Difference between revisions of "Mock AIME 3 Pre 2005 Problems"
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A function <math>f(x)</math> is defined for all real numbers <math>x</math>. For all non-zero values <math>x</math>, we have | A function <math>f(x)</math> is defined for all real numbers <math>x</math>. For all non-zero values <math>x</math>, we have | ||
− | <cmath>2f\left(x\right) + f\left(\frac{1}{x}\right) = 5x + 4</cmath> | + | <cmath>2f\left(x\right) + f\left(\frac{1}{x}\right) = 5x + 4.</cmath> |
Let <math>S</math> denote the sum of all of the values of <math>x</math> for which <math>f(x) = 2004</math>. Compute the integer nearest to <math>S</math>. | Let <math>S</math> denote the sum of all of the values of <math>x</math> for which <math>f(x) = 2004</math>. Compute the integer nearest to <math>S</math>. |
Revision as of 23:52, 24 March 2019
Contents
Problem 1
Three circles are mutually externally tangent. Two of the circles have radii and
. If the area of the triangle formed by connecting their centers is
, then the area of the third circle is
for some integer
. Determine
.
Problem 2
Let denote the number of
digit positive integers have the property that their digits are in increasing order. Determine the remainder obtained when
is divided by
. (Repeated digits are allowed.)
Problem 3
A function is defined for all real numbers
. For all non-zero values
, we have
Let denote the sum of all of the values of
for which
. Compute the integer nearest to
.
Problem 4
and
are complex numbers such that
Compute .
Problem 5
In Zuminglish, all words consist only of the letters and
. As in English,
is said to be a vowel and
and
are consonants. A string of
and
is a word in Zuminglish if and only if between any two
there appear at least two consonants. Let
denote the number of
-letter Zuminglish words. Determine the remainder obtained when
is divided by
.
Problem 6
Let denote the value of the sum
can be expressed as
, where
and
are positive integers and
is not divisible by the square of any prime. Determine
.
Problem 7
is a cyclic quadrilateral that has an inscribed circle. The diagonals of
intersect at
. If
and
then the area of the inscribed circle of
can be expressed as
, where
and
are relatively prime positive integers. Determine
.
Problem 8
Let denote the number of
-tuples
of real numbers such that
and
Determine the remainder obtained when is divided by
.
Problem 9
is an isosceles triangle with base
.
is a point on
and
is the point on the extension of
past
such that
is right. If
and
, then
can be expressed as
, where
and
are relatively prime positive integers. Determine
.
Problem 10
is a sequence of positive integers such that
for all integers . Compute the remainder obtained when
is divided by
if
.
Problem 11
is an acute triangle with perimeter
.
is a point on
. The circumcircles of triangles
and
intersect
and
at
and
respectively such that
and
. If
, then the value of
can be expressed as
, where
and
are relatively prime positive integers. Compute
.
Problem 12
Determine the number of integers such that
and
is divisible by
.
Problem 13
Let denote the value of the sum
Determine the remainder obtained when is divided by
.
Problem 14
Circles and
are centered on opposite sides of line
, and are both tangent to
at
.
passes through
, intersecting
again at
. Let
and
be the intersections of
and
, and
and
respectively.
and
are extended past
and intersect
and
at
and
respectively. If
and
, then the area of triangle
can be expressed as
, where
and
are positive integers such that
and
are coprime and
is not divisible by the square of any prime. Determine
.
Problem 15
Let denote the value of the sum
The value of can be expressed as
, where
and
are relatively prime positive integers. Compute
.