1995 AIME Problems
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| 1995 AIME (Answer Key) Printable version: AoPS Resources • PDF | MathLinks Resources • PDF | ||
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Instructions
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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 
Problem 10
What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer?
Problem 11
A right rectangular prism 
(i.e., a rectangular parallelpiped) has sides of integral length with 
A plane parallel to one of the faces of cuts into two prisms, one of which is similar to 
and both of which have nonzero volume. Given that 
for how many ordered triples 
does such a plane exist?
Problem 12
Pyramid 
has square base 
congruent edges 
and 
and 
Let 
be the measure of the dihedral angle formed by faces 
and 
Given that 
where and are integers, find
Problem 13
Let 
be the integer closest to ![\sqrt[4]{n}.](http://alt2.artofproblemsolving.com/Forum/latexrender/pictures/b/3/4/b340eeed6a67c7ab3aeb645742e4d9e30bf73187.gif)
Find 
Problem 14
In a circle of radius 42, two chords of length 78 intersect at a point whose distance from the center is 18. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lenghts, and the area of either of them can be expressed uniquley in the form 
where 
and 
are positive integers and is not divisible by the square of any prime number. Find 
Problem 15
Let 
be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails. Given that can be written in the form 
where and are relatively prime positive integers, find 
.
