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Difference between revisions of "Mock AIME 2 Pre 2005/Problems"

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==Problem 1==
  
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1. Compute the largest integer k such that 2004^k divides 2004!.
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Compute the largest integer <math>k</math> such that <math>2004^k</math> divides <math>2004!</math>.
  
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2. x is a real number with the property that x + \frac{1}{x} = 3. Let S_m = x^m + \frac{1}{x^m}. Determine the value of S_7.
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==Problem 2==
  
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3. In a box, there are 4 green balls, 4 blue balls, 2 red balls, a brown ball, a white ball, and a black ball. These balls are randomly drawn out of the box one at a time (without replacement) until two of the same color have been removed. This process requires that at most 7 balls be removed. The probability that 7 balls are drawn can be expressed as \frac{m}{n}, where m and n are relatively prime positive integers. Compute m + n.
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<math>x</math> is a real number with the property that <math>x + \frac{1}{x} = 3</math>. Let <math>S_m = x^m + \frac{1}{x^m}</math>.  Determine the value of <math>S_7</math>.
  
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4. Let S := \{5^k | k \in \mathbb{Z}, 0 \le k \le 2004\}. Given that 5^{2004} = 5443 \cdots 0625 has 1401 digits, how many elements of S begin with the digit 1?
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==Problem 3==
  
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5. Let S be the set of integers n > 1 for which \frac{1}{n} = 0.d_1d_2d_3d_4\dots, an infinite decimal that has the property that d_i = d_{i+12} for all positive integers i. Given that 9901 is prime, how many positive integers are in S? (The d_i are digits.)
+
In a box, there are <math>4</math> green balls, <math>4</math> blue balls, <math>2</math> red balls, a brown ball, a white ball, and a black ball. These balls are randomly drawn out of the box one at a time (without replacement) until two of the same color have been removed.  This process requires that at most <math>7</math> balls be removed.  The probability that <math>7</math> balls are drawn 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>.
  
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6. ABC is a scalene triangle. Points D, E, and F are selected on sides BC, CA, and AB respectively. The cevians AD, BE, and CF concur at point P. If the areas of triangles AFP, FBP, and CEP are 126, 63, and 24 respectively, then determine the area of triangle ABC.
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==Problem 4==
  
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7. Anders, Po-Ru, Reid, and Aaron are playing Bridge. After one hand, they notice that all of the cards of two suits are split between Reid and Po-Ru's hands. Let N denote the number of ways 13 cards can be dealt to each player such that this is the case. Determine the remainder obtained when N is divided by 1000. (Bridge is a card game played with the standard 52-card deck.)
+
Let <math>S := \{5^k | k \in \mathbb{Z}, 0 \le k \le 2004\}</math>. Given that <math>5^{2004}</math> has <math>1401</math> digits, how many elements of <math>S</math> begin with the digit <math>1</math>?
  
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8. Determine the remainder obtained when the expression
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==Problem 5==
  
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<br /> <br /> \Large{2004^{2003^{2002^{2001}}}} <br />
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Let <math>S</math> be the set of integers <math>n > 1</math> for which <math>\frac{1}{n} = 0.d_1d_2d_3d_4\dots</math>, an infinite decimal that has the property that <math>d_i = d_{i+12}</math> for all positive integers <math>i</math>.  Given that <math>9901</math> is prime, how many positive integers are in <math>S</math>?  (The <math>d_i</math> are digits.)
  
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is divided by 1000.
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==Problem 6==
  
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9. Let
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<math>ABC</math> is a scalene triangle.  Points <math>D</math>, <math>E</math>, and <math>F</math> are selected on sides <math>\overline{BC}</math>, <math>\overline{CA}</math>, and <math>\overline{AB}</math> respectively.  The cevians <math>\overline{AD}</math>, <math>\overline{BE}</math>, and <math>\overline{CF}</math> concur at point <math>P</math>.  If the areas of triangles <math>AFP</math>, <math>FBP</math>, and <math>CEP</math> are <math>126</math>, <math>63</math>, and <math>24</math> respectively, then determine the area of triangle <math>ABC</math>.
  
