2017 AIME I真题
Fifteen distinct points are designated on : the 3 vertices , , and ; other points on side ; other points on side ; and other points on side . Find the number of triangles with positive area whose vertices are among these points.
When each of , , and is divided by the positive integer , the remainder is always the positive integer . When each of , , and is divided by the positive integer , the remainder is always the positive integer . Find .
For a positive integer , let be the units digit of . Find the remainder whenis divided by .
A pyramid has a triangular base with side lengths , , and . The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length . The volume of the pyramid is , where and are positive integers, and is not divisible by the square of any prime. Find .
A rational number written in base eight is , where all digits are nonzero. The same number in base twelve is . Find the base-ten number .
A circle circumscribes an isosceles triangle whose two congruent angles have degree measure . Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is . Find the difference between the largest and smallest possible values of .
For nonnegative integers and with , let . Let denote the sum of all , where and are nonnegative integers with . Find the remainder when is divided by .
Two real numbers and are chosen independently and uniformly at random from the interval . Let and be two points on the plane with . Let and be on the same side of line such that the degree measures of and are and respectively, and and are both right angles. The probability that is equal to , where and are relatively prime positive integers. Find .
Let , and for each integer let . Find the least such that is a multiple of .
Let , and where . Let be the unique complex number with the properties that is a real number and the imaginary part of is the greatest possible. Find the real part of .
Consider arrangements of the numbers in a array. For each such arrangement, let , , and be the medians of the numbers in rows , , and respectively, and let be the median of . Let be the number of arrangements for which . Find the remainder when is divided by .
Call a set product-free if there do not exist (not necessarily distinct) such that . For example, the empty set and the set are product-free, whereas the sets and are not product-free. Find the number of product-free subsets of the set .
For every , let be the least positive integer with the following property: For every , there is always a perfect cube in the range . Find the remainder whenis divided by 1000.
Let and satisfy and . Find the remainder when is divided by .
The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths , , and , as shown, is , where , , and are positive integers, and are relatively prime, and is not divisible by the square of any prime. Find .