# 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: | ||

+ | == Problem 7 == | ||

+ | An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region <math> \mathcal{C} </math> to the area of shaded region <math> \mathcal{B} </math> is 11/5. Find the ratio of shaded region <math> \mathcal{D} </math> to the area of shaded region <math> \mathcal{A}. </math> | ||

− | + | [[Image:2006AimeA7.PNG]] | |

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+ | [[2006 AIME I Problems/Problem 7|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 2 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 4 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 | + | [[2007 AIME I Problems/Problem 2|Solution]] |

− | + | == Problem 3 == | |

+ | Let <math> P </math> be the product of the first 100 positive odd integers. Find the largest integer <math> k </math> such that <math> P </math> is divisible by <math> 3^k </math>. | ||

− | [[ | + | [[2006 AIME II Problems/Problem 3|Solution]] |

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− | + | == Problem 9== | |

− | == | + | The value of the sum |

+ | <cmath>\sum_{n=1}^{\infty} \frac{(7n+32)\cdot 3^n}{n\cdot(n+1)\cdot 4^n}</cmath> | ||

+ | can be expressed in the form <math>\frac{p}{q}</math>, for some relatively prime positive integers <math>p</math> and <math>q</math>. Compute the value of <math>p + q</math>. | ||

− | + | == Problem 8== | |

+ | Determine the remainder obtained when the expression | ||

+ | <cmath>2004^{2003^{2002^{2001}}}</cmath> | ||

+ | is divided by <math>1000</math>. | ||

− | <cmath> | + | ==Problem 9== |

− | + | Let | |

− | + | <cmath>(a+x^3)(a+2x^{3^2}) ... (1+kx^{3^k}) ... (1+1997x^{3^{1997}}) = 1+a_1x^{k_1}+a_2x^{k_2}+...+a_mx^{k_m}</cmath> | |

− | + | where <math>a_i \neq 0</math> and <math>k_1 < k_2 < ... < k_m</math>. Determine the remainder obtained when <math>a_1997</math> is divided by <math>1000</math>. | |

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− | + | == Problem 11 == | |

+ | A sequence is defined as follows <math> a_1=a_2=a_3=1, </math> and, for all positive integers <math> n, a_{n+3}=a_{n+2}+a_{n+1}+a_n. </math> Given that <math> a_{28}=6090307, a_{29}=11201821, </math> and <math> a_{30}=20603361, </math> find the remainder when <math>\sum^{28}_{k=1} a_k </math> is divided by 1000. | ||

− | + | [[2006 AIME II Problems/Problem 11|Solution]] | |

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

− | <math> | + | <math>p, q</math>, and <math>r</math> are positive real numbers such that |

− | + | <cmath>p^2+pq+q^2 = 211</cmath> | |

− | + | <cmath>q^2+qr+r^2 = 259</cmath> | |

+ | <cmath>r^2+rp+p^2 = 307</cmath> | ||

+ | Compute the value of <math>pq + qr + rp</math>. | ||

− | |||

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

− | + | <math>x_1</math>, <math>x_2</math>, and <math>x_3</math> are complex numbers such that | |

+ | <cmath>x_1 + x_2 + x_3 = 0</cmath> | ||

+ | <cmath>x_1^2+x_2^2+x_3^2 = 16</cmath> | ||

+ | <cmath>x_1^3+x_2^3+x_3^3 = -24</cmath> | ||

− | + | Let <math>\gamma = min(|x1| , |x2| , |x3|)</math>, where <math>|a + bi| = \sqrt{a^2+b^2}</math>. Determine the value of <math>\gamma^6-15\gamma^4+\gamma^3+56\gamma^2</math>. | |

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− | + | ==Problem 12== | |

− | + | <math>ABC</math> is a scalene triangle. The circle with diameter <math>AB</math> intersects <math>BC</math> at <math>D</math>, and <math>E</math> is the foot of the altitude from <math>C</math>. <math>P</math> is the intersection of <math>AD</math> and <math>CE</math>. Given that <math>AP = 136</math>, <math>BP = 80</math>, and <math>CP = 26</math>, determine the circumradius of <math>ABC</math>. | |

− | ==Problem | ||

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## Revision as of 18:29, 1 June 2020

Here's the AIME compilation I will be doing:

## Contents

## Problem 7

An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region to the area of shaded region is 11/5. Find the ratio of shaded region to the area of shaded region

## 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 2 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 4 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

Let be the product of the first 100 positive odd integers. Find the largest integer such that is divisible by .

## Problem 9

The value of the sum can be expressed in the form , for some relatively prime positive integers and . Compute the value of .

## Problem 8

Determine the remainder obtained when the expression is divided by .

## Problem 9

Let where and . Determine the remainder obtained when is divided by .

## Problem 11

A sequence is defined as follows and, for all positive integers Given that and find the remainder when is divided by 1000.

## Problem 10

, and are positive real numbers such that Compute the value of .

## Problem 11

, , and are complex numbers such that

Let , where . Determine the value of .

## Problem 12

is a scalene triangle. The circle with diameter intersects at , and is the foot of the altitude from . is the intersection of and . Given that , , and , determine the circumradius of .