Difference between revisions of "2013 AIME II Problems"
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[[2013 AIME II Problems/Problem 15|Solution]] | [[2013 AIME II Problems/Problem 15|Solution]] | ||
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+ | {{AIME box|year=2013|n=II|before=[[2013 AIME I Problems]]|after=[[2014 AIME I Problems]]}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 12:17, 13 March 2020
2013 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
Suppose that the measurement of time during the day is converted to the metric system so that each day has metric hours, and each metric hour has
metric minutes. Digital clocks would then be produced that would read
just before midnight,
at midnight,
at the former
AM, and
at the former
PM. After the conversion, a person who wanted to wake up at the equivalent of the former
AM would set his new digital alarm clock for
, where
,
, and
are digits. Find
.
Problem 2
Positive integers and
satisfy the condition
Find the sum of all possible values of
.
Problem 3
A large candle is centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes
seconds to burn down the first centimeter from the top,
seconds to burn down the second centimeter, and
seconds to burn down the
-th centimeter. Suppose it takes
seconds for the candle to burn down completely. Then
seconds after it is lit, the candle's height in centimeters will be
. Find
.
Problem 4
In the Cartesian plane let and
. Equilateral triangle
is constructed so that
lies in the first quadrant. Let
be the center of
. Then
can be written as
, where
and
are relatively prime positive integers and
is an integer that is not divisible by the square of any prime. Find
.
Problem 5
In equilateral let points
and
trisect
. Then
can be expressed in the form
, where
and
are relatively prime positive integers, and
is an integer that is not divisible by the square of any prime. Find
.
Problem 6
Find the least positive integer such that the set of
consecutive integers beginning with
contains no square of an integer.
Problem 7
A group of clerks is assigned the task of sorting files. Each clerk sorts at a constant rate of
files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a similar assignment occurs at the end of the third hour. The group finishes the sorting in
hours and
minutes. Find the number of files sorted during the first one and a half hours of sorting.
Problem 8
A hexagon that is inscribed in a circle has side lengths ,
,
,
,
, and
in that order. The radius of the circle can be written as
, where
and
are positive integers. Find
.
Problem 9
A board is completely covered by
tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let
be the number of tilings of the
board in which all three colors are used at least once. For example, a
red tile followed by a
green tile, a
green tile, a
blue tile, and a
green tile is a valid tiling. Note that if the
blue tile is replaced by two
blue tiles, this results in a different tiling. Find the remainder when
is divided by
.
Problem 10
Given a circle of radius , let
be a point at a distance
from the center
of the circle. Let
be the point on the circle nearest to point
. A line passing through the point
intersects the circle at points
and
. The maximum possible area for
can be written in the form
, where
,
,
, and
are positive integers,
and
are relatively prime, and
is not divisible by the square of any prime. Find
.
Problem 11
Let , and let
be the number of functions
from set
to set
such that
is a constant function. Find the remainder when
is divided by
.
Problem 12
Let be the set of all polynomials of the form
, where
,
, and
are integers. Find the number of polynomials in
such that each of its roots
satisfies either
or
.
Problem 13
In ,
, and point
is on
so that
. Let
be the midpoint of
. Given that
and
, the area of
can be expressed in the form
, where
and
are positive integers and
is not divisible by the square of any prime. Find
.
Problem 14
For positive integers and
, let
be the remainder when
is divided by
, and for
let
. Find the remainder when
is divided by
.
Problem 15
Let be angles of an acute triangle with
There are positive integers
,
,
, and
for which
where
and
are relatively prime and
is not divisible by the square of any prime. Find
.
2013 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2013 AIME I Problems |
Followed by 2014 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.