Mock AIME 5 2005-2006 Problems
Contents
Problem 1
Suppose is a positive integer. Let be the sum of the distinct positive prime divisors of less than (e.g. and ). Evaluate the remainder when is divided by .
Problem 2
A circle of radius is internally tangent to a larger circle of radius such that the center of lies on . A diameter of is drawn tangent to . A second line is drawn from tangent to . Let the line tangent to at intersect at . Find the area of .
Problem 3
A number , where denotes the th digit in the base- representation of for , is a positive integer with distinct nonzero digits such that if is even and if is odd for (and ). Let be the number of four-digit hailstone numbers and be the number of three-digit hailstone numbers. Find .
Problem 4
Let and be integers such that and . Given that and , find the number of ordered pairs such that . ( is the greatest integer less than or equal to and is the least integer greater than or equal to ).
Problem 5
Find the largest prime divisor of .
Problem 6
, , and are polynomials defined by:
Find the number of distinct complex roots of .
Problem 7
A coin of radius is flipped onto an square grid divided into equal squares. Circles are inscribed in of these squares. Let be the probability that, given that the coin lands completely within one of the smaller squares, it also lands completely within one of the circles. Let be the probability that, when flipped onto the grid, the coin lands completely within one of the smaller squares. Let smallest value of such that . Find the value of .
Problem 8
Let be a polyhedron with faces, all of which are equilateral triangles, squares, or regular pentagons with equal side length. Given there is at least one of each type of face and there are twice as many pentagons as triangles, what is the sum of all the possible number of vertices can have?
Problem 9
nondistinguishable residents are moving into distinct houses in Conformistville, with at least one resident per house. In how many ways can the residents be assigned to these houses such that there is at least one house with residents?
Problem 10
Find the smallest positive integer such that is divisible by all the primes between and .
Problem 11
Let be a subset of consecutive elements of where is a positive integer. Define , where if has an odd number of divisors and if has an even number of divisors. For how many does there exist an such that and ? ( denotes the cardinality of the set , or the number of elements in )
Problem 12
Let be a triangle with , , and . Let be the foot of the altitude from to and be the point on between and such that . Extend to meet the circumcircle of at . If the area of triangle is , where and are relatively prime positive integers, find .
Problem 13
Let be the set of positive integers with only odd digits satisfying the following condition: any with digits must be divisible by . Let be the sum of the smallest elements of . Find the remainder upon dividing by .
Problem 14
Let be a triangle such that , , and . Let be the orthocenter of (intersection of the altitudes). Let be the midpoint of , be the midpoint of , and be the midpoint of . Points , , and are constructed on , , and , respectively, such that is the midpoint of , is the midpoint of , and is the midpoint of . Find .
Problem 15
colored beads are placed on a necklace (circular ring) such that each bead is adjacent to two others. The beads are labeled , , , around the circle in order. Two beads and , where and are non-negative integers, satisfy if and only if the color of is the same as the color of . Given that there exists no non-negative integer and positive integer such that , where all subscripts are taken , find the minimum number of different colors of beads on the necklace.
See also
Mock AIME 5 2005-2006 (Problems, Source) | ||
Preceded by Mock AIME 4 2005-2006 |
Followed by Mock AIME 1 2006-2007 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |