Mock AIME 6 Pre 2005/Problems
. Find the remainder when is divided by .
. Suppose , , and . Given that and , where is a prime, find .
. Suppose there are red points and blue points on a circle. Let be the probability that a convex polygon whose vertices are among the points has at least one blue vertex, where and are relatively prime. Find .
. Let be the circumcircle of and let , , and be points on sides , , and , respectively, such that , , , , and , and , , and are concurrent. Let be a point on minor arc . Extend to such that is tangent to the circle at and . Find .
. Let where . Find the remainder when is divided by .
. Let and for . Find the smallest such that is an integer.
. Let be the region inside the graph and to the right of the line . If the area of is in the form where is squarefree, find .
. If and is real, evaluate .
. You have boxes and balls. , , and of these balls are blue, green, and red, respectively. Suppose the boxes are numbered through . You place blue ball, green balls, and red balls in box . Then blue balls, green balls, and red balls in box . Similarly, you put blue balls, green balls, and red balls in box for . Repeat the entire process (from boxes to ) until you run out of one color of balls. How many red balls are in boxes , , and ? (NOTE: After placing the last ball of a certain color in a box, you still place the balls of the other colors in that box. You do not, however, place balls in the following box.)
. There are two ants on opposite corners of a cube. On each move, they can travel along an edge to an adjacent vertex. If the probability that they both return to their starting position after moves is , where and are relatively prime, find . (NOTE: They do not stop if they collide.)
. Evaluate .
. A rigged coin has the property that when it is flipped times the probability of getting heads times is equal to the probability of getting heads times. If is the probability of getting a two heads in a row, where and are relatively prime, find .
. In , , , and . Let be on side and be on side such that is perpendicular to and . If , find . (NOTE: denotes the area of .)
. Let be a polynomial of degree with leading coefficient such that for , . Find the number of 's at the end of .
. An AIME has questions, of each of three difficulties: easy, medium, and hard. Let denote the number of easy questions up to question (including question ). Similarly define and . Let be the number of ways to arrange the questions in the AIME such that, for any , and if a easy and hard problem are consecutive, the easy always comes first. Find the remainder when is divided by .