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\left(1 + x^3\right)\left(1 + 2x^{3^2}\right)\cdots\left(1 + kx^{3^k}\right)\cdots\left(1 + 1997x^{3^{1997}}\right) = 1 + a_1x^{k_1} + a_2x^{k_2} + \cdots + a_mx^{k_m}
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==Problem 7==
  
 +
Anders, Po-Ru, Reid, and Aaron are playing Bridge.  After one hand, they notice that all of the cards of two suits are split between Reid and Po-Ru's hands.  Let <math>N</math> denote the number of ways <math>13</math> cards can be dealt to each player such that this is the case.  Determine the remainder obtained when <math>N</math> is divided by <math>1000</math>. (Bridge is a card game played with the standard <math>52-</math>card deck.)
  
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where a_i \ne 0 and k_1 < k_2 < \cdots < k_m. Determine the remainder obtained when a_{1997} is divided by 1000.
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==Problem 8==
  
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10. ABCDE is a cyclic pentagon with BC = CD = DE. The diagonals AC and BE intersect at M. N is the foot of the altitude from M to AB. We have MA = 25, MD = 113, and MN = 15. [ABE] can be expressed as \frac{m}{n} where m and n are relatively prime positive integers. Determine the remainder obtained when m+n is divided by 1000.
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Determine the remainder obtained when the expression
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<cmath>2004^{2003^{2002^{2001}}}</cmath>
 +
is divided by <math>1000</math>.
  
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11. \alpha, \beta, and \gamma are the roots of x(x-200)(4x+1) = 1. Let
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==Problem 9==
  
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<br /> <br /> \omega = \tan^{-1}(\alpha) + \tan^{-1}(\beta) + \tan^{-1}(\gamma) <br />
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Let
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<cmath>\left(1 + x^3\right)\left(1 + 2x^{3^2}\right)\cdots\left(1 + 1997x^{3^{1997}}\right) = 1 + a_1x^{k_1} + a_2x^{k_2} + \cdots + a_mx^{k_m}</cmath>
 +
where <math>a_i \neq 0</math> and <math>k_1 < k_2 < \cdots < k_m</math>.  Determine the remainder obtained when <math>a_{1997}</math> is divided by <math>1000</math>.
  
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The value of \tan(\omega) can be written as \frac{m}{n} where m and n are relatively prime positive integers. Determine the value of m + n.
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==Problem 10==
  
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12. ABCD is a cyclic quadrilateral with AB = 8, BC = 4, CD = 1, and DA = 7. Let O and P denote the circumcenter and intersection of AC and BD respectively. The value of OP^2 can be expressed as \frac{m}{n}, where m and n are relatively prime, positive integers. Determine the remainder obtained when m+n is divided by 1000.
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<math>ABCDE</math> is a cyclic pentagon with <math>BC = CD = DE</math>.  The diagonals <math>\overline{AC}</math> and <math>\overline{BE}</math> intersect at <math>M</math>. <math>N</math> is the foot of the altitude from <math>M</math> to <math>\overline{AB}</math>.  We have <math>MA = 25</math>, <math>MD = 113</math>, and <math>MN = 15</math>. [<math>ABE</math>] can be expressed as <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Determine the remainder obtained when <math>m+n</math> is divided by <math>1000</math>.
  
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13. P(x) is the polynomial of minimal degree that satisfies
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==Problem 11==
  
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<br /> P(k) = \frac{1}{k(k+1)} <br />
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<math>\alpha</math>, <math>\beta</math>, and <math>\gamma</math> are the roots of <math>x(x-200)(4x+1) = 1</math>.  Let
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<cmath>\omega = \tan^{-1}(\alpha) + \tan^{-1}(\beta) + \tan^{-1}(\gamma).</cmath>
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The value of <math>\tan(\omega)</math> can be written as <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers.  Determine the value of <math>m + n</math>.
  
