2005 AIME I Problems
|2005 AIME I (Answer Key)|
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Six congruent circles form a ring with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle with radius 30. Let be the area of the region inside circle and outside of the six circles in the ring. Find
For each positive integer let denote the increasing arithmetic sequence of integers whose first term is 1 and whose common difference is For example, is the sequence For how many values of does contain the term 2005?
How many positive integers have exactly three proper divisors, each of which is less than 50?
The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are 5 members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. Find the maximum number of members this band can have.
Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the 8 coins.
Let be the product of the nonreal roots of Find
In quadrilateral and Given that where and are positive integers, find
The equation has three real roots. Given that their sum is where and are relatively prime positive integers, find
Twenty seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. The 27 cubes are randomly arranged to form a cube. Given the probability of the entire surface area of the larger cube is orange is where and are distinct primes and and are positive integers, find
Triangle lies in the Cartesian Plane and has an area of 70. The coordinates of and are and respectively, and the coordinates of are The line containing the median to side has slope Find the largest possible value of
A semicircle with diameter is contained in a square whose sides have length 8. Given the maximum value of is find
For positive integers let denote the number of positive integer divisors of including 1 and For example, and Define by Let denote the number of positive integers with odd, and let denote the number of positive integers with even. Find
A particle moves in the Cartesian Plane according to the following rules:
- From any lattice point the particle may only move to or
- There are no right angle turns in the particle's path.
How many different paths can the particle take from to ?
Consider the points and There is a unique square such that each of the four points is on a different side of Let be the area of Find the remainder when is divided by 1000.
Triangle has The incircle of the triangle evenly trisects the median If the area of the triangle is where and are integers and is not divisible by the square of a prime, find