Difference between revisions of "Mock AIME 1 2010 Problems"
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== Problem 2 == | == Problem 2 == | ||
− | Find the last three digits of the number of 7-tuples of positive integers <math>(a_1, a_2, a_3, a_4, a_5, a_6, a_7)</math> such that <math>a_1 \, | \, a_2 \, | \, a_3 \, | \, a_4 \, | \, a_5 \, | \, a_6 \, | \, a_7 \, | \, 6468</math>, that is, <math>a_1</math> divides <math>a_2</math>, <math>a_2</math> divides <math>a_3</math>, <math>a_3</math> divides <math>a_4</math>, <math>a_4</math> divides <math>a_5</math>, <math>a_5</math> divides <math>a_6</math>, <math>a_6</math> divides <math>a_7</math>, and <math>a_7</math> divides 6468. | + | Find the last three digits of the number of 7-tuples of positive integers <math>(a_1, a_2, a_3, a_4, a_5, a_6, a_7)</math> such that <math>a_1 \, | \, a_2 \, | \, a_3 \, | \, a_4 \, | \, a_5 \, | \, a_6 \, | \, a_7 \, | \, 6468</math>, that is, <math>a_1</math> divides <math>a_2</math>, <math>a_2</math> divides <math>a_3</math>, <math>a_3</math> divides <math>a_4</math>, <math>a_4</math> divides <math>a_5</math>, <math>a_5</math> divides <math>a_6</math>, <math>a_6</math> divides <math>a_7</math>, and <math>a_7</math> divides <math>6468</math>. |
[[Mock AIME 1 2010 Problems/Problem 2|Solution]] | [[Mock AIME 1 2010 Problems/Problem 2|Solution]] |
Revision as of 19:57, 1 August 2024
Contents
Problem 1
Let . Find the number of perfect squares among
.
Problem 2
Find the last three digits of the number of 7-tuples of positive integers such that
, that is,
divides
,
divides
,
divides
,
divides
,
divides
,
divides
, and
divides
.
Problem 3
Let be a line segment of length
, and let
be the set of all points
such that
. Find the last three digits of the largest integer less than the area of
.
Problem 4
A round robin tournament is a tournament in which every player plays every other player exactly once. There is a round robin tournament with 2010 people. In each match, the winner scores one point, and the loser scores no points. There are no ties. Find the last three digits of the greatest possible difference between the first and second highest scores appearing among the players.
Problem 5
For every integer , the
representation of
is defined to be the unique sequence of integers
, with
and
such that
. We represent
as
, where
if
is 0 or 1, and
if
. For example,
. Find the last three digits of the sum of all integers
with
such that
has at least one zero when written in balanced ternary form.
Problem 6
Find the number of Gaussian integers with magnitude less than 10000 such that there exists a different Gaussian integer
such that
. (The magnitude of a complex
, where
and
are reals, is defined to be
. A Gaussian integer is defined to be a complex number whose real and imaginary parts are both integers.)
Problem 7
Find the number of positive integers for which there exists a positive integer
such that
is the square of an integer.
Problem 8
In the context of this problem, a is a
block, a
is a
block, and a
is a
block. If
is the number of ways George can place one square, two identical dominoes, and three identical trominoes on a
chessboard such that no two overlap, find the remainder when
is divided by 1000.
Problem 9
Let and
be circles of radii 5 and 7, respectively, and suppose that the distance between their centers is 10. There exists a circle
that is internally tangent to both
and
, and tangent to the line joining the centers of
and
. If the radius of
can be expressed in the form
, where
,
, and
are integers, and
is not divisible by the square if any prime, find the value of
.
Problem 10
Find the last three digits of the largest possible value of where
and
are positive reals.
Problem 11
Let be such that
,
, and
. Let
,
, and
be points such that
,
,
,
,
, and
. If the perimeter of hexagon
can be expressed in the form
, where
and
are relatively prime positive integers and
is an integer not divisible by the square of any prime, find
.
Problem 12
Suppose ,
, and
for all integers
. Find the last three digits of
.
Problem 13
Suppose is inscribed in circle
.
and
are the feet of the altitude from
to
and
to
, respectively. Let
be the intersection of lines
and
, let
be the point of intersection of
and line
distinct from
, and let
be the foot of the perpendicular from
to
. Given that
,
, and
, and that
can be expressed in the form
, where
and
are relatively prime positive integers and
is an integer not divisible by the square of any prime, find the last three digits of
.
Problem 14
Let , and let
be a subset of
with
and
. For such a set
, let
denote the number of sets
with
such that
(i) is a subset of
with the same number of elements as
,
(ii) for
,
(iii) for
.
Let . What is the smallest positive integer
such that
is over 9000?
Problem 15
Let be the set of all integers less than or equal to 2010 such that when its divisors are listed in increasing order, they are alternatingly odd and even. For example, 6 belongs to
, since the divisors of 6 are 1, 2, 3, and 6, which are odd, even, odd, and even in that order. Find the last three digits of the largest possible value of
, where
lies in
and
denotes the number of divisors of
.