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for k = 1, 2, 3, \dots, 10. The value of P(11) can be written as -\frac{m}{n}, where m and n are relatively prime positive integers. Determine m+n.
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==Problem 12==
  
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14. 3 Elm trees, 4 Dogwood trees, and 5 Oak trees are to be planted in a line in front of a library such that
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<math>ABCD</math> is a cyclic quadrilateral with <math>AB = 8</math>, <math>BC = 4</math>, <math>CD = 1</math>, and <math>DA = 7</math>.  Let <math>O</math> and <math>P</math> denote the circumcenter and intersection of <math>AC</math> and <math>BD</math> respectively.  The value of <math>OP^2</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers.  Determine the remainder obtained when <math>m+n</math> is divided by <math>1000</math>.
  
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<br /> <br /> \begin{eqnarray*} <br /> i)\phantom{iiM}\textrm{No two Elm trees are next to each other.}\phantom{xxx} \\ <br /> ii)\phantom{iM}\textrm{No Dogwood tree is adjacent to an Oak tree.}\phantom{} \\ <br /> iii)\phantom{M}\textrm{All of the trees are planted.}\phantom{xxxxxxxxxxxxx|} <br /> \end{eqnarray*} <br />
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==Problem 13==
  
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<math>P(x)</math> is the polynomial of minimal degree that satisfies
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<cmath>P(k) = \frac{1}{k(k+1)}</cmath>
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for <math>k = 1, 2, 3, \dots, 10</math>.  The value of <math>P(11)</math> can be written as <math>-\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers.  Determine <math>m+n</math>.
 +
 +
==Problem 14==
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<math>3</math> Elm trees, <math>4</math> Dogwood trees, and <math>5</math> Oak trees are to be planted in a line in front of a library such that
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<cmath>\begin{eqnarray*}
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\text{i)} & \text{No two Elm trees are next to each other.} \hspace{6 mm} \\
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\text{ii)} & \text{No Dogwood tree is adjacent to an Oak tree.} \\
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\text{iii)} & \text{All of the trees are planted.} \hspace{27 mm}
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\end{eqnarray*}</cmath>
 
How many ways can the trees be situated in this manner?
 
How many ways can the trees be situated in this manner?
  
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15. In triangle ABC, we have BC = 13, CA = 37, and AB = 40. Points D, E, and F are selected on BC, CA, and AB respectively such that AD, BE, and CF concur at the circumcenter of ABC. The value of
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==Problem 15==  
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<br /> \frac{1}{AD} + \frac{1}{BE} + \frac{1}{CF} <br />
 
  
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can be expressed as \frac{m}{n} where m and n are relatively prime positive integers. Determine m + n.
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In triangle <math>ABC</math>, we have <math>BC = 13</math>, <math>CA = 37</math>, and <math>AB = 40</math>.  Points <math>D</math>, <math>E</math>, and <math>F</math> are selected on <math>\overline{BC}</math>, <math>\overline{CA}</math>, and <math>\overline{AB}</math> respectively such that <math>\overline{AD}</math>, <math>\overline{BE}</math>, and <math>\overline{CF}</math> concur at the circumcenter of <math>ABC</math>. The value of
 +
<cmath>\frac{1}{AD} + \frac{1}{BE} + \frac{1}{CF}</cmath>
 +
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>.

Latest revision as of 15:23, 11 November 2009

Problem 1

Compute the largest integer $k$ such that $2004^k$ divides $2004!$.

Problem 2

$x$ is a real number with the property that $x + \frac{1}{x} = 3$. Let $S_m = x^m + \frac{1}{x^m}$. Determine the value of $S_7$.

Problem 3

In a box, there are $4$ green balls, $4$ blue balls, $2$ red balls, a brown ball, a white ball, and a black ball. These balls are randomly drawn out of the box one at a time (without replacement) until two of the same color have been removed. This process requires that at most $7$ balls be removed. The probability that $7$ balls are drawn can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Problem 4

Let $S := \{5^k | k \in \mathbb{Z}, 0 \le k \le 2004\}$. Given that $5^{2004}$ has $1401$ digits, how many elements of $S$ begin with the digit $1$?

