Mock AIME 4 Pre 2005/Problems
Contents
[hide]Problem 1
For how many integers is it possible to express as the sum of distinct positive integers?
Problem 2
is a sequence of real numbers where is the arithmetic mean of the previous terms for and is a sequence of real numbers in which is the geometric mean of the previous terms for and If for and then compute the value of .
Problem 3
Compute the largest integer such that is divisible by .
Problem 4
is a regular heptagon, and is a point in its interior such that is equilateral. There exists a unique pair of relatively prime positive integers such that . Compute the value of
Problem 5
Compute, to the nearest integer, the area of the region enclosed by the graph of
Problem 6
Determine the remainder when is divided by .
Problem 7
is a pyramid consisting of a square base and four slanted triangular faces such that all of its edges are equal in length. is a cube of edge length . Six pyramids similar to are constructed by taking points (all outside of ) where and using the nearest face of as the base of each pyramid exactly once. The volume of the octahedron formed by the (taking the convex hull) can be expressed as for some positive integers , , and , where is not divisible by the square of any prime. Determine the value of .
Problem 8
A single atom of Uranium-238 rests at the origin. Each second, the particle has a chance of moving one unit in the negative direction and a chance of moving in the positive direction. If the particle reaches , it ignites a fission that will consume the earth. If it reaches , it is harmlessly diffused. The probability that, eventually, the particle is safely contained can be expressed as for some relatively prime positive integers and . Determine the remainder obtained when is divided by .
Problem 9
The value of the sum can be expressed in the form , for some relatively prime positive integers and . Compute the value of .
Problem 10
blocks are selected from a crate containing blocks of each of the following dimensions: and The chosen blocks are stacked on top of each other (one per cross section) forming a tower of height . Compute the number of possible values of .
Problem 11
lines and circles divide the plane into at most disjoint regions. Compute .
Problem 12
Determine the number of permutations of such that if divides , the th number divides the th number.
Problem 13
, , and are distinct non-zero integers such that Compute the number of solutions to the equation
Problem 14
In triangle and is the unique point such that the perimeters of triangles and are equal. The value of can be expressed as where and are positive integers such that there is no prime divisor common to and and is not divisible by the square of any prime. Determine the value of
Problem 15
is a convex quadrilateral in which Let denote the intersection of the extensions of and is the circle tangent to line segment which also passes through and , and is the circle tangent to which passes through and Call the points of tangency and Let and be the points of intersection between and Finally, intersects at . If and then the value of is some integer Determine the remainder obtained when is divided by