1995 AIME Problems
1995 AIME (Answer Key) | AoPS Contest Collections | ||
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 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 lengths, and the area of either of them can be expressed uniquely 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 .