Difference between revisions of "Mock AIME 4 2006-2007 Problems"
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[[Mock AIME 4 2006-2007 Problems/Problem 8 | Solution]] | [[Mock AIME 4 2006-2007 Problems/Problem 8 | Solution]] | ||
==Problem 9== | ==Problem 9== | ||
− | Compute the smallest integer <math>k</math> such that the fraction <center><p><math>\frac{7k+100}{5k-3}</math></p></center> is reducible. | + | Compute the smallest positive integer <math>k</math> such that the fraction <center><p><math>\frac{7k+100}{5k-3}</math></p></center> is reducible. |
[[Mock AIME 4 2006-2007 Problems/Problem 9 | Solution]] | [[Mock AIME 4 2006-2007 Problems/Problem 9 | Solution]] | ||
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==Problem 10== | ==Problem 10== | ||
Compute the remainder when <center><p><math>{2007 \choose 0} + {2007 \choose 3} + \cdots + {2007 \choose 2007}</math></p></center> is divided by 1000. | Compute the remainder when <center><p><math>{2007 \choose 0} + {2007 \choose 3} + \cdots + {2007 \choose 2007}</math></p></center> is divided by 1000. | ||
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[[Mock AIME 4 2006-2007 Problems/Problem 15 | Solution]] | [[Mock AIME 4 2006-2007 Problems/Problem 15 | Solution]] | ||
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+ | == See also == | ||
+ | |||
+ | * [[Mock AIME 4 2006-2007]] |
Latest revision as of 13:17, 16 January 2007
Contents
[hide]Problem 1
Albert starts to make a list, in increasing order, of the positive integers that have a first digit of 1. He writes but by the 1,000th digit he (finally) realizes that the list would contain an infinite number of elements. Find the three-digit number formed by the last three digits he wrote (the 998th, 999th, and 1000th digits, in that order).
Problem 2
Two points and
are chosen on the graph of
, with
. The points
and
trisect
, with
. Through
a horizontal line is drawn to cut the curve at
. Find
if
and
.
Problem 3
Find the largest prime factor of the smallest positive integer such that
are distinct integers such that the polynomial
has exactly
nonzero coefficients.
Problem 4
Points ,
, and
are on the circumference of a unit circle so that the measure of
is
, the measure of
is
, and the measure of
is
. The area of the triangular shape bounded by
and line segments
and
can be written in the form
, where
and
are relatively prime positive integers. Find
.
Problem 5
How many 10-digit positive integers have all digits either 1 or 2, and have two consecutive 1's?
Problem 6
For how many positive integers does there exist a regular
-sided polygon such that the number of diagonals is a nonzero perfect square?
Problem 7
Find the remainder when is divided by 1000.
Problem 8
The number of increasing sequences of positive integers such that
is even for
can be expressed as
for some positive integers
. Compute the remainder when
is divided by 1000.
Problem 9
Compute the smallest positive integer such that the fraction
is reducible.
Problem 10
Compute the remainder when
is divided by 1000.
Problem 11
Let be an equilateral triangle. Two points
and
are chosen on
and
, respectively, such that
. Let
be the intersection of
and
. The area of
is 13 and the area of
is 3. If
, where
,
, and
are relatively prime positive integers, compute
.
Problem 12
The number of partitions of 2007 that have an even number of even parts can be expressed as , where
and
are positive integers and
is prime. Find the sum of the digits of
.
Problem 13
The sum
can be written in the form , where
. Compute the remainder when
is divided by 100.
Problem 14
Let be the arithmetic mean of all positive integers
such that
.
Find the greatest integer less than or equal to .
Problem 15
Triangle has sides
,
, and
of length 43, 13, and 48, respectively. Let
be the circle circumscribed around
and let
be the intersection of
and the perpendicular bisector of
that is not on the same side of
as
. The length of
can be expressed as
, where
and
are positive integers and
is not divisible by the square of any prime. Find the greatest integer less than or equal to
.