Difference between revisions of "2010 AIME I Problems"
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== Problem 9 == | == Problem 9 == | ||
− | Let <math>(a,b,c)</math> be | + | Let <math>(a,b,c)</math> be a real solution of the system of equations <math>x^3 - xyz = 2</math>, <math>y^3 - xyz = 6</math>, <math>z^3 - xyz = 20</math>. The greatest possible value of <math>a^3 + b^3 + c^3</math> can be written in the form <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. |
[[2010 AIME I Problems/Problem 9|Solution]] | [[2010 AIME I Problems/Problem 9|Solution]] |
Revision as of 21:38, 22 February 2012
2010 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
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1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
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 .
Problem 2
Find the remainder when is divided by .
Problem 3
Suppose that and . The quantity can be expressed as a rational number , where and are relatively prime positive integers. Find .
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 .
Problem 5
Positive integers , , , and satisfy , , and . Find the number of possible values of .
Problem 6
Let be a quadratic polynomial with real coefficients satisfying for all real numbers , and suppose . Find .
Problem 7
Define an ordered triple of sets to be minimally intersecting 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 .
Problem 8
For a real number , let denominate 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 .
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 .
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 .
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 .
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 , .
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 .
Problem 14
For each positive integer n, let . Find the largest value of n for which .
Note: is the greatest integer less than or equal to .
Problem 15
In with , , and , let be a point on such that the incircles of and have equal radii. Let and be positive relatively prime integers such that . Find .