# Difference between revisions of "User:Rowechen"

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[[2006 AIME I Problems/Problem 7|Solution]] | [[2006 AIME I Problems/Problem 7|Solution]] | ||

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

− | A | + | 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>. |

− | [[2007 AIME I Problems/Problem | + | [[2007 AIME I Problems/Problem 12|Solution]] |

− | |||

− | |||

− | [[2006 AIME II Problems/Problem | + | == Problem 13 == |

+ | How many integers <math> N </math> less than 1000 can be written as the sum of <math> j </math> consecutive positive odd integers from exactly 5 values of <math> j\ge 1 </math>? | ||

+ | |||

+ | [[2006 AIME II Problems/Problem 13|Solution]] | ||

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

## Revision as of 19:22, 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 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

How many integers less than 1000 can be written as the sum of consecutive positive odd integers from exactly 5 values of ?

## 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 .

## Problem 13

Point lies on side of so that bisects The perpendicular bisector of intersects the bisectors of and in points and respectively. Given that and the area of can be written as where and are relatively prime positive integers, and is a positive integer not divisible by the square of any prime. Find

## Problem 15

Let be an acute triangle with circumcircle and let be the intersection of the altitudes of Suppose the tangent to the circumcircle of at intersects at points and with and The area of can be written as where and are positive integers, and is not divisible by the square of any prime. Find

## Problem 14

Let be a quadratic polynomial with complex coefficients whose coefficient is Suppose the equation has four distinct solutions, Find the sum of all possible values of

## Problem 13

For each integer , let be the number of -element subsets of the vertices of a regular -gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of such that .

## Problem 15

In triangle , we have , , and . Points , , and are selected on , , and respectively such that , , and concur at the circumcenter of . The value of can be expressed as where and are relatively prime positive integers. Determine .