2015 AIME II真题
Let be the least positive integer that is both percent less than one integer and percent greater than another integer. Find the remainder when is divided by .
In a new school, percent of the students are freshmen, percent are sophomores, percent are juniors, and percent are seniors. All freshmen are required to take Latin, and percent of sophomores, percent of the juniors, and percent of the seniors elect to take Latin. The probability that a randomly chosen Latin student is a sophomore is , where and are relatively prime positive integers. Find .
Let be the least positive integer divisible by whose digits sum to . Find .
In an isosceles trapezoid, the parallel bases have lengths and , and the altitude to these bases has length . The perimeter of the trapezoid can be written in the form , where and are positive integers. Find .
Two unit squares are selected at random without replacement from an grid of unit squares. Find the least positive integer such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than .
Steve says to Jon, "I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form for some positive integers and . Can you tell me the values of and ?"
After some calculations, Jon says, "There is more than one such polynomial."
Steve says, "You're right. Here is the value of ." He writes down a positive integer and asks, "Can you tell me the value of ?"
Jon says, "There are still two possible values of ."
Find the sum of the two possible values of .
Triangle has side lengths , , and . Rectangle has vertex on , vertex on , and vertices and on . In terms of the side length , the area of can be expressed as the quadratic polynomial
Then the coefficient , where and are relatively prime positive integers. Find .
Let and be positive integers satisfying . The maximum possible value of is , where and are relatively prime positive integers. Find .
A cylindrical barrel with radius feet and height feet is full of water. A solid cube with side length feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is cubic feet. Find .
Call a permutation of the integers quasi-increasing if for each . For example, and are quasi-increasing permutations of the integers , but is not. Find the number of quasi-increasing permutations of the integers .
The circumcircle of acute has center . The line passing through point perpendicular to intersects lines and at and , respectively. Also , , , and , where and are relatively prime positive integers. Find .
There are possible -letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than adjacent letters that are identical.
Define the sequence by , where represents radian measure. Find the index of the 100th term for which .
Let and be real numbers satisfying and . Evaluate .
Circles and have radii and , respectively, and are externally tangent at point . Point is on and point is on such that is a common external tangent of the two circles. A line through intersects again at and intersects again at . Points and lie on the same side of , and the areas of and are equal. This common area is , where and are relatively prime positive integers. Find .