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[[2016 AIME II Problems/Problem 4 | Solution]] | [[2016 AIME II Problems/Problem 4 | Solution]] | ||
==Problem 5== | ==Problem 5== | ||
− | Triangle <math>ABC_0</math> has a right angle at <math>C_0</math>. Its side lengths are pairwise relatively prime positive integers, and its perimeter is <math>p</math>. Let <math>C_1</math> be the foot of the altitude to <math>\overline{AB}</math>, and for <math>n \geq 2</math>, let <math>C_n</math> be the foot of the altitude to <math>\overline{C_{n-2}B}</math> in <math>\triangle C_{n-2}C_{n-1}B</math>. The sum <math>\sum_{i= | + | Triangle <math>ABC_0</math> has a right angle at <math>C_0</math>. Its side lengths are pairwise relatively prime positive integers, and its perimeter is <math>p</math>. Let <math>C_1</math> be the foot of the altitude to <math>\overline{AB}</math>, and for <math>n \geq 2</math>, let <math>C_n</math> be the foot of the altitude to <math>\overline{C_{n-2}B}</math> in <math>\triangle C_{n-2}C_{n-1}B</math>. The sum <math>\sum_{i=2}^\infty C_{n-2}C_{n-1} = 6p</math>. Find <math>p</math>. |
[[2016 AIME II Problems/Problem 5 | Solution]] | [[2016 AIME II Problems/Problem 5 | Solution]] |
Revision as of 14:25, 17 August 2016
2016 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
Initially Alex, Betty, and Charlie had a total of peanuts. Charlie had the most peanuts, and Alex had the least. The three numbers of peanuts that each person had formed a geometric progression. Alex eats
of his peanuts, Betty eats
of her peanuts, and Charlie eats
of his peanuts. Now the three numbers of peanuts each person has forms an arithmetic progression. Find the number of peanuts Alex had initially.
Problem 2
There is a chance of rain on Saturday and a
chance of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is
, where
and
are relatively prime positive integers. Find
.
Problem 3
Let and
be real numbers satisfying the system
Find the value of
.
Problem 4
An rectangular box is built from
unit cubes. Each unit cube is colored red, green, or yellow. Each of the
layers of size
parallel to the
faces of the box contains exactly
red cubes, exactly
green cubes, and some yellow cubes. Each of the
layers of size
parallel to the
faces of the box contains exactly
green cubes, exactly
yellow cubes, and some red cubes. Find the smallest possible volume of the box.
Problem 5
Triangle has a right angle at
. Its side lengths are pairwise relatively prime positive integers, and its perimeter is
. Let
be the foot of the altitude to
, and for
, let
be the foot of the altitude to
in
. The sum
. Find
.
Problem 6
For polynomial , define
.
Then
, where
and
are relatively prime positive integers. Find
.
Problem 7
Squares and
have a common center at
. The area of
is 2016, and the area of
is a smaller positive integer. Square
is constructed so that each of its vertices lies on a side of
and each vertex of
lies on a side of
. Find the difference between the largest and smallest positive integer values for the area of
.
Problem 8
Find the number of sets of three distinct positive integers with the property that the product of
and
is equal to the product of
.
Problem 9
The sequences of positive integers and
are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let
. There is an integer
such that
and
. Find
.
Problem 10
Triangle is inscribed in circle
. Points
and
are on side
with
. Rays
and
meet
again at
and
(other than
), respectively. If
and
, then
, where
and
are relatively prime positive integers. Find
.
Problem 11
For positive integers and
, define
to be
-nice if there exists a positive integer
such that
has exactly
positive divisors. Find the number of positive integers less than
that are neither
-nice nor
-nice.
Problem 12
The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
Problem 13
Beatrix is going to place six rooks on a chessboard where both the rows and columns are labeled
to
; the rooks are placed so that no two rooks are in the same row or the same column. The
of a square is the sum of its row number and column number. The
of an arrangement of rooks is the least value of any occupied square.The average score over all valid configurations is
, where
and
are relatively prime positive integers. Find
.
Problem 14
Equilateral has side length
. Points
and
lie outside the plane of
and are on opposite sides of the plane. Furthermore,
, and
, and the planes of
and
form a
dihedral angle (the angle between the two planes). There is a point
whose distance from each of
and
is
. Find
.
Problem 15
For let
and
. Let
be positive real numbers such that
and
. The maximum possible value of
, where
and
are relatively prime positive integers. Find
.
2016 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2016 AIME I |
Followed by 2017 AIME I | |
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.