Difference between revisions of "Mock AIME 3 Pre 2005 Problems"
Line 1: | Line 1: | ||
− | + | ==Problem 1== | |
+ | Three circles are mutually externally tangent. Two of the circles have radii <math>3</math> and <math>7</math>. If the area of the triangle formed by connecting their centers is <math>84</math>, then the area of the third circle is <math>k\pi</math> for some integer <math>k</math>. Determine <math>k</math>. | ||
− | + | [[Mock AIME 3 Pre 2005/Problem 1|Solution]] | |
− | <math> | + | ==Problem 2== |
+ | Let <math>N</math> denote the number of <math>7</math> digit positive integers have the property that their digits are in increasing order. Determine the remainder obtained when <math>N</math> is divided by <math>1000</math>. (Repeated digits are allowed.) | ||
+ | |||
+ | [[Mock AIME 3 Pre 2005/Problem 2|Solution]] | ||
+ | |||
+ | ==Problem 3== | ||
+ | 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 | ||
<math>2f\left(x\right) + f\left(\frac{1}{x}\right) = 5x + 4</math> | <math>2f\left(x\right) + f\left(\frac{1}{x}\right) = 5x + 4</math> | ||
Line 9: | Line 16: | ||
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>. | ||
− | + | [[Mock AIME 3 Pre 2005/Problem 3|Solution]] | |
+ | |||
+ | ==Problem 4== | ||
+ | <math>\zeta_1, \zeta_2,</math> and <math>\zeta_3</math> are complex numbers such that | ||
<math>\zeta_1 + \zeta_2 + \zeta_3 = 1</math> | <math>\zeta_1 + \zeta_2 + \zeta_3 = 1</math> | ||
Line 20: | Line 30: | ||
Compute <math>\zeta_1^{7} + \zeta_2^{7} + \zeta_3^{7}</math>. | Compute <math>\zeta_1^{7} + \zeta_2^{7} + \zeta_3^{7}</math>. | ||
− | + | [[Mock AIME 3 Pre 2005/Problem 4|Solution]] | |
− | <math> | + | ==Problem 5== |
+ | In Zuminglish, all words consist only of the letters <math>M, O,</math> and <math>P</math>. As in English, <math>O</math> is said to be a vowel and <math>M</math> and <math>P</math> are consonants. A string of <math>M's, O's,</math> and <math>P's</math> is a word in Zuminglish if and only if between any two <math>O's</math> there appear at least two consonants. Let <math>N</math> denote the number of <math>10</math>-letter Zuminglish words. Determine the remainder obtained when <math>N</math> is divided by <math>1000</math>. | ||
+ | |||
+ | [[Mock AIME 3 Pre 2005/Problem 5|Solution]] | ||
+ | |||
+ | ==Problem 6== | ||
+ | Let <math>S</math> denote the value of the sum | ||
<math>\sum_{n = 1}^{9800} \frac{1}{\sqrt{n + \sqrt{n^2 - 1}}}</math> | <math>\sum_{n = 1}^{9800} \frac{1}{\sqrt{n + \sqrt{n^2 - 1}}}</math> | ||
Line 28: | Line 44: | ||
<math>S</math> can be expressed as <math>p + q \sqrt{r}</math>, where <math>p, q,</math> and <math>r</math> are positive integers and <math>r</math> is not divisible by the square of any prime. Determine <math>p + q + r</math>. | <math>S</math> can be expressed as <math>p + q \sqrt{r}</math>, where <math>p, q,</math> and <math>r</math> are positive integers and <math>r</math> is not divisible by the square of any prime. Determine <math>p + q + r</math>. | ||
− | + | [[Mock AIME 3 Pre 2005/Problem 6|Solution]] | |
− | <math>8.</math> Let <math>N</math> denote the number of <math>8</math>-tuples <math>(a_1, a_2, \dots, a_8)</math> of real numbers such that <math>a_1 = 10</math> and | + | ==Problem 7== |
+ | <math>ABCD</math> is a cyclic quadrilateral that has an inscribed circle. The diagonals of <math>ABCD</math> intersect at <math>P</math>. If <math>AB = 1, CD = 4,</math> and <math>BP : DP = 3 : 8,</math> then the area of the inscribed circle of <math>ABCD</math> can be expressed as <math>\frac{p\pi}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Determine <math>p + q</math>. | ||
+ | |||
+ | [[Mock AIME 3 Pre 2005/Problem 7|Solution]] | ||
+ | |||
+ | ==Problem 8== | ||
+ | Let <math>N</math> denote the number of <math>8</math>-tuples <math>(a_1, a_2, \dots, a_8)</math> of real numbers such that <math>a_1 = 10</math> and | ||
<math>\left|a_1^{2} - a_2^{2}\right| = 10</math> | <math>\left|a_1^{2} - a_2^{2}\right| = 10</math> | ||
Line 42: | Line 64: | ||
<math>\left|a_8^{2} - a_1^{2}\right| = 80</math> | <math>\left|a_8^{2} - a_1^{2}\right| = 80</math> | ||
+ | Determine the remainder obtained when <math>N</math> is divided by <math>1000</math>. | ||
+ | |||
+ | [[Mock AIME 3 Pre 2005/Problem 8|Solution]] | ||
− | + | ==Problem 9== | |
+ | <math>ABC</math> is an isosceles triangle with base <math>\overline{AB}</math>. <math>D</math> is a point on <math>\overline{AC}</math> and <math>E</math> is the point on the extension of <math>\overline{BD}</math> past <math>D</math> such that <math>\angle{BAE}</math> is right. If <math>BD = 15, DE = 2,</math> and <math>BC = 16</math>, then <math>CD</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Determine <math>m + n</math>. | ||
− | + | [[Mock AIME 3 Pre 2005/Problem 9|Solution]] | |
− | + | ==Problem 10== | |
+ | <math>\{A_n\}_{n \ge 1}</math> is a sequence of positive integers such that | ||
<math>a_{n} = 2a_{n-1} + n^2</math> | <math>a_{n} = 2a_{n-1} + n^2</math> | ||
Line 53: | Line 80: | ||
for all integers <math>n > 1</math>. Compute the remainder obtained when <math>a_{2004}</math> is divided by <math>1000</math> if <math>a_1 = 1</math>. | for all integers <math>n > 1</math>. Compute the remainder obtained when <math>a_{2004}</math> is divided by <math>1000</math> if <math>a_1 = 1</math>. | ||
− | + | [[Mock AIME 3 Pre 2005/Problem 10|Solution]] | |
+ | |||
+ | ==Problem 11== | ||
+ | <math>ABC</math> is an acute triangle with perimeter <math>60</math>. <math>D</math> is a point on <math>\overline{BC}</math>. The circumcircles of triangles <math>ABD</math> and <math>ADC</math> intersect <math>\overline{AC}</math> and <math>\overline{AB}</math> at <math>E</math> and <math>F</math> respectively such that <math>DE = 8</math> and <math>DF = 7</math>. If <math>\angle{EBC} \cong \angle{BCF}</math>, then the value of <math>\frac{AE}{AF}</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Compute <math>m + n</math>. | ||
+ | |||
+ | [[Mock AIME 3 Pre 2005/Problem 11|Solution]] | ||
+ | |||
+ | ==Problem 12== | ||
+ | Determine the number of integers <math>n</math> such that <math>1 \le n \le 1000</math> and <math>n^{12} - 1</math> is divisible by <math>73</math>. | ||
− | + | [[Mock AIME 3 Pre 2005/Problem 12|Solution]] | |
− | + | ==Problem 13== | |
+ | Let <math>S</math> denote the value of the sum | ||
<math>\left(\frac{2}{3}\right)^{2005} \cdot \sum_{k=1}^{2005} \frac{k^2}{2^k} \cdot {2005 \choose k}</math> | <math>\left(\frac{2}{3}\right)^{2005} \cdot \sum_{k=1}^{2005} \frac{k^2}{2^k} \cdot {2005 \choose k}</math> | ||
Line 63: | Line 99: | ||
Determine the remainder obtained when <math>S</math> is divided by <math>1000</math>. | Determine the remainder obtained when <math>S</math> is divided by <math>1000</math>. | ||
− | + | [[Mock AIME 3 Pre 2005/Problem 13|Solution]] | |
+ | |||
+ | ==Problem 14== | ||
+ | Circles <math>\omega_1</math> and <math>\omega_2</math> are centered on opposite sides of line <math>l</math>, and are both tangent to <math>l</math> at <math>P</math>. <math>\omega_3</math> passes through <math>P</math>, intersecting <math>l</math> again at <math>Q</math>. Let <math>A</math> and <math>B</math> be the intersections of <math>\omega_1</math> and <math>\omega_3</math>, and <math>\omega_2</math> and <math>\omega_3</math> respectively. <math>AP</math> and <math>BP</math> are extended past <math>P</math> and intersect <math>\omega_2</math> and <math>\omega_1</math> at <math>C</math> and <math>D</math> respectively. If <math>AD = 3, AP = 6, DP = 4,</math> and <math>PQ = 32</math>, then the area of triangle <math>PBC</math> can be expressed as <math>\frac{p\sqrt{q}}{r}</math>, where <math>p, q,</math> and <math>r</math> are positive integers such that <math>p</math> and <math>r</math> are coprime and <math>q</math> is not divisible by the square of any prime. Determine <math>p + q + r</math>. | ||
+ | |||
+ | [[Mock AIME 3 Pre 2005/Problem 14|Solution]] | ||
− | + | ==Problem 15== | |
+ | Let <math>\Omega</math> denote the value of the sum | ||
<math>\sum_{k=1}^{40} \cos^{-1}\left(\frac{k^2 + k + 1}{\sqrt{k^4 + 2k^3 + 3k^2 + 2k + 2}}\right)</math> | <math>\sum_{k=1}^{40} \cos^{-1}\left(\frac{k^2 + k + 1}{\sqrt{k^4 + 2k^3 + 3k^2 + 2k + 2}}\right)</math> | ||
The value of <math>\tan\left(\Omega\right)</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Compute <math>m + n</math>. | The value of <math>\tan\left(\Omega\right)</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Compute <math>m + n</math>. | ||
+ | |||
+ | [[Mock AIME 3 Pre 2005/Problem 15|Solution]] | ||
+ | |||
+ | ==See Also== | ||
+ | *[[Mock AIME 3 Pre 2005]] | ||
+ | *[[Mock AIME]] | ||
+ | *[[AIME]] |
Revision as of 07:27, 14 February 2008
Contents
[hide]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
.