# Difference between revisions of "1995 AIME Problems"

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

+ | Triangle <math>\displaystyle ABC</math> is isosceles, with <math>\displaystyle AB=AC</math> and altitude <math>\displaystyle AM=11.</math> Suppose that there is a point <math>\displaystyle D</math> on <math>\displaystyle \overline{AM}</math> with <math>\displaystyle AD=10</math> and <math>\displaystyle \angle BDC=3\angle BAC.</math> Then the perimeter of <math>\displaystyle \triangle ABC</math> may be written in the form <math>\displaystyle a+\sqrt{b},</math> where <math>\displaystyle a</math> and <math>\displaystyle b</math> are integers. Find <math>\displaystyle a+b.</math> | ||

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+ | [[Image:AIME_1995_Problem_9.png]] | ||

[[1995 AIME Problems/Problem 9|Solution]] | [[1995 AIME Problems/Problem 9|Solution]] |

## Revision as of 01:21, 22 January 2007

## Contents

## Problem 1

Square is For the lengths of the sides of square are half the lengths of the sides of square two adjacent sides of square are perpendicular bisectors of two adjacent sides of square and the other two sides of square are the perpendicular bisectors of two adjacent sides of square The total area enclosed by at least one of can be written in the form where and are relatively prime positive integers. Find

## Problem 2

Find the last three digits of the product of the positive roots of

## Problem 3

Starting at an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Let be the probability that the object reaches in six or fewer steps. Given that can be written in the form where and are relatively prime positive integers, find

## Problem 4

Circles of radius and are externally tangent to each other and are internally tangent to a circle of radius . The circle of radius has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord.

## Problem 5

For certain real values of and the equation has four non-real roots. The product of two of these roots is and the sum of the other two roots is where Find

## Problem 6

Let How many positive integer divisors of are less than but do not divide ?

## Problem 7

Given that and

where and are positive integers with and relatively prime, find

## Problem 8

For how many ordered pairs of positive integers with are both and integers?

## Problem 9

Triangle is isosceles, with and altitude Suppose that there is a point on with and Then the perimeter of may be written in the form where and are integers. Find