Problem 1

Maya lists all the positive divisors of . She then randomly selects two distinct divisors from this list. Let be the probability that exactly one of the selected divisors is a perfect square. The probability can be expressed in the form , where and are relatively prime positive integers. Find .

Solution

Problem 2

Find the remainder when is divided by .

Solution

Problem 3

Suppose that and . The quantity can be expressed as a rational number , where and are relatively prime positive integers. Find .

Solution

Problem 4

Jackie and Phil have two fair coins and a third coin that comes up heads with probability . Jackie flips the three coins, and then Phil flips the three coins. Let be the probability that Jackie gets the same number of heads as Phil, where and are relatively prime positive integers. Find .

Solution

Problem 5

Positive integers , , , and satisfy , , and . Find the number of possible values of .

Solution

Problem 6

Let be a quadratic polynomial with real coefficients satisfying for all real numbers , and suppose . Find .

Solution

Problem 7

Define an ordered triple of sets to be if and . For example, is a minimally intersecting triple. Let be the number of minimally intersecting ordered triples of sets for which each set is a subset of . Find the remainder when is divided by .

Note: represents the number of elements in the set .

Solution

Problem 8

For a real number , let denote the greatest integer less than or equal to . Let denote the region in the coordinate plane consisting of points such that . The region is completely contained in a disk of radius (a disk is the union of a circle and its interior). The minimum value of can be written as , where and are integers and is not divisible by the square of any prime. Find .

Solution

Problem 9

Let be a real solution of the system of equations , , . The greatest possible value of can be written in the form , where and are relatively prime positive integers. Find .

Solution

Problem 10

Let be the number of ways to write in the form , where the 's are integers, and . An example of such a representation is . Find .

Solution

Problem 11

Let be the region consisting of the set of points in the coordinate plane that satisfy both and . When is revolved around the line whose equation is , the volume of the resulting solid is , where , , and are positive integers, and are relatively prime, and is not divisible by the square of any prime. Find .

Solution

Problem 12

Let be an integer and let . Find the smallest value of such that for every partition of into two subsets, at least one of the subsets contains integers , , and (not necessarily distinct) such that .

Note: a partition of is a pair of sets , such that , .

Solution

Problem 13

Rectangle and a semicircle with diameter are coplanar and have nonoverlapping interiors. Let denote the region enclosed by the semicircle and the rectangle. Line meets the semicircle, segment , and segment at distinct points , , and , respectively. Line divides region into two regions with areas in the ratio . Suppose that , , and . Then can be represented as , where and are positive integers and is not divisible by the square of any prime. Find .

Solution

Problem 14

For each positive integer let . Find the largest value of for which .

Note: is the greatest integer less than or equal to .

Solution

Problem 15

In with , , and , let be a point on such that the incircles of and have equal radii. Then , where and are relatively prime positive integers. Find .

Solution

See also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.