Difference between revisions of "2015 AIME II Problems"
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==Problem 8== | ==Problem 8== | ||
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+ | Let <math>a</math> and <math>b</math> be positive integers satisfying <math>\frac{ab+1}{a+b} \lt \frac{3}{2}</math>. The maximum possible value of <math>\frac{a^3b^3+1}{a^3+b^3}</math> is <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. | ||
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+ | [[2015 AIME II Problems/Problem 8 | Solution]] | ||
==Problem 9== | ==Problem 9== |
Revision as of 18:02, 26 March 2015
2015 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
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
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 $\frac{ab+1}{a+b} \lt \frac{3}{2}$ (Error compiling LaTeX. Unknown error_msg). The maximum possible value of
is
, where
and
are relatively prime positive integers. Find
.
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
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.