Difference between revisions of "2009 AIME II Problems"
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== Problem 11 == | == Problem 11 == | ||
− | For certain pairs <math>(m,n)</math> of positive integers with <math>m\geq n</math> there are exactly <math>50</math> distinct positive integers <math>k</math> such that <math>|\log m - \log k| < \log n</math>. Find the sum of all possible values of the product <math> | + | For certain pairs <math>(m,n)</math> of positive integers with <math>m\geq n</math> there are exactly <math>50</math> distinct positive integers <math>k</math> such that <math>|\log m - \log k| < \log n</math>. Find the sum of all possible values of the product <math>m \cdot n</math>. |
[[2009 AIME II Problems/Problem 11|Solution]] | [[2009 AIME II Problems/Problem 11|Solution]] |
Revision as of 01:08, 27 July 2020
2009 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
Before starting to paint, Bill had ounces of blue paint,
ounces of red paint, and
ounces of white paint. Bill painted four equally sized stripes on a wall, making a blue stripe, a red stripe, a white stripe, and a pink stripe. Pink is a mixture of red and white, not necessarily in equal amounts. When Bill finished, he had equal amounts of blue, red, and white paint left. Find the total number of ounces of paint Bill had left.
Problem 2
Suppose that ,
, and
are positive real numbers such that
,
, and
. Find
Problem 3
In rectangle ,
. Let
be the midpoint of
. Given that line
and line
are perpendicular, find the greatest integer less than
.
Problem 4
A group of children held a grape-eating contest. When the contest was over, the winner had eaten grapes, and the child in
-th place had eaten
grapes. The total number of grapes eaten in the contest was
. Find the smallest possible value of
.
Problem 5
Equilateral triangle is inscribed in circle
, which has radius
. Circle
with radius
is internally tangent to circle
at one vertex of
. Circles
and
, both with radius
, are internally tangent to circle
at the other two vertices of
. Circles
,
, and
are all externally tangent to circle
, which has radius
, where
and
are relatively prime positive integers. Find
.
Problem 6
Let be the number of five-element subsets that can be chosen from the set of the first
natural numbers so that at least two of the five numbers are consecutive. Find the remainder when
is divided by
.
Problem 7
Define to be
for
odd and
for
even. When
is expressed as a fraction in lowest terms, its denominator is
with
odd. Find
.
Problem 8
Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let and
be relatively prime positive integers such that
is the probability that the number of times Dave rolls his die is equal to or within one of the number of times Linda rolls her die. Find
.
Problem 9
Let be the number of solutions in positive integers to the equation
, and let
be the number of solutions in positive integers to the equation
. Find the remainder when
is divided by
.
Problem 10
Four lighthouses are located at points ,
,
, and
. The lighthouse at
is
kilometers from the lighthouse at
, the lighthouse at
is
kilometers from the lighthouse at
, and the lighthouse at
is
kilometers from the lighthouse at
. To an observer at
, the angle determined by the lights at
and
and the angle determined by the lights at
and
are equal. To an observer at
, the angle determined by the lights at
and
and the angle determined by the lights at
and
are equal. The number of kilometers from
to
is given by
, where
,
, and
are relatively prime positive integers, and
is not divisible by the square of any prime. Find
.
Problem 11
For certain pairs of positive integers with
there are exactly
distinct positive integers
such that
. Find the sum of all possible values of the product
.
Problem 12
From the set of integers , choose
pairs
with
so that no two pairs have a common element. Suppose that all the sums
are distinct and less than or equal to
. Find the maximum possible value of
.
Problem 13
Let and
be the endpoints of a semicircular arc of radius
. The arc is divided into seven congruent arcs by six equally spaced points
. All chords of the form
or
are drawn. Let
be the product of the lengths of these twelve chords. Find the remainder when
is divided by
.
Problem 14
The sequence satisfies
and
for
. Find the greatest integer less than or equal to
.
Problem 15
Let be a diameter of a circle with diameter
. Let
and
be points on one of the semicircular arcs determined by
such that
is the midpoint of the semicircle and
. Point
lies on the other semicircular arc. Let
be the length of the line segment whose endpoints are the intersections of diameter
with the chords
and
. The largest possible value of
can be written in the form
, where
,
, and
are positive integers and
is not divisible by the square of any prime. Find
.
See also
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.