# Difference between revisions of "User:Rowechen"

(→Problem 5) |
|||

Line 10: | Line 10: | ||

[[2007 AIME I Problems/Problem 4|Solution]] | [[2007 AIME I Problems/Problem 4|Solution]] | ||

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

− | + | If | |

<cmath>\frac{1}{0!10!}+\frac{1}{1!9!}+\frac{1}{2!8!}+\frac{1}{3!7!}+\frac{1}{4!6!}+\frac{1}{5!5!}</cmath> | <cmath>\frac{1}{0!10!}+\frac{1}{1!9!}+\frac{1}{2!8!}+\frac{1}{3!7!}+\frac{1}{4!6!}+\frac{1}{5!5!}</cmath> |

## Revision as of 14:04, 31 May 2020

Here's the AIME compilation I will be doing:

## Contents

## Problem 3

A triangle has vertices , , and . The probability that a randomly chosen point inside the triangle is closer to vertex than to either vertex or vertex can be written as , where and are relatively prime positive integers. Find .

## 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

If

is written as a common fraction reduced to lowest terms, the result is . Compute the sum of the prime divisors of plus the sum of the prime divisors of .

## Problem 9

Let , and for each integer let . Find the least such that is a multiple of .

## Problem 8

Two real numbers and are chosen independently and uniformly at random from the interval . Let and be two points on the plane with . Let and be on the same side of line such that the degree measures of and are and respectively, and and are both right angles. The probability that is equal to , where and are relatively prime positive integers. Find .

## Problem 7

Triangle has side lengths , , and . Points are on segment with between and for , and points are on segment with between and for . Furthermore, each segment , , is parallel to . The segments cut the triangle into regions, consisting of trapezoids and triangle. Each of the regions has the same area. Find the number of segments , , that have rational length.

## Problem 10

Find the number of functions from to that satisfy for all in .

## Problem 11

Find the number of permutations of such that for each with , at least one of the first terms of the permutation is greater than .

## Problem 14

The incircle of triangle is tangent to at . Let be the other intersection of with . Points and lie on and , respectively, so that is tangent to at . Assume that , , , and , where and are relatively prime positive integers. Find .

## Problem 10

Four lighthouses are located at points , , , and . The lighthouse at is kilometers from the lighthouse at , the lighthouse at is kilometers from the lighthouse at , and the lighthouse at is kilometers from the lighthouse at . To an observer at , the angle determined by the lights at and and the angle determined by the lights at and are equal. To an observer at , the angle determined by the lights at and and the angle determined by the lights at and are equal. The number of kilometers from to is given by , where , , and are relatively prime positive integers, and is not divisible by the square of any prime. Find .

## Problem 11

lines and circles divide the plane into at most disjoint regions. Compute .

## Problem 15

Find the number of functions from to the integers such that , , and

for all and in .

## Problem 14

The sequence satisfies and for . Find the greatest integer less than or equal to .

## Problem 15

Let be a diameter of a circle with diameter . Let and be points on one of the semicircular arcs determined by such that is the midpoint of the semicircle and . Point lies on the other semicircular arc. Let be the length of the line segment whose endpoints are the intersections of diameter with the chords and . The largest possible value of can be written in the form , where , , and are positive integers and is not divisible by the square of any prime. Find .

## Problem 14

Let and be real numbers satisfying and . Evaluate .