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[[2015 AIME II Problems/Problem 15 | Solution]] | [[2015 AIME II Problems/Problem 15 | Solution]] | ||
− | {{AIME box|year=2015|n=II|before=[[2015 AIME I]]|after=[[2016 AIME I]]}} | + | {{AIME box|year=2015|n=II|before=[[2015 AIME I Problems]]|after=[[2016 AIME I Problems]]}} |
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Latest revision as of 19:15, 13 January 2024
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
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
Steve says to Jon, "I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form for some positive integers
and
. Can you tell me the values of
and
?"
After some calculations, Jon says, "There is more than one such polynomial."
Steve says, "You're right. Here is the value of ." He writes down a positive integer and asks, "Can you tell me the value of
?"
Jon says, "There are still two possible values of ."
Find the sum of the two possible values of .
Problem 7
Triangle has side lengths
,
, and
. Rectangle
has vertex
on
, vertex
on
, and vertices
and
on
. In terms of the side length
, the area of
can be expressed as the quadratic polynomial
Then the coefficient , where
and
are relatively prime positive integers. Find
.
Problem 8
Let and
be positive integers satisfying
. The maximum possible value of
is
, where
and
are relatively prime positive integers. Find
.
Problem 9
A cylindrical barrel with radius feet and height
feet is full of water. A solid cube with side length
feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is
cubic feet. Find
.
Problem 10
Call a permutation of the integers
quasi-increasing if
for each
. For example,
and
are quasi-increasing permutations of the integers
, but
is not. Find the number of quasi-increasing permutations of the integers
.
Problem 11
The circumcircle of acute has center
. The line passing through point
perpendicular to
intersects lines
and
at
and
, respectively. Also
,
,
, and
, where
and
are relatively prime positive integers. Find
.
Problem 12
There are possible
-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
adjacent letters that are identical.
Problem 13
Define the sequence by
, where
represents radian measure. Find the index of the 100th term for which
.
Problem 14
Let and
be real numbers satisfying
and
. Evaluate
.
Problem 15
Circles and
have radii
and
, respectively, and are externally tangent at point
. Point
is on
and point
is on
such that
is a common external tangent of the two circles. A line
through
intersects
again at
and intersects
again at
. Points
and
lie on the same side of
, and the areas of
and
are equal. This common area is
, where
and
are relatively prime positive integers. Find
.
2015 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2015 AIME I Problems |
Followed by 2016 AIME I Problems | |
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.