# Difference between revisions of "1985 AIME Problems"

m (→Problem 9: added) |
m (→Problem 10: added) |
||

Line 45: | Line 45: | ||

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

+ | How many of the first 1000 positive integers can be expressed in the form | ||

+ | <math>\lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor</math>, | ||

+ | |||

+ | where <math>x</math> is a real number, and <math>\lfloor z \rfloor</math> denotes the greatest integer less than or equal to <math>z</math>? | ||

[[1985 AIME Problems/Problem 10 | Solution]] | [[1985 AIME Problems/Problem 10 | Solution]] | ||

+ | |||

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

## Revision as of 09:28, 3 December 2006

## Contents

## Problem 1

Let , and for let. Calculate the product .

## Problem 2

When a right triangle is rotated about one leg, the volume of the cone produced is . When the triangle is rotated about the other leg, the volume of the cone produced is . What is the length (in cm) of the hypotenuse of the triangle?

## Problem 3

Find if , , and are positive integers which satisfy , where .

## Problem 4

A small square is constructed inside a square of area 1 by dividing each side of the unit square into equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of if the the area of the small square is exactly .

## Problem 5

A sequence of integers is chosen so that for each . What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492?

## Problem 6

As shown in the figure, triangle is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of triangle .

*An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.*

## Problem 7

Assume that , , , and are positive integers such that , , and . Determine .

## Problem 8

The sum of the following seven numbers is exactly 19: , , , , , , . It is desired to replace each by an integer approximation , , so that the sum of the 's is also 19 and so that , the maximum of the "errors" , the maximum absolute value of the difference, is as small as possible. For this minimum , what is ?

## Problem 9

In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of , , and radians, respectively, where . If , which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?

## Problem 10

How many of the first 1000 positive integers can be expressed in the form

,

where is a real number, and denotes the greatest integer less than or equal to ?