2009 AIME I真题
Call a -digit number geometric if it has distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers.
There is a complex number with imaginary part and a positive integer such that
A coin that comes up heads with probability and tails with probability independently on each flip is flipped eight times. Suppose the probability of three heads and five tails is equal to of the probability of five heads and three tails. Let , where and are relatively prime positive integers. Find .
In parallelogram , point is on so that and point is on so that . Let be the point of intersection of and . Find .
Triangle has and . Points and are located on and respectively so that , and is the angle bisector of angle . Let be the point of intersection of and , and let be the point on line for which is the midpoint of . If , find .
How many positive integers less than are there such that the equation has a solution for ? (The notation denotes the greatest integer that is less than or equal to .)
The sequence satisfies and for . Let be the least integer greater than for which is an integer. Find .
Let . Consider all possible positive differences of pairs of elements of. Let be the sum of all of these differences. Find the remainder when is divided by .
A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from to inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, the digits given were . Find the total number of possible guesses for all three prizes consistent with the hint.
The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from to in clockwise order. Committee rules state that a Martian must occupy chair and an Earthling must occupy chair . Furthermore, no Earthling can sit immediately to the left of a Martian, no Martian can sit immediately to the left of a Venusian, and no Venusian can sit immediately to the left of an Earthling. The number of possible seating arrangements for the committee is . Find .
Consider the set of all triangles where is the origin and and are distinct points in the plane with nonnegative integer coordinates such that . Find the number of such distinct triangles whose area is a positive integer.
In right with hypotenuse , , , and is the altitude to . Let be the circle having as a diameter. Let be a point outside such that and are both tangent to circle . The ratio of the perimeter of to the length can be expressed in the form , where and are relatively prime positive integers. Find .
The terms of the sequence defined by for are positive integers. Find the minimum possible value of .
For , define , where . If and , find the minimum possible value for .
In triangle , , , and . Let be a point in the interior of . Let and denote the incenters of triangles and , respectively. The circumcircles of triangles and meet at distinct points and . The maximum possible area of can be expressed in the form , where , , and are positive integers and is not divisible by the square of any prime. Find .