2013 Mock AIME I Problems
Two circles and , each of unit radius, have centers and such that . Let be the midpoint of and let be a circle externally tangent to both and . and have a common tangent that passes through . If this tangent is also a common tangent to and , find the radius of circle .
Find the number of ordered positive integer triplets such that evenly divides , evenly divides , and .
Let be the greatest integer less than or equal to , and let . If , compute .
Compute the number of ways to fill in the following magic square such that:
1. the product of all rows, columns, and diagonals are equal (the sum condition is waived),
2. all entries are nonnegative integers less than or equal to ten, and
3. entries CAN repeat in a column, row, or diagonal.
In quadrilateral , . Also, , and . The perimeter of can be expressed in the form where and are relatively prime, and is not divisible by the square of any prime number. Find .
Find the number of integer values can have such that the equation has a solution.
Let be the set of all th primitive roots of unity with imaginary part greater than . Let be the set of all th primitive roots of unity with imaginary part greater than . (A primitive th root of unity is a th root of unity that is not a th root of unity for any .)Let . The absolute value of the real part of can be expressed in the form where and are relatively prime numbers. Find .
Let and be two perpendicular vectors in the plane. If there are vectors for in the same plane having projections of and along and respectively, then find (Note: and are unit vectors such that and , and the projection of a vector onto is the length of the vector that is formed by the origin and the foot of the perpendicular of onto .)
In a magic circuit, there are six lights in a series, and if one of the lights short circuit, then all lights after it will short circuit as well, without affecting the lights before it. Once a turn, a random light that isn’t already short circuited is short circuited. If is the expected number of turns it takes to short circuit all of the lights, find .
Let denote the th triangular number, i.e. . Let and be relatively prime positive integers so that Find .
Let and be the roots of the equation , and let and be the two possible values of Find .
In acute triangle , the orthocenter lies on the line connecting the midpoint of segment to the midpoint of segment . If , and the altitude from has length , find .
In acute , is the orthocenter, is the centroid, and is the midpoint of . It is obvious that , but does not always hold. If , , then the value of which produces the smallest value of such that can be expressed in the form , for squarefree. Compute .
Let If are its roots, then compute the remainder when is divided by 997.
Let be the set of integers such that for all integers . Compute the remainder when the sum of the elements in is divided by .