Difference between revisions of "2015 AIME II Problems"
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==Problem 14== | ==Problem 14== | ||
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+ | Let <math>x</math> and <math>y</math> be real numbers satisfying <math>x^4y^5+y^4x^5=810</math> and <math>x^3y^6+y^3x^6=945</math>. Evaluate <math>2x^3+(xy)^3+2y^3</math>. | ||
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+ | [[2015 AIME II Problems/Problem 14 | Solution]] | ||
==Problem 15== | ==Problem 15== |
Revision as of 18:27, 26 March 2015
2015 AIME II (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
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 .
Problem 2
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 .
Problem 3
Let be the least positive integer divisible by whose digits sum to . Find .
Problem 4
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 .
Problem 5
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 .
Problem 6
Problem 7
Problem 8
Let and be positive integers satisfying . The maximum possible value of is , where and are relatively prime positive integers. Find .
Problem 9
Problem 10
Call a permutation of the integers quasi-increasing if for each . For example, 53421 and 14253 are quasi-increasing permutations of the integers , but 45123 is not. Find the number of quasi-increasing permutations of the integers .
Problem 11
Problem 12
There are possible 10-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 3 adjacent letters that are identical.
Problem 13
Problem 14
Let and be real numbers satisfying and . Evaluate .
Problem 15
2015 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2014 AIME I, 2014 AIME II |
Followed by 2016 AIME | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.