Difference between revisions of "1995 AIME Problems"
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== Problem 5 == | == Problem 5 == | ||
| − | For certain real values of <math>\displaystyle a, b, c,</math> and <math>\displaystyle | + | For certain real values of <math>\displaystyle a, b, c,</math> and <math>\displaystyle d_{},</math> the equation <math>\displaystyle x^4+ax^3+bx^2+cx+d=0</math> has four non-real roots. The product of two of these roots is <math>\displaystyle 13+i</math> and the sum of the other two roots is <math>\displaystyle 3+4i,</math> where <math>i=\sqrt{-1}.</math> Find <math>\displaystyle b.</math> |
[[1995 AIME Problems/Problem 5|Solution]] | [[1995 AIME Problems/Problem 5|Solution]] | ||
Revision as of 01:09, 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 
