <math>n</math> is not divisible by the square of any prime. Find <math>m + n</math>. | ||

− | [[ | + | [[2002 AIME I Problems/Problem 13|Solution]] |

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

− | + | The perimeter of triangle <math>APM</math> is <math>152</math>, and the angle <math>PAM</math> is a right angle. A circle of radius <math>19</math> with center <math>O</math> on <math>\overline{AP}</math> is drawn so that it is tangent to <math>\overline{AM}</math> and <math>\overline{PM}</math>. Given that <math>OP=m/n</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>m+n</math>. | |

− | [[ | + | [[2002 AIME II Problems/Problem 14|Solution]] |

− | == Problem | + | == Problem 13 == |

− | + | Let <math> N </math> be the number of positive integers that are less than or equal to 2003 and whose base-2 representation has more 1's than 0's. Find the remainder when <math> N </math> is divided by 1000. | |

− | [[ | + | [[2003 AIME I Problems/Problem 13|Solution]] |

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

− | + | The decimal representation of <math> m/n, </math> where <math> m </math> and <math> n </math> are relatively prime positive integers and <math> m < n, </math> contains the digits 2, 5, and 1 consecutively, and in that order. Find the smallest value of <math> n </math> for which this is possible. | |

− | |||

− | |||

− | |||

− | |||

− | [[ | + | [[2003 AIME I Problems/Problem 14|Solution]] |

## Revision as of 08:52, 27 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

Find the number of ordered pairs of positive integers that satisfy and .

## Problem 5

Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is where and are positive integers, and and are relatively prime, find

## Problem 6

The cards in a stack of cards are numbered consecutively from 1 through from top to bottom. The top cards are removed, kept in order, and form pile The remaining cards form pile The cards are then restacked by taking cards alternately from the tops of pile and respectively. In this process, card number becomes the bottom card of the new stack, card number 1 is on top of this card, and so on, until piles and are exhausted. If, after the restacking process, at least one card from each pile occupies the same position that it occupied in the original stack, the stack is named magical. Find the number of cards in the magical stack in which card number 131 retains its original position.

## Problem 8

How many different arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0 and the sum of the entries in each column is 0?

## Problem 9

Given a nonnegative real number , let denote the fractional part of ; that is, , where denotes the greatest integer less than or equal to . Suppose that is positive, , and . Find the value of .

## Problem 10

Let be the set of points whose coordinates and are integers that satisfy and Two distinct points are randomly chosen from The probability that the midpoint of the segment they determine also belongs to is where and are relatively prime positive integers. Find

## Problem 11

In a rectangular array of points, with 5 rows and columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through the second row is numbered through and so forth. Five points, and are selected so that each is in row Let be the number associated with Now renumber the array consecutively from top to bottom, beginning with the first column. Let be the number associated with after the renumbering. It is found that and Find the smallest possible value of

## Problem 12

A sphere is inscribed in the tetrahedron whose vertices are and The radius of the sphere is where and are relatively prime positive integers. Find

## Problem 9

The system of equations

has two solutions and . Find .

## Problem 12

Given a function for which

holds for all real what is the largest number of different values that can appear in the list ?

## Problem 14

There are complex numbers that satisfy both and . These numbers have the form , where and angles are measured in degrees. Find the value of .

## Problem 13

In triangle the medians and have lengths 18 and 27, respectively, and . Extend to intersect the circumcircle of at . The area of triangle is , where and are positive integers and is not divisible by the square of any prime. Find .

## Problem 14

The perimeter of triangle is , and the angle is a right angle. A circle of radius with center on is drawn so that it is tangent to and . Given that where and are relatively prime positive integers, find .

## Problem 13

Let be the number of positive integers that are less than or equal to 2003 and whose base-2 representation has more 1's than 0's. Find the remainder when is divided by 1000.

## Problem 14

The decimal representation of where and are relatively prime positive integers and contains the digits 2, 5, and 1 consecutively, and in that order. Find the smallest value of for which this is possible.