2010 AIME II真题
Let be the greatest integer multiple of all of whose digits are even and no two of whose digits are the same. Find the remainder when is divided by .
A point is chosen at random in the interior of a unit square . Let denote the distance from to the closest side of . The probability that is equal to , where and are relatively prime positive integers. Find .
Let be the product of all factors (not necessarily distinct) where and are integers satisfying . Find the greatest positive integer such that divides .
Dave arrives at an airport which has twelve gates arranged in a straight line with exactly feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Dave walks feet or less to the new gate be a fraction , where and are relatively prime positive integers. Find .
Positive numbers , , and satisfy and . Find .
Find the smallest positive integer with the property that the polynomial can be written as a product of two nonconstant polynomials with integer coefficients.
Let , where , , and are real. There exists a complex number such that the three roots of are , , and , where . Find .
Let be the number of ordered pairs of nonempty sets and that have the following properties:
- The number of elements of is not an element of ,
- The number of elements of is not an element of .
Let be a regular hexagon. Let , , , , , and be the midpoints of sides , , , , , and , respectively. The segments , , , , , and bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of be expressed as a fraction where and are relatively prime positive integers. Find .
Find the number of second-degree polynomials with integer coefficients and integer zeros for which .
Define a T-grid to be a matrix which satisfies the following two properties:
- Exactly five of the entries are 's, and the remaining four entries are 's.
- Among the eight rows, columns, and long diagonals (the long diagonals are and , no more than one of the eight has all three entries equal.
Find the number of distinct T-grids.
Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is . Find the minimum possible value of their common perimeter.
The cards in a deck are numbered . Alex, Blair, Corey, and Dylan each pick a card from the deck randomly and without replacement. The two people with lower numbered cards form a team, and the two people with higher numbered cards form another team. Let be the probability that Alex and Dylan are on the same team, given that Alex picks one of the cards and , and Dylan picks the other of these two cards. The minimum value of for which can be written as , where and are relatively prime positive integers. Find .
Triangle with right angle at , and . Point on is chosen such that and . The ratio can be represented in the form , where , , are positive integers and is not divisible by the square of any prime. Find .
In triangle , , , and . Points and lie on with and . Points and lie on with and . Let be the point, other than , of intersection of the circumcircles of and . Ray meets at . The ratio can be written in the form , where and are relatively prime positive integers. Find .