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Difference between revisions of "Mock AIME 3 Pre 2005 Problems"

 
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<math>1.</math> 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>.
+
==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>.
  
−
<math>2.</math> 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 1|Solution]]
  
−
<math>3.</math> 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
+
==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>.
  
−
<math>4.</math> <math>\zeta_1, \zeta_2,</math> and <math>\zeta_3</math> are complex numbers such that
+
[[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>.
  
−
<math>5.</math> 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 4|Solution]]
  
−
<math>6.</math> Let <math>S</math> denote the value of the sum
+
==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>.
  
−
<math>7.</math> <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 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]]
  
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Determine the remainder obtained when <math>N</math> is divided by <math>1000</math>.
+
==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>.
  
−
<math>9.</math> <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]]
  
−
<math>10.</math> <math>\{A_n\}_{n \ge 1}</math> is a sequence of positive integers such that
+
==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>.
  
−
<math>11.</math> <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 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>.
  
−
<math>12.</math> 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]]
  
−
<math>13.</math> Let <math>S</math> denote the value of the sum
+
==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>.
  
−
<math>14.</math> 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 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]]
  
−
<math>15.</math> Let <math>\Omega</math> denote the value of the sum
+
==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 08:27, 14 February 2008

Problem 1

Three circles are mutually externally tangent. Two of the circles have radii $3$ and $7$. If the area of the triangle formed by connecting their centers is $84$, then the area of the third circle is $k\pi$ for some integer $k$. Determine $k$.

Solution

Problem 2

Let $N$ denote the number of $7$ digit positive integers have the property that their digits are in increasing order. Determine the remainder obtained when $N$ is divided by $1000$. (Repeated digits are allowed.)

Solution

Problem 3

A function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have

$2f\left(x\right) + f\left(\frac{1}{x}\right) = 5x + 4$

Let $S$ denote the sum of all of the values of $x$ for which $f(x) = 2004$. Compute the integer nearest to $S$.

Solution

Problem 4

$\zeta_1, \zeta_2,$ and $\zeta_3$ are complex numbers such that

$\zeta_1 + \zeta_2 + \zeta_3 = 1$

$\zeta_1^{2} + \zeta_2^{2} + \zeta_3^{2} = 3$

$\zeta_1^{3} + \zeta_2^{3} + \zeta_3^{3} = 7$


Compute $\zeta_1^{7} + \zeta_2^{7} + \zeta_3^{7}$.

Solution

Problem 5

In Zuminglish, all words consist only of the letters $M, O,$ and $P$. As in English, $O$ is said to be a vowel and $M$ and $P$ are consonants. A string of $M's, O's,$ and $P's$ is a word in Zuminglish if and only if between any two $O's$ there appear at least two consonants. Let $N$ denote the number of $10$-letter Zuminglish words. Determine the remainder obtained when $N$ is divided by $1000$.

Solution

Problem 6

Let $S$ denote the value of the sum

$\sum_{n = 1}^{9800} \frac{1}{\sqrt{n + \sqrt{n^2 - 1}}}$

$S$ can be expressed as $p + q \sqrt{r}$, where $p, q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime. Determine $p + q + r$.

Solution

Problem 7

$ABCD$ is a cyclic quadrilateral that has an inscribed circle. The diagonals of $ABCD$ intersect at $P$. If $AB = 1, CD = 4,$ and $BP : DP = 3 : 8,$ then the area of the inscribed circle of $ABCD$ can be expressed as $\frac{p\pi}{q}$, where $p$ and $q$ are relatively prime positive integers. Determine $p + q$.

Solution

Problem 8

Let $N$ denote the number of $8$-tuples $(a_1, a_2, \dots, a_8)$ of real numbers such that $a_1 = 10$ and

$\left|a_1^{2} - a_2^{2}\right| = 10$

$\left|a_2^{2} - a_3^{2}\right| = 20$

$\cdots$

$\left|a_7^{2} - a_8^{2}\right| = 70$

$\left|a_8^{2} - a_1^{2}\right| = 80$

Determine the remainder obtained when $N$ is divided by $1000$.

Solution

Problem 9

$ABC$ is an isosceles triangle with base $\overline{AB}$. $D$ is a point on $\overline{AC}$ and $E$ is the point on the extension of $\overline{BD}$ past $D$ such that $\angle{BAE}$ is right. If $BD = 15, DE = 2,$ and $BC = 16$, then $CD$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine $m + n$.

Solution

Problem 10

$\{A_n\}_{n \ge 1}$ is a sequence of positive integers such that

$a_{n} = 2a_{n-1} + n^2$

for all integers $n > 1$. Compute the remainder obtained when $a_{2004}$ is divided by $1000$ if $a_1 = 1$.

Solution

Problem 11

$ABC$ is an acute triangle with perimeter $60$. $D$ is a point on $\overline{BC}$. The circumcircles of triangles $ABD$ and $ADC$ intersect $\overline{AC}$ and $\overline{AB}$ at $E$ and $F$ respectively such that $DE = 8$ and $DF = 7$. If $\angle{EBC} \cong \angle{BCF}$, then the value of $\frac{AE}{AF}$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Solution

Problem 12

Determine the number of integers $n$ such that $1 \le n \le 1000$ and $n^{12} - 1$ is divisible by $73$.

Solution

Problem 13

Let $S$ denote the value of the sum

$\left(\frac{2}{3}\right)^{2005} \cdot \sum_{k=1}^{2005} \frac{k^2}{2^k} \cdot {2005 \choose k}$

Determine the remainder obtained when $S$ is divided by $1000$.

Solution

Problem 14

Circles $\omega_1$ and $\omega_2$ are centered on opposite sides of line $l$, and are both tangent to $l$ at $P$. $\omega_3$ passes through $P$, intersecting $l$ again at $Q$. Let $A$ and $B$ be the intersections of $\omega_1$ and $\omega_3$, and $\omega_2$ and $\omega_3$ respectively. $AP$ and $BP$ are extended past $P$ and intersect $\omega_2$ and $\omega_1$ at $C$ and $D$ respectively. If $AD = 3, AP = 6, DP = 4,$ and $PQ = 32$, then the area of triangle $PBC$ can be expressed as $\frac{p\sqrt{q}}{r}$, where $p, q,$ and $r$ are positive integers such that $p$ and $r$ are coprime and $q$ is not divisible by the square of any prime. Determine $p + q + r$.

Solution

Problem 15

Let $\Omega$ denote the value of the sum

$\sum_{k=1}^{40} \cos^{-1}\left(\frac{k^2 + k + 1}{\sqrt{k^4 + 2k^3 + 3k^2 + 2k + 2}}\right)$

The value of $\tan\left(\Omega\right)$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Solution

See Also

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