Problem 5

Let $S$ be the set of integers $n > 1$ for which $\frac{1}{n} = 0.d_1d_2d_3d_4\dots$, an infinite decimal that has the property that $d_i = d_{i+12}$ for all positive integers $i$. Given that $9901$ is prime, how many positive integers are in $S$? (The $d_i$ are digits.)

Problem 6

$ABC$ is a scalene triangle. Points $D$, $E$, and $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ respectively. The cevians $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ concur at point $P$. If the areas of triangles $AFP$, $FBP$, and $CEP$ are $126$, $63$, and $24$ respectively, then determine the area of triangle $ABC$.

Problem 7

Anders, Po-Ru, Reid, and Aaron are playing Bridge. After one hand, they notice that all of the cards of two suits are split between Reid and Po-Ru's hands. Let $N$ denote the number of ways $13$ cards can be dealt to each player such that this is the case. Determine the remainder obtained when $N$ is divided by $1000$. (Bridge is a card game played with the standard $52-$card deck.)

Problem 8

Determine the remainder obtained when the expression \[2004^{2003^{2002^{2001}}}\] is divided by $1000$.

Problem 9

Let \[\left(1 + x^3\right)\left(1 + 2x^{3^2}\right)\cdots\left(1 + 1997x^{3^{1997}}\right) = 1 + a_1x^{k_1} + a_2x^{k_2} + \cdots + a_mx^{k_m}\] where $a_i \neq 0$ and $k_1 < k_2 < \cdots < k_m$. Determine the remainder obtained when $a_{1997}$ is divided by $1000$.

Problem 10

$ABCDE$ is a cyclic pentagon with $BC = CD = DE$. The diagonals $\overline{AC}$ and $\overline{BE}$ intersect at $M$. $N$ is the foot of the altitude from $M$ to $\overline{AB}$. We have $MA = 25$, $MD = 113$, and $MN = 15$. [$ABE$] can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Determine the remainder obtained when $m+n$ is divided by $1000$.

Problem 11

$\alpha$, $\beta$, and $\gamma$ are the roots of $x(x-200)(4x+1) = 1$. Let \[\omega = \tan^{-1}(\alpha) + \tan^{-1}(\beta) + \tan^{-1}(\gamma).\] The value of $\tan(\omega)$ can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Determine the value of $m + n$.

Problem 12

$ABCD$ is a cyclic quadrilateral with $AB = 8$, $BC = 4$, $CD = 1$, and $DA = 7$. Let $O$ and $P$ denote the circumcenter and intersection of $AC$ and $BD$ respectively. The value of $OP^2$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine the remainder obtained when $m+n$ is divided by $1000$.

Problem 13

$P(x)$ is the polynomial of minimal degree that satisfies \[P(k) = \frac{1}{k(k+1)}\] for $k = 1, 2, 3, \dots, 10$. The value of $P(11)$ can be written as $-\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine $m+n$.

Problem 14

$3$ Elm trees, $4$ Dogwood trees, and $5$ Oak trees are to be planted in a line in front of a library such that \begin{eqnarray*} \text{i)} & \text{No two Elm trees are next to each other.} \hspace{6 mm} \\ \text{ii)} & \text{No Dogwood tree is adjacent to an Oak tree.} \\ \text{iii)} & \text{All of the trees are planted.} \hspace{27 mm} \end{eqnarray*} How many ways can the trees be situated in this manner?

Problem 15

In triangle $ABC$, we have $BC = 13$, $CA = 37$, and $AB = 40$. Points $D$, $E$, and $F$ are selected on $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ respectively such that $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ concur at the circumcenter of $ABC$. The value of \[\frac{1}{AD} + \frac{1}{BE} + \frac{1}{CF}\] can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Determine $m + n$